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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\input{text\/sec_intro}\n\n\\section{Observations}\n\\label{sec:obs}\n\\input{text\/sec_obs}\n\n\\section{Construction of the Wideband\\\\Data Set}\n\\label{sec:wb}\n\\input{text\/sec_wb}\n\\vspace{1in}\n\\section{Results \\& Discussion}\n\\label{sec:results}\n\\input{text\/sec_results}\n\n\\section{Summary \\& Conclusions}\n\\label{sec:conclusion}\n\\input{text\/sec_conclusion}\n\n\n\\input{text\/sec_acknowledgement}\n\n\\facilities{Arecibo, GBT}\n\n\\software{\\texttt{ENTERPRISE} \\citep{enterprise}, \\texttt{libstempo} \\citep{libstempo}, \\texttt{matplotlib} \\citep{matplotlib}, \\texttt{nanopipe} \\citep{nanopipe}, \\texttt{PSRCHIVE} \\citep{vS11}, \\texttt{PTMCMC} \\citep{ptmcmc}, \\texttt{PulsePortraiture} \\citep{pulseportraiture}, \\texttt{PyPulse} \\citep{pypulse}, \\texttt{Tempo} \\citep{tempo}, \\texttt{Tempo2} \\citep{tempo2}, \\texttt{tempo\\_utils}\\footnote{\\href{https:\/\/github.com\/demorest\/tempo\\_utils}{https:\/\/github.com\/demorest\/tempo\\_utils}}}\n\\\\\\\\\n\\input{text\/sec_contribution}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Overview}\n\\label{subsec:wideband_overview}\n\nThe measurement of TOAs from pulsar data with a large instantaneous bandwidth was first developed in \\citet{Liu14} and \\citet{PDR14}, and further explored in \\citet{PennucciPhDT} and \\citet{Pennucci19}.\nWe refer the reader to those works for details and here briefly summarize the important points.\n\nA single narrowband TOA corresponds to the time of arrival of a pulse profile observed in a single frequency channel\\footnote{Another similar protocol used in the pulsar timing community is to produce band-averaged TOAs, in which the detected profiles are summed over the observing bandwidth, creating a single profile from which to extract the TOA.} (sometimes referred to as a ``subband''); in contrast, a single wideband measurement is composed of both the time of arrival of a pulse at some reference frequency and an estimate of the dispersion measure at the time of observation.\nThe difference can be conceptualized thusly: narrowband TOAs from a single subintegration are like the individual, scattered measurements around a linear relationship, whereas the fitted intercept and slope to this relationship are like the wideband TOA and DM, respectively.\nThe log-likelihood function for the wideband measurements is reproduced in Section~\\ref{subsec:wb_toa_lnlike}.\n\nThe second important difference in the new wideband data set is not fundamental to the measurement of the TOA.\nHeretofore we have used a single, frequency-independent template profile for each receiver band to generate narrowband TOAs and have used FD parameters \\citep{Arzoumanian2015b} to account for constant phase offsets originating from the mismatch between the template and the evolving shape of the profiles.\nFor the measurement of wideband TOAs, we explicitly account for pulse profile evolution by using a high-fidelity, noise-free, frequency-dependent model for each receiver band.\nSee Section~\\ref{subsec:prof_evol} for a brief description of how these models are created.\n\nAlthough the narrowband and wideband data sets were developed in parallel, the established techniques in preparing the former allowed us to use some information from its final products to facilitate the production of the latter.\nIn particular, some of the curating performed, including flagging bad epochs, as well as the initial timing, was borrowed from the narrowband analysis.\nIn this way, the wideband data set is not completely independent, as is detailed in Sections~\\ref{subsec:wb_cleaning}~\\&~\\ref{subsec:timing}.\n\nIt is important to underscore that the wideband data set for each pulsar is composed of TOAs that are paired with estimates of the instantaneous DM.\nWhat makes the analysis of the wideband data set truly unique is that these DM estimates inform the portion of the timing model that accounts for DM variability (for our analyses, this is ``DMX''; see Section~\\ref{subsec:timing}).\nIn Section~\\ref{subsec:timing}, we describe our approach, with greater detail in Appendix~\\ref{sec:wb_like}; the results are examined in Section~\\ref{sec:results}.\n\nPublicly available code\\footnote{\\url{https:\/\/github.com\/pennucci\/PulsePortraiture}} is used for both the generation of frequency-dependent templates and the measurement of the wideband TOAs \\citep{pulseportraiture}.\n\\\\\n\\subsection{Wideband TOA Log-Likelihood Function}\n\\label{subsec:wb_toa_lnlike}\n\nAll of our narrowband and wideband TOAs are measured using what is now referred to as the ``Fourier phase-gradient shift algorithm'' \\citep[][historically known as ``FFTFIT'']{Taylor92}, which makes use of the Fourier shift theorem to achieve a phase offset precision much better than a single rotational phase bin, and which is computationally efficient by virtue of avoiding the time-domain cross-correlation calculation between the data and template pulse profiles.\nWe use a similar notation as Appendix~B of \\nineyr, but see also \\citet{DemorestPhDT} and \\citet{PDR14} for details of what follows.\nThe time-domain model has the assumed form\n\\begin{equation}\n\\label{eqn:toa_signal}\n    D(\\nu, \\varphi) = B(\\nu) + a(\\nu)\\,T(\\nu, \\varphi-\\phi(\\nu)) + N(\\nu),\n\\end{equation}\nThat is, for each subintegration in an observation, we assume that the data profiles $D$ as a function of rotational phase $\\varphi$ and frequency $\\nu$ can be described by a template $T$ that is shifted in phase by $\\phi$ and scaled in amplitude by $a$, with added Gaussian-distributed phase-independent noise $N$; the term $B$ represents the bandpass shape.\nAfter discretizing these quantities, taking the discrete Fourier transform (DFT), making use of the Fourier shift theorem, and rearranging terms, we can reformulate Equation~\\ref{eqn:toa_signal} into our TOA log-likelihood,\n\\begin{equation}\n\\label{eqn:toa_lnlike}\n   \n    \\chi^{2} = \\sum_{n,k} \\frac{\\lvert d_{nk} - a_{n} t_{nk}e^{-2\\pi ik\\phi_n}\\rvert^{2}}{\\sigma^2_n}.\n\\end{equation}\nIn Equation~\\ref{eqn:toa_lnlike}, the integer index $k$ is the Fourier frequency (conjugate to rotational phase or time), $t_{nk}$ is the DFT of the template profile for the frequency channel indexed by $n$ (with frequency center $\\nu_n$), $a_{n}$ is the scaling amplitude parameter for the template, $\\phi_n$ is the phase offset for the template, and $d_{nk}$ is the DFT of the data profile for frequency channel $n$, which has the corresponding Fourier-domain noise level $\\sigma^2_n$\\,\\footnote{$\\sigma^2_n$ is the noise for either the real or imaginary part, and is larger than its (real) time-domain counterpart by a factor of $n_{\\textrm{bin}}\/2$.}.\n\nFor conventional TOAs, the optimization of this function takes place on an individual channel basis, in which case there is no index $n$ in Equation~\\ref{eqn:toa_lnlike} over which a summation occurs.\nMoreover, for our narrowband TOAs, $t_{nk}$ is not a function of $n$; that is, profile evolution is not accounted for by changing the shape of the template across a single receiver's frequency band.\nInstead, in \\twelveyr, a single template profile is used for each receiver band and constant phase offsets arising from the mismatch between the template shape and the evolving pulse shape are accounted for via FD parameters in the narrowband timing models.\n\nThe crucial difference in wideband TOAs is that the phase offsets $\\phi_n$ in Equation~\\ref{eqn:toa_lnlike} are constrained to follow the cold-plasma dispersion law, proportional to $\\nu^{-2}$:\n\\begin{equation}\n\\label{eqn:wb_constraint}\n    \\phi_n(\\nu_n) = \\phi_\\circ}%{\\phi^\\circ_{ref} + \\frac{K\\times\\textrm{DM}}{P_s}\\Big(\\nu_n^{-2}-\\nu_{\\phi_\\circ}^{-2}\\Big),\n\\end{equation}\nwhere $P_s$ is the instantaneous spin period of the pulsar, $K$ is the dispersion constant (a combination of fundamental physical constants approximately equal to $4.148808 \\times 10^3$~MHz$^2$~cm$^3$~pc$^{-1}$~s), $\\textrm{DM}$ is the dispersion measure, and $\\phi_\\circ}%{\\phi^\\circ_{ref}$ is the phase offset at reference frequency $\\nu_{\\phi_\\circ}$.\nEquation~\\ref{eqn:toa_lnlike} can be recast using the maximum-likelihood values of $a_{n}$ and rewritten as a function of only the two parameters $\\phi_\\circ}%{\\phi^\\circ_{ref}$ and DM (see \\citet{PDR14}), which can then be readily optimized numerically.\nWe calculate the parameter uncertainties using the Fisher matrix and choose $\\nu_{\\phi_\\circ}$ such that there is zero covariance between the DM and $\\phi_\\circ}%{\\phi^\\circ_{ref}$, the latter of which is directly related to the TOA.\n\nAdditional terms to the wideband TOA log-likelihood are currently being explored (Pennucci et al., \\textit{in prep.}), which include accounting for pulse broadening from multi-path propagation through the turbulent ISM (i.e., ``scattering'') in a similar fashion to \\citet{lkd+17}, as well incorporating a higher-order delay term besides $\\nu^{-2}$, the motivation for which are discrete ISM ``events'' \\citep{Lam2018}.\nThe low-frequency, high-cadence capabilities offered by CHIME\/Pulsar will make tracking the interstellar weather in this way an exciting endeavor, following in the footsteps of studies like \\citet{Ramachandran06} and \\citet{Driessen19} (long-term ISM tracking of B1937$+$21 and the Crab pulsar, respectively).\n\n\\subsection{Frequency-dependent Template Profiles}\n\\label{subsec:prof_evol}\n\nThe evolving template $t_{nk}$ in Equation~\\ref{eqn:toa_lnlike} can be freely chosen, and in this work we employ the modeling method from \\citet{Pennucci19}, which describes a generalized, frequency-dependent version of our usual protocol for making template profiles.\nIn contrast, to make the conventional noise-free templates used in \\twelveyr\\ for narrowband TOA measurement, all profiles for each combination of pulsar and receiver are averaged together to build a single, high S\/N mean profile, which is then smoothed.\nWe direct the reader to \\citet{Pennucci19} for details, but we summarize its novel procedure as follows.\n\nAn analogous averaging of the data for each combination of pulsar and receiver is performed, but frequency resolution is maintained to arrive at a high S\/N mean ``portrait'' (a collection of nominally aligned mean pulse profiles across a contiguous frequency band); only the PUPPI and GUPPI data were averaged for this purpose.\nA principal component analysis is performed on the average portrait, and the most significant, highest S\/N eigenvectors (and mean profile) are smoothed to become noise-free basis functions (``eigenprofiles'').\nThe mean-profile-subtracted profiles from the average portrait are projected onto each of the eigenprofiles, producing a set of coefficients for each.\nThese coefficients are simultaneously fit to a slowly varying spline function that is parameterized by frequency and encapsulates the evolution of the pulse profile shape.\n\nIn this manner, a template profile $T$ at any frequency $\\nu$ can be constructed by evaluating the $n_{\\textrm{\\scriptsize{eig}}}$ coefficient spline functions $B_i$ at $\\nu$, linearly combining the eigenprofiles $\\hat{e}_i$ using these coefficients, and adding the result to the mean profile $\\widetilde{p}$,\n\\begin{equation}\n\\label{eqn:Tprof}\n    T(\\nu) = \\sum^{n_{\\textrm{\\scriptsize{eig}}}}_{i=1} B_{i}(\\nu) \\, \\hat{e}_i ~ + ~ \\widetilde{p}.\n\\end{equation}\n\n\nIn summary, a single model for generating high-fidelity, noise-free template profiles is composed of the smoothed mean profile, the smoothed basis eigenprofiles, and a function to describe the profile evolution curve in that basis.\n\nThese models were made for each combination of pulsar and receiver, and then used to measure wideband TOAs according to Equation~\\ref{eqn:toa_lnlike}.\nThe modeling procedure attempts to guess the true, unknown profile alignment by starting with the same Occam assumption used in the narrowband analysis: there is no profile evolution, neither in the shape nor alignment of the profiles.\nThis assumption is used to initially align and average the profile data by using the fixed, mean profile shape as a reference for the alignment.\nAfter iteratively aligning and averaging the profile data and then creating a model, it should not come as a surprise that the absolute, average DM measured in each receiver band will differ slightly.\nWe minimize this difference by measuring the weighted-mean DM offset relative to the DM measured in the lowest frequency band.\nThe DM offset was then applied as a rotation proportional to $\\nu^{-2}$ to the average portrait, the profile evolution model was recreated, and the TOAs were remeasured; this process was iterated a total of three times.\nThe reference DM choice was made relative to the lowest frequency band because, except in the cases of Arecibo pulsars observed only at L- and S-bands, this will be a frequency band with lower fractional bandwidth than L-band, but from which reasonably precise DM measurements are made.\nThis choice gave better modeling results than rotating the averaged low frequency data relative to the L-band alignment, which may be ambiguous due to profile evolution.\nFor the other sources, S-band generally does not give precise DM measurements, and so L-band is used as the reference.\nSee Section~\\ref{subsec:model_results} for more discussion on this topic.\n\nThe initial set of wideband TOAs used in the timing and noise analyses were measured with these DM-aligned models, and instrumental time offsets were applied to TOAs from ASP and GASP profiles, as detailed in Appendix~A of \\nineyr.\nMetadata in the TOA files take the form of ``flags'', which get appended to each TOA line in the files.\nA number of new TOA flags have been added to aid wideband timing analyses, and a few of the usual TOA flags have different meanings from their narrowband TOA counterparts; these are listed in the top portion of Table~\\ref{tab:toa_flags}.\n\nThe choice of DM alignment in wideband profile models is analogous to the ambiguity of absolute phase between TOAs measured with different template profiles in the narrowband analysis.\nThose constant phase offsets are modeled in the timing model with so-called ``JUMP'' parameters and are also present in the wideband analysis.\nOur fiducial DM alignment is an attempt at getting the simplest profile evolution models, but a new, analogous timing model parameter is necessary when using multi-band DM measurements as data for the timing model.\nTo this end, we implemented ``DMJUMP'' parameters for wideband timing in the extended likelihood introduced in Section~\\ref{subsec:timing}.\nAppendix~\\ref{sec:wb_like} contains details about how these parameters influence the timing model.\n\n\\input{tables\/tab_toa_flags}\n\n\\subsection{Cleaning \\& Curating the Wideband Data Set}\n\\label{subsec:wb_cleaning}\n\nThe narrowband data set was prepared in advance of the wideband data set, and as a part of its creation we kept track of bad observations that were corrupted by instrumentation or calibration issues, or were so affected by RFI that we excised them outright (250 of 11,178 observations). \nThese observations (which are included in the narrowband data set as commented TOAs with the flag \\texttt{-cut badepoch}) were simply not introduced into the wideband pipeline.\nThere are also a small number of observations (36) for which data were taken on a pulsar using a different receiver than usual, often for testing purposes (these are included in the narrowband data set as commented TOAs with the flag \\texttt{-cut orphaned}).\nThese data are generally not sufficient to create good profile evolution models, and would add very few degrees of freedom; we similarly excluded them from the wideband analysis at the start.\n\nThere was one other additional step in curating the profile data set used to make wideband TOAs.\nUpon finishing the modeling procedure described in Section~\\ref{subsec:prof_evol}, we calculated goodness-of-fit statistics for each profile in the data set based on its predicted pulse shape from the corresponding model.\nProfiles in a given subintegration were zero-weighted if their goodness-of-fit exceeded a threshold ($\\chi^2_{\\textrm{reduced}} > 1.25$), which was empirically determined after examining the distributions for each combination of pulsar and receiver.\n\nFor most combinations, the number of discarded profiles in this manner was of order a few percent.\nAfter zero-weighting these profiles, the data were re-averaged and the profile evolution models were recreated.\nThis step was necessary because, as with the ADC artifact mentioned in Section~\\ref{sec:obs}, unmitigated RFI can corrupt the modeling procedure.\nMore general RFI-flagging techniques based on template-matching using the wideband profile models are in development within NANOGrav and elsewhere (MeerTime collaboration, private communication).\nSuch techniques could potentially identify irregularities in the profiles, be it from RFI or other sources, earlier in the reduction pipeline.\n\nThe remainder of the cleaning of the wideband data set was performed on the measured TOAs; any TOAs ``cut'' from further analysis were given one of the flags listed in Table~\\ref{tab:toa_flags}, but are included as commented TOAs in the publicly available text files.\nMost of the cuts described in the table have counterparts in the preparation of the narrowband data set, and we refer the reader to \\twelveyr\\ for details beyond those offered in the table and those that follow.\n\nThe S\/N threshold used for the wideband TOAs was set at $25$, compared to the value of $8$ used for narrowband TOAs.\nThe main reason for this was empirical and related to the fact that the estimated S\/N for wideband TOAs is subject to significant bias in the low S\/N regime, favoring a higher threshold than is naively derived.\nWe justify this choice in Appendix~\\ref{sec:low_snr}.\n\nNote that in \\elevenyr\\ and \\twelveyr\\ a numerical TOA outlier analysis is performed \\citep{Vallisneri2017}.\nSome of the narrowband TOAs identified in this way are from profiles corrupted by RFI or instrumental problems that were not otherwise identified.\nOur goodness-of-fit filter of the profile data described earlier served a similar purpose, and no separate outlier TOA analysis was performed.\nWe found that after filtering the profiles in this way and thresholding the TOAs based on the S\/N cutoff of $25$, the initial timing results were remarkably clean; there were only a handful of additional TOAs that were culled based on a large timing residual ($>100~\\mu$s) or were otherwise identified by eye (see Table~\\ref{tab:toa_flags}).\n\n\\input{tables\/tab_ntoa_nchan}\n\nOverall, despite the procedural differences in preparing the two data sets, the quality control for the wideband data set resulted in $\\sim$~16\\% more profiles used for TOA measurement, as can be seen in Table~\\ref{tab:ntoa_nchan}.\nThis difference is largely due to the inclusion of low S\/N ratio profiles that are discarded in the narrowband data set (see Appendix~\\ref{sec:low_snr}); as such, it is unsurprising that these additional data in general do not carry a proportionally large impact on the timing results, as will be shown.\nHowever, see Section~\\ref{subsec:pulsar_results} for specific examples.\n\nAfter curation, the resulting wideband data set has 12,598 TOAs, corresponding to 480,474 profiles; this is compared to the 415,122 TOAs in the narrowband data set, a factor of $\\sim$~33 larger in TOA volume, which will only grow as the ASP and GASP TOAs become a fractionally smaller subset of the entire data set, and as new wideband facilities and receivers come into use. \nNote, however, that the overall wideband data set volume is only a factor of $33\/2 \\sim 16$ smaller, after including the DM measurements in the analysis.\n\n\\begin{table*}[th!]\n    \\caption{Basic Pulsar Parameters and TOA Statistics\\label{tab:toa_summary}}\n\\centering\n\\input{tables\/tab_toa_summary}\n\\end{table*}\n\n\\input{text\/fig_wb_toa_dm_err}\n\nA summary of the TOA uncertainties are presented in two forms.\nFirst, the median uncertainties are listed, along with other basic pulsar parameters, in Table~\\ref{tab:toa_summary}.\nThere is an analogous table in \\twelveyr\\ for the narrowband TOAs; in both cases, the uncertainties have been scaled to estimate the median TOA uncertainty from a 1800~second observation of the pulsar with 100~MHz of bandwidth.\nOverall, the values are comparable to their narrowband counterparts, but differences may be attributable to any of: unmodeled profile evolution in the narrowband data set, the inclusion of very low S\/N profiles in the wideband data set, the additional fit parameter (DM) in the wideband measurement, or other subtle discrepancies.\nSecond, in Figure~\\ref{fig:wb_toa_dm_err} we graphically present the ``raw'' median TOA and DM uncertainties with central intervals covering the central 68\\% of the distribution, ranking pulsars by their median PUPPI or GUPPI L-band TOA uncertainty.\nWe use ``raw'' to mean that these are the formal, estimated uncertainties from the template-matching procedure, which do not include any other sources of uncertainty and are not scaled in any way.\nIt is obvious from this plot that, depending on the pulsar, the improvement in raw TOA precision after moving from ASP and GASP to PUPPI and GUPPI is a factor of 2--3 or more in many cases, but the DM precision improves by an order of magnitude or more in all receiver bands except 327 and 430~MHz.\nThis improvement is due to the increase in bandwidth covered by PUPPI and GUPPI (see Table~\\ref{tab:observing_systems}).\n\n\\subsection{Obtaining Timing Solutions}\n\\label{subsec:timing}\n\nWe used the 12.5-year data set results from \\twelveyr\\ as initial timing solutions instead of deriving completely new timing results from the extended baselines of the 11-year data set.\nThis was done in part to facilitate comparisons and in part to reduce the need for redundant analyses.\nSpecifically, any new spin, astrometric, or binary timing model parameters found to be significant in \\twelveyr\\ were retained, but FD parameters were removed, as were the parameters that describe the DM model, called DMX.\n\nDMX is a piecewise-constant characterization of DM variability that is part of the timing model.\nSimpler models of DM variability, such as low-order polynomials, do not describe the data well, but more advanced models, such as those that use a stochastic description of variability \\citep[e.g., as a Gaussian process,][]{Lentati13}, are currently being investigated.\nThe criteria for dividing up the TOAs into DMX epochs defined by Modified Julian Dates (MJDs) can be found in \\twelveyr.\nFor each DMX epoch, a DM is measured based on the $\\nu^{-2}$ dependence of the TOAs that fall within the epoch, and all of these DMX model parameters are measured simultaneously with the fit for the rest of the timing model.\n\nIf we were to ignore the wideband DM measurements, the wideband TOA data set would be significantly hampered in the following ways.\nThere are a large number of DMX epochs which contain data from a single receiver.\nIn the cases where such an epoch has a single wideband TOA (instead of the dozens of analogous narrowband TOAs), the corresponding single DMX parameter removes the single degree of freedom, artificially zeroing out the timing residual for this epoch.\nIf there are a few wideband TOAs from the same receiver band in such an epoch, they will have similar reference frequencies, and so the DMX parameter will be poorly constrained and perhaps biased.\nFinally, even for the majority of DMX epochs for which there are multi-frequency wideband TOAs from dual receiver observations, DMX only has access to the TOAs, their uncertainties, and reference frequencies.\nThat is, the information about the dispersive delays across the individual receiver bands (captured by the wideband DM measurements, or, equivalently, the multi-frequency TOAs in the narrowband data set) is lost, and DMX only sees the dispersive delay between the bands.\nAs can be seen in most pulsars' DM and DMX time series (see Appendix~\\ref{sec:resid}), the wideband TOAs and their inter-band dispersive delay carry more weight in the DMX model than do the intra-band delays characterized by their corresponding wideband DM measurements.\nHow much more so depends on the pulsar and receiver bands in question, but it is important to highlight that disregarding the DM data is not a viable option for analyzing this data set.\nIndeed, we attempted several such analyses, which yielded significantly worse results in many pulsars.\n\nTherefore, not only was it appropriate, but it was also necessary to expand the likelihood used to fit our timing models so that the wideband DM measurements inform the DM model.\nIn effect, in the new likelihood, the wideband DM measurements influence the timing model as prior information on the DMX values.\nEach of the TOAs falling within a DMX epoch have a corresponding DM measurement; the weighted average of these measurements is used as the mean of a Gaussian prior on the DMX value for that epoch, while the standard error of the weighted average is the prior distribution's standard deviation.\nThe details of this new likelihood and its implementation in the pulsar timing software packages \\texttt{Tempo} \\citep{tempo} and \\texttt{ENTERPRISE} \\citep{enterprise} can be found in Appendix~\\ref{sec:wb_like}.\n\n\\afterpage{\\clearpage}\n\nThe timing models from \\twelveyr\\ were first refit with \\texttt{Tempo} using the wideband TOAs only, omitting the DM measurements, to setup the DMX epochs and to get initial DMX values.\nIncluding the DM measurements at this point sometimes resulted in poor timing results because there is currently no way to fit the DMJUMP parameters simultaneously with the timing model within \\texttt{Tempo}.\nIt is at this stage that TOAs were excluded from further analysis if they did not meet the frequency ratio criterion described in Table~\\ref{tab:toa_flags} or if the entire epoch was removed based on a new analysis performed in \\twelveyr\\ (also mentioned in Table~\\ref{tab:toa_flags}).\n\nThe wideband TOAs, DMs, and timing models were then subject to a Bayesian analysis with \\texttt{ENTERPRISE} using the new wideband likelihood.\nThis analysis optimizes the probability of the observed data by characterizing the noise in the timing residuals, which has both white and red components, much in the same way as in \\twelveyr, \\elevenyr, and \\nineyr, with a few important differences:\n\n\\textit{No ECORR --}\nThere is one parameter in the standard white noise model that is not used in the wideband analyses.\nThis parameter, called ECORR, accounts for the (assumed 100\\%) correlation between multi-frequency TOAs taken at the same time and is used in the narrowband analyses of \\nineyr, \\elevenyr, and \\twelveyr\\ \\citep{Arzoumanian2014,Arzoumanian2015b}.\nSince wideband TOAs effectively consolidate the many narrowband TOAs into one, any physical effects contributing to this parameter (such as pulse jitter or ISM effects; see Section~\\ref{subsubsec:wn_compare}) would be absorbed by the standard EQUAD noise parameter, which is added in quadrature to the measured TOA uncertainty \\citep{Edwards06,Lentati14}.\nAlternatively, any effects contributing to ECORR in the narrowband analysis may be modeled by a larger and shallower red noise process in the wideband analysis.\nA comparison of the detected excess white noise in the two data sets is presented in Section~\\ref{subsubsec:wn_compare}.\n\n\\textit{DMEFAC \\& DMJUMP --}\nTwo additional parameters are needed in the new wideband likelihood.\nThe first, which we call ``DMEFAC'', is analogous to the standard TOA EFAC: it is a factor that scales the estimated wideband DM measurement uncertainty.\nIn a similar fashion to the other white noise parameters, a DMEFAC is assigned for each combination of receiver and backend in each pulsar's noise model.\nThe second was introduced in Section~\\ref{subsec:prof_evol}, which we call ``DMJUMP''.\nThis parameter is analogous to standard JUMP parameters, but instead of modeling an achromatic phase offset between TOAs measured in different receiver bands, DMJUMP is a DM offset between wideband DMs measured in different bands.\nThese parameters account for the differences in alignment between profile evolution models in disparate bands, and amount to making a choice for the absolute DM.\nIt is important to stress that this ambiguity in absolute DM, as well as the offsets in DMs measured in disparate bands, exist also in the narrowband analyses; in \\twelveyr, the choice of having fixed templates in each band, coupled with using FD parameters to account for constant TOA biases as a function of frequency, amount to addressing the analogous problems.\nWe assign one DMJUMP parameter per receiver in each pulsar's timing model, since the profile evolution models are independent of backend.\nIt may seem that we should use one less DMJUMP parameter than there are receivers in each pulsar's analysis, as is done for standard phase JUMP parameters.\nHowever, because the DMX model is separately informed by the TOAs, it is not an overdetermined problem.\nThis fact is borne out by examining the posterior chains; although we see that the DMJUMP parameters are often highly covariant, they are not completely degenerate.\nWe used a uniform prior distribution on DMJUMP parameters in the range $[-0.01,0.01]~\\textrm{cm}^{-3}~\\textrm{pc}$; virtually all of the values are $\\lvert\\textrm{DMJUMP}\\rvert < 0.004~\\textrm{cm}^{-3}~\\textrm{pc}$.\n\n\\textit{White noise priors --}\nIn the analyses of all of our other data sets, we have used large, uniform priors on EFAC between 0.1 and 10.0.\nEFAC was originally implemented to account for instances when the profile data poorly matched the template profile in the TOA fit, which would underestimate the TOA uncertainty.\nIn the present analysis, we expect EFAC to be near 1.0 because we are using evolving profile templates and have carefully excised RFI at a number of stages in the pipeline.\nWe have found that allowing extreme EFAC values can inadvertently over- or down-weight subsets of the data when it is not justified.\nOne reason for this is that there is a larger amount of covariance between EFAC and EQUAD parameters in the wideband analysis because the formal TOA uncertainties (of which there are far fewer) are more homoscedastic; EFAC and EQUAD parameters can only be differentiated if there is variance in the uncertainties.\nEquation~\\ref{eqn:N_matrix} describes how EFAC and EQUAD parameters are related and affect the TOA measurement uncertainty.\nTherefore, we used a Gaussian prior on all EFAC parameters with a mean of 1.0 and standard deviation of 0.25; for similar reasons, we applied the same prior to DMEFAC parameters.\nThis choice is further justified in Appendix~\\ref{sec:low_snr}, where we show that the estimated TOA and DM uncertainties based on calculating the Fisher matrix of Equation~\\ref{eqn:toa_lnlike} are accurate down to very low S\/N.\nIt should also be noted that these uncertainties, being based on the Fisher information matrix, are equal to the Cram\\'{e}r-Rao lower bound, which motivates the continued use of EFAC parameters.\nWe use the same prior on EQUAD parameters as is used for both EQUAD and ECORR in \\twelveyr, which is a uniform distribution on log$_{10}$(EQUAD~[s])~$\\in [-8.5, -5.0]$.\nDue to our use of non-uniform priors for EFAC and DMEFAC parameters, we refer to all point estimates from the noise modeling as maximum a posteriori (MAP) values, instead of maximum-likelihood values.\n\n\\textit{Red noise priors --}\nWe use the exact same red noise model and priors as in \\twelveyr, but because the determination of red noise significance differs slightly from \\elevenyr\\ and \\nineyr, and because it will be relevant in the discussion of results, we summarize it here.\nThe red noise is assumed to be a stationary Gaussian process, which we parameterize with a power-law power spectral density $P$ of the form\n\\begin{equation}\n    \\label{eqn:red_pl}\n    P(f_m) = A^2_{\\small \\textrm{red}} \\left(\\frac{f_m}{1~\\textrm{yr}^{-1}}\\right)^{\\gamma_{\\small \\textrm{red}}},\n\\end{equation}\nwhere $A_{\\small \\textrm{red}}$ is the amplitude of the red noise at a frequency of 1\\,yr$^{-1}$ in units of $\\mu$s~yr$^{1\/2}$, and $\\gamma_{\\small \\textrm{red}}$ is the spectral index.\nThe spectrum is evaluated at thirty linearly spaced frequencies $f_m$ indexed by $m$, incremented by 1\/$T_{\\textrm{span}}$, where $T_{\\textrm{span}}$ is the span of the pulsar's data set.\nThe prior on the red noise amplitude is uniform on log$_{10}$($A_{\\small \\textrm{red}}$~[yr$^{3\/2}$])~$\\in [-20, -12]$, whereas the prior on the red noise index has been constrained in both 12.5-year analyses to be uniform on $\\gamma \\in [-7, -1.2]$.\nA pulsar is deemed to have ``significant red noise'' in these analyses if the Savage-Dickey density ratio \\citep[a proxy for the Bayes factor,][]{dickey1971} estimated from the posterior distribution of log$_{10}$($A_{\\small \\textrm{red}}$) is greater than one hundred.\nVery low-index red noise is thought to primarily arise from imperfect modeling of various effects from the ISM \\citep{ShannonCordes2017} and will be covariant with the white noise parameters.\nIncluding shallow red noise instead of modeling it with only white noise parameters will not significantly change the timing model.\nThe analyses here and in \\twelveyr\\ are only indicative of the presence of red noise, which may or may not be wholly intrinsic to the pulsar; a comparison of the red noise models is presented in Section~\\ref{subsubsec:rn_compare}.\nAdvanced noise modeling of the 11- and 12.5-year data sets, in which we explore bespoke models for each pulsar specifically in the context of GW analyses, is underway and will be presented elsewhere (Simon et al. \\textit{in prep.}).\n\nUpon completion of the noise analysis, following the same protocol as in \\twelveyr, the MAP noise model is included as fixed parameters in the timing model, which is re-optimized using the generalized least squares implementation of \\texttt{Tempo}, now using the augmented, wideband likelihood.\nThe large majority of the reduced chi-squared (goodness-of-fit) values fall between 0.9 and 1.1, with a few larger values.\nSome of these are to be expected because the additional DM data may not be particularly informative, or they may not be modeled well by DMX (e.g., see Section~\\ref{subsec:DM_nu}).\nAs in \\twelveyr, we examined the significance of adding and removing various timing model parameters, but after finding no strong evidence favoring change, we kept the identical set of timing model parameters for ease of comparison.\nThe differences with respect to crossing the significance threshold for including or excluding parameters are marginal, and in several cases are a function of the difference in red noise model (see Section~\\ref{subsubsec:rn_compare}).\n\nThe timing models are summarized in Table~\\ref{tab:timing_model_summary}, which also lists the Bayes factor, $B$, indicating the significance of red noise.\nThere is an analogous table in \\twelveyr\\ containing the results from the analyses of the narrowband data set.\nAs mentioned in Table~\\ref{tab:toa_flags}, we removed ASP and GASP TOAs that were taken simultaneously with concurrent PUPPI or GUPPI observations from the final TOA data sets.\nThe final timing models with noise parameters, curated wideband TOAs, and related auxiliary files are the furnished products comprising this data release.\nWe present the timing residuals and DM time series for these data in Appendix~\\ref{sec:resid}, which includes visual comparisons with the counterpart averaged residuals and DMX models from \\twelveyr.\n\n\\begin{table*}[th!]\n\\centering\n\\caption{Summary of Timing Model Fits\\label{tab:timing_model_summary}}\n\\input{tables\/tab_timing_model_summary}\n\\end{table*}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Average Portraits \\& Flux Density Measurements}\n\\label{subsec:port_results}\n\nA by-product of the profile evolution modeling procedure is a calibrated high S\/N average portrait with a nominal profile alignment and full polarization information.\nThe polarization portraits contain a wealth of information and are of interest to model in their own right; their models could potentially be used to improve the TOA measurement in cases of significant polarization.\nFor sufficiently polarized, large bandwidth, high S\/N data, the rotation measure (RM) could be measured as part of the wideband TOA measurement.\nSuch a development would combine the techniques summarized in Section~\\ref{subsec:prof_evol} with those from \\citet{vS06}, \\cite{vS13}, and \\citet{Oslowski13}, and is an active field of research.\n\nWe also estimated the phase- and frequency-averaged flux density for each of our PUPPI and GUPPI TOA measurements; ASP and GASP data were excluded because the profile data from which TOA measurements were made had been rescaled from their original flux calibration (see \\nineyr\\ for details).\nThe two main assumptions that go into the estimate and its formal, statistical uncertainty are that the profile evolution model sufficiently describes the data (i.e., no model error) and that it has a correct baseline of zero flux density; all phases contribute to the measurement.\nThe frequency-averaged flux density and uncertainty are calculated from the weighted-mean of the phase-averaged flux densities.\nSince the scaling parameters $a_n$ enter the calculation in the same way as for the S\/N estimate, the flux density estimates may contain similar biases (see Appendix~\\ref{sec:low_snr}).\nThe relevant flags for these measurements are listed in Table~\\ref{tab:toa_flags}, including a reference frequency for the flux density estimate.\nNo additional sources of uncertainties are considered, and the interpretation of these measurements should be treated with caution.\n\n\\subsection{Profile Evolution Models}\n\\label{subsec:model_results}\n\nWe find that for the majority of our pulsars, the profile evolution model for a given receiver band requires a single eigenprofile (62 of 102 pulsar-receiver combinations), which can be thought of as the gradient of the mean profile.\nMost of the remainder required two (20 of 102) or zero (13 of 102; i.e., those data are consistent with a constant, non-evolving profile).\nThe few cases in which more than three basis eigenprofiles are used to describe profile evolution arise in two very high S\/N pulsars (3 of 102 have three, the remaining 4 cases have more). \nB1937$+$21 shows spectral leakage from the overlapping, finite-attenuation filters used to subband the data\\footnote{We note that a better choice of filter appears to drastically improve this situation \\citep{Bailes20}.}, which results in the increased number of eigenprofiles in three of its models, and the imperfect correction of the ADC artifact image described in Section~\\ref{sec:obs} has the same consequence for one model for J1713$+$0747.\nRemoving the perhaps spurious eigenprofiles for these pulsars does not appear to significantly change the timing results in Section~\\ref{sec:results}, so we leave them for completeness.\nFurthermore, these two pulsars are observed with both observatories at L-band, and we find that the first two eigenprofiles (which contribute the most to profile evolution) are qualitatively the same between the models from each receiver\n\nProfile broadening from scattering in the ISM or other drastic, intrinsic profile evolution may be responsible for second and third eigenprofiles in the cases where either of those are detected.\nHowever, ``incorrect'' profile alignment with respect to a constant rotation proportional to $\\nu^{-2}$ (corresponding to a small, constant DM offset, generally not larger than, but at most a few times $\\sim$10$^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$) may also be the culprit for additional eigenprofiles.\n\nIt is important to highlight that this subtle issue exists in the narrowband analysis as well; the implicit assumption there is perhaps the most parsimonious one, that the profile shape does not evolve with frequency and that the profiles are aligned in phase.\nThe choice of profile alignment sets the value of the absolute DMs measured and will not have an effect on the timing analyses, though a detailed study of this question is beyond the scope of this paper.\nMore interesting questions about disentangling profile evolution from ISM variations and possible magnetospheric effects are still open \\citep{Hassall12}.\nA possible future development in the context of the present work is to take a similarly parsimonious approach and simultaneously model profile evolution across all observed bands while minimizing the number of significant eigenprofiles as a function of dispersive rotation.\nFurthermore, the underlying physical description of the observed profile evolution also warrants its own investigation.\n\nOne might expect a correlation between the total number of eigenprofiles for each pulsar and the number of FD parameters in the timing models from \\twelveyr.\nWe see a rough correspondence between these two numbers, but its interpretation is dubious.\nFor example, the FD parameters for B1855$+$09 (a.k.a. J1857$+$0943) from \\twelveyr\\ account for an approximate 20~$\\mu$s delay across the profiles in its 430~MHz band, purportedly from unmodeled frequency evolution of the profile shape.\nCareful inspection reveals that its 430~MHz profiles show no evidence for profile evolution, neither in the number of significant eigenprofiles (zero), nor in the profile residuals after subtracting the model, nor by direct comparison of the profiles, whereas there is prominent profile evolution across the L-wide bandwidth.\nEven though the 430~MHz band is a factor of three lower in frequency than L-wide, the latter's narrowband TOAs will be more influential in DM estimation.\nThis can be understood by the much larger fractional bandwidth of the L-wide receiver (see Table~\\ref{tab:observing_systems}): although the dispersive delay across both receiver bands is comparable, the median raw wideband TOA uncertainty from L-wide is an order of magnitude more precise, and its median raw wideband DM uncertainty is $\\sim$5 times smaller (see Figure~\\ref{fig:wb_toa_dm_err}).\nThe spurious FD prediction may arise from the interplay between the relative weighting of the L-band and 430~MHz data in the DMX model, the covariance between FD parameters and DMX values, or perhaps something more interesting; most likely, the FD parameters are filling in for the role of DMJUMP, as mentioned in Section~\\ref{subsec:timing}.\nThe details are beyond the scope of this paper and are under investigation elsewhere.\n\n\\subsection{Frequency-dependent DMs}\n\\label{subsec:DM_nu}\n\nFor a handful of our highest DM pulsars, the DM time series from each frequency band appear significantly different from one another.\nThese trends are apparent in the panels second from the top in Appendix~\\ref{sec:resid} for pulsars J1600$-$3053, J1643$-$1224, J1747$-$4036, and J1903$+$0327 (Figures \\ref{fig:summary-J1600-3053}, \\ref{fig:summary-J1643-1224}, \\ref{fig:summary-J1747-4036}, and \\ref{fig:summary-J1903+0327}, with DMs $\\sim$~52.3, 62.3, 153.0, and 297.5~$\\textrm{cm}^{-3}~\\textrm{pc}$, respectively).\nIt is also readily apparent in these panels, and in many other pulsars' DM time series, that the DM measurements are only significant after the switchover from the older generation of backend instruments (ASP and GASP) to the newer ones (PUPPI and GUPPI) due to their ability to process a larger bandwidth in real time (see Table~\\ref{tab:observing_systems}).\n\nAll four of these pulsars have clear pulse broadening in the form of frequency-dependent tails on the trailing edges of their profile components.\nTo estimate the amount of scattering present in these pulsars, we decomposed their concatenated average portraits into a small number of fixed Gaussian components and an evolving one-sided exponential function \\citep{PDR14}.\nIn this way we estimated the scattering timescale $\\tau$ at 1400~MHz for each of these four pulsars to be $\\tau_{\\textrm{1400}} \\sim$~26, 52, 22, and 130~$\\mu$s, respectively.\n\nIf the scattering timescale is changing with time and is not accounted for in the TOA measurement, the wideband DM measurements will be biased similarly as a function of time.\nAs mentioned in Section~\\ref{subsec:wb_toa_lnlike}, a forthcoming publication will present extensions to the wideband TOA measurement that will be better able to segregate time-variable profile broadening from classical DM variations (Pennucci et al. \\textit{in prep.}).\nThe scattering timescale scales more steeply with frequency than does the dispersive delay (approximately, $\\tau \\propto \\nu^{-4}$), and therefore the wideband DMs measured at lower frequencies will incur a greater bias, since the centroids of scattered pulse components shift by a greater amount.\nHowever, one expects that these biases, even if they are different in magnitude, will be correlated in time.\nConditioned on that assumption, it is difficult to explain the DM time series of these pulsars arising solely from time-variable scattering.\nIn all four instances, there are periods of correlation \\textit{and} anti-correlation between the DM time series measured in each frequency band.\n\nThis sort of behavior is, however, predicted by the phenomenon of ``frequency-dependent DM'' \\citep{Cordes16}, and very similar behavior has been seen in at least one other (canonical) pulsar \\citep{Lam19,Donner19}, although earlier indications existed in B1937$+$21 \\citep{DemorestPhDT,Ramachandran06,Cordes90} and in sparse multi-frequency measurements of the highest DM pulsar \\citep{Pennucci15}.\nThe dispersion measure is defined as the path integral of the free-electron density sampled by a propagating electromagnetic wave.\nDue to the refractive nature of the ISM, the path will vary as a function of the frequency of the wave, and due to the density inhomogeneities in the ISM, the integrated density -- the DM -- will therefore also be a function of frequency.\nHowever, these differences are expected to be small, with root-mean-square (RMS) values typically $\\ll 10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$, and thus only high-precision observations (e.g., bright MSPs, or bright low-frequency sources) of high-DM pulsars over long periods of time are expected to convincingly show this phenomenon.\n\nTo substantiate the claim that the DM trends seen in these four pulsars may arise from this peculiar ISM effect, we can calculate the predicted RMS difference between DMs measured at a fiducial frequency $\\nu$ and a lower frequency $\\nu'$, $\\sigma_{\\overline{\\textrm{DM}}}(\\nu,\\nu')$, using Equations~12 and 15 of \\citet{Cordes16}.\nUsing our rough scattering timescales to estimate the scintillation bandwidths at $\\nu$, and using the appropriate frequencies for each pulsar, we find $\\sigma_{\\overline{\\textrm{DM}}}(\\nu,\\nu') \\approx 2,~4,~2,~\\textrm{and~} 3~\\times~10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$ for J1600$-$3053, J1643$-$1224, J1747$-$4036, and J1903$+$0327, respectively.\nThese values are all within a factor of $\\sim$~2--3 of the RMS differences measured in the observed DM time series: 0.6, 1.7, 2.8, and 5.9~$\\times~10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$, respectively, where we only considered the PUPPI and GUPPI data for these measurements.\nGiven that this quick assessment involves the assumptions that the density inhomogeneities in the ISM are Kolmogorov in nature, and that the scattering occurs in a single thin screen, we find this level of agreement suggestive.\nA more in depth analysis is beyond the scope of this work, but these results indicate that long-term timing of high-DM MSPs in the context of PTA experiments offer a unique opportunity to study this phenomenon, as well as time-variable scattering; the low-frequency, high-cadence observations of CHIME\/Pulsar are especially promising in these areas.\n\n\nIn the two largest DM pulsars (J1747$-$4036 and J1903$+$0327), there are obvious chromatic trends in the timing residuals from \\twelveyr\\ that are ameliorated in the wideband analysis.\nThe narrowband noise analyses compensate for this by having larger white noise parameters and slightly larger, shallower red noise, which helps to explain the timing improvements seen in the wideband data set.\nSimilarly, because the ISM effects appear as apparently chromatic DM measurements in the wideband data set, the DMEFAC parameters are larger than expected ($\\sim$1.5$-$2.0).\nThat is, the boilerplate DMX model may not be good representation of these data, even with DMEFAC and DMJUMP parameters, and more advanced DM and noise models are required.\n\nIn addition to these four pulsars, there are four more in our sample that have a DM $\\gtrsim$~50~$\\textrm{cm}^{-3}~\\textrm{pc}$: J0340$+$4130, B1937$+$21, J1946$+$3417, and B1953$+$29 (a.k.a. J1955$+$2908; Figures \\ref{fig:summary-J0340+4130}, \\ref{fig:summary-B1937+21}, \\ref{fig:summary-J1946+3417}, and \\ref{fig:summary-B1953+29}, with DMs $\\sim$~49.6, 71.1, 110.2, and 104.5~$\\textrm{cm}^{-3}~\\textrm{pc}$, respectively).\nNone of their DM time series show the clear chromatic trends seen in the other four, but they all have some amount of additional variance that inflates their DMEFAC parameters.\nUsing the measured scintillation parameters from \\citet{Levin16} for the three lower DM pulsars and repeating similar calculations as above, we find that the RMS differences predicted from \\citet{Cordes16} are much smaller ($\\sim$ an order of magnitude or more) than what is seen in the data.\nWe could not find a published value for J1946$+$3417, so we estimated its scattering timescale by modeling its profile with Gaussian components in the same fashion as the first four pulsars and find $\\tau_{\\textrm{1400}} \\sim 64~\\mu$s.\nThe predicted and observed RMS DM differences are again similar, $\\sim$~4 and 2~$\\times~10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$, respectively, and so frequency dependent DM effects could also be playing a role here.\n\nA couple of other well-timed pulsars with intermediate DM values show the same kind of extra DM variance (e.g., J0613$-$0200; Figure~\\ref{fig:summary-J0613-0200}, DM $\\sim$~38.8~$\\textrm{cm}^{-3}~\\textrm{pc}$, and see also Section~\\ref{subsubsec:rn_compare}).\nNeither frequency-dependent dispersion nor time-variable scattering (in the form of profile broadening) appears to be playing a role here or in the three pulsars mentioned above.\nAnother subtle effect may be at play in some of these pulsars, which is a manifestation of short-timescale variations referred to as ``pulse jitter''.\nPulse jitter arises due to the fact that any finite collection of real single pulses will produce a mean profile with a slightly different shape and location between realizations, despite the long-term stability of the average profile \\citep{Helfand75}.\nFor broadband observations that are significantly influenced by pulse jitter, the wideband DM estimate will be biased (cf. Parthasarathy et al. \\textit{in prep.}).\nDepending on the frequency dependence of pulse jitter \\citep{Lam19b}, the bias may also be strongly frequency dependent.\nAlternatively, the ``finite scintle effect'', in which the relevant time-variable scattering effects occur in the strong diffractive regime \\citep{CordShan10,Cordes90}, can have the same effect as pulse jitter and similarly bias the DM measurements.\nAn investigation of the observed DM variance in some of our pulsars is beyond the scope of this paper, but it will play a role in considerations of future wideband data sets.\n\n\\subsection{Comparison of Collective Timing Results}\n\\label{subsec:timing_results}\n\nIn this section we give an overview of the wideband timing results by assessing the overall characteristics in comparison with the those found in \\twelveyr\\ and addressing a few pulsars individually.\nBesides those already mentioned, and those that will be included in Section~\\ref{subsec:pulsar_results}, specific interesting, astrophysical results from the 12.5-year data set can be found in \\twelveyr; in particular, these include new or improved astrometric and binary timing model parameters.\nThe timing residuals from both data sets are presented in Appendix~\\ref{sec:resid}.\n\n\\subsubsection{Timing Model Parameters}\n\\label{subsubsec:param_compare}\n\n\\input{text\/fig_param_compare}\n\nAs mentioned in Section~\\ref{subsec:timing}, the set of spin, astrometric, and binary timing model parameters used in our analyses is identical to that in \\twelveyr; the phrase ``timing model parameters'' used for the remainder of the text refers to this collection, excluding DMX parameters, which are compared separately.\nThe ensemble of differences in these parameters is shown in Figure~\\ref{fig:param_compare}.\nThe differences between parameter values are plotted, where each difference has been normalized by the parameter uncertainty from \\twelveyr\\ ($\\sigma_{\\textrm{NB}}$), and the ``error bar'' on each difference has a length equal to the ratio of parameter uncertainties, with the uncertainty from \\twelveyr\\ in the denominator (i.e., $\\sigma_{\\textrm{WB}}\/\\sigma_{\\textrm{NB}}$).\nSuch a convention allows us to discuss the relative differences we see in parameters and address their consistency without having to reference their absolute units.\nIn the discussions that follow, we suppress the subscript on $\\sigma_{\\textrm{NB}}$ and use the standalone symbol $\\sigma$ to refer to the units of these normalized differences.\nJUMP parameters are not included in Figure~\\ref{fig:param_compare}, as they are not meaningful, and parameters that reference an epoch were excluded if the epochs differed.\nGenerally, this was not the case; 526 of 549 total parameters (96\\%), not including JUMPs or DMX parameters, were directly compared.\n\nAt a glance, we see that the timing model parameters are in very good agreement, almost entirely $<2\\sigma$ different (99\\% of the parameters), and with very similar parameter uncertainties.\nThe cases in which red noise is detected, or is detected in only one analysis, can be harder to interpret (these parameters are semi-transparent in Figure~\\ref{fig:param_compare}); due to covariance with the MAP red noise model, especially if the red noise is shallow, the parameter uncertainties can differ by a large factor.\nOnly 1 of the 526 parameters (0.2\\%) is $>5\\sigma$ away, while in total 4 (0.8\\%) are $>3\\sigma$ away; for context, if these were independent experiments and we interpreted these differences as random samples from the unit normal distribution, we would ``expect'' $\\sim$~0 deviations $>5\\sigma$ and $\\sim$~1 deviation $>3\\sigma$.\nThe only parameter larger than $5\\sigma$ different is a known peculiarity in our data set (see Section~\\ref{subsubsec:J1640+2224}).\nTwo of the remaining three parameters with a difference larger than $3\\sigma$ belong to J2234$+$0611, but also have larger uncertainties by $\\sim$30\\% (see Section~\\ref{subsubsec:J2234+0611}).\nThe last differing parameter is the parallax measurement of the black widow pulsar J2234$+$0944 (see Section~\\ref{subsubsec:J2234+0944}).\nNevertheless, the parameters agree remarkably well across the board, even in cases where red noise is detected.\n\n\\subsubsection{DMX Parameters}\n\\label{subsubsec:dmx_compare}\n\n\\input{text\/fig_dmx_compare}\n\nWe compare the mean-subtracted DMX model parameters in Figure~\\ref{fig:dmx_compare}, which has the same presentation as Figure~\\ref{fig:param_compare}.\nOf the 4,685 differences, 13 (0.3\\%) are $>5\\sigma$ away ($\\sim$~0 ``expected''), a total of 104 (2.2\\%) are $>3\\sigma$ away ($\\sim$~6 ``expected''), and 94\\% agree to better than $2\\sigma$.\nB1937$+$21 is responsible for 35 of the differences $>3\\sigma$, which are due to the scatter in its DM measurements and the large influence they have on the DMX model (see Sections~\\ref{subsec:DM_nu}~\\&~\\ref{subsubsec:B1937+21}).\nAnother 37 of these belong to the combination of J1713$+$0747 (see below), J1903$+$0327 (Section~\\ref{subsec:DM_nu}), and J2234$+$0944 (Section~\\ref{subsubsec:J2234+0944}).\nThe remaining 32 differences are distributed among 11 pulsars for which we have no particular suspicions.\n\nBesides the influence of differing red noise models that was already mentioned, the FD parameters in the narrowband data set are covariant with all DMX parameters, which makes the interpretation of the uncertainty ratios in Figure~\\ref{fig:dmx_compare} difficult.\nNevertheless, 92\\% of the DMX uncertainties agree to within a factor of 1.5, 99\\% agree to within a factor of two, and the median DMX uncertainty for each pulsar is comparable between the data sets.\n\nThe number of DMX parameters often differs slightly between the data sets by one, two, or three parameters; there are 10, 6, and 4 such instances, respectively, with 26 pulsars having the same number of DMX parameters.\nThe discrepancies in the number of DMX epochs arise from the slight differences in curating the data sets (Section~\\ref{subsec:wb_cleaning}).\nThe exception to this is J1713$+$0747, where we opted to use a higher density of DMX bins during and after the second dip in its DM time series \\citep{Lam2018}, resulting in 37 additional DMX values; a similar binning exception is made for the same reason in \\twelveyr.\nHowever, this means the relevant DMX epochs in \\twelveyr\\ average over a greater span of time and will be biased in comparison to their counterparts here.\nThe DMX time series are plotted in the topmost panels of the figures in Appendix~\\ref{sec:resid}.\nIn most instances, the DMX parameters from \\twelveyr\\ are hidden by those from the present analysis, demonstrating their close agreement.\n\n\\subsubsection{Excess White Noise}\n\\label{subsubsec:wn_compare}\n\n\\input{text\/fig_wn_compare}\n\nA pulse TOA is a proxy for the moment a fixed point of longitude on the neutron star passes over the line of sight.\nThere are a number of sources of uncertainty that obfuscate and bias the determination of this moment in time, even if the formal arrival time of the pulse can be very precisely determined.\nThese additional uncertainties introduce either time-uncorrelated (white) scatter or time-correlated (red) trends into the timing residuals, and can originate from a wide variety of sources local to the observatory, Earth, the solar system, the pulsar, or the intervening ISM.\nFor thorough reviews of the sources of these uncertainties, we direct the reader to \\citet{Verbiest18} and \\citet{CordShan10}.\nHere, we compare the excess white noise seen in both data sets, followed by the red noise in Section~\\ref{subsubsec:rn_compare}.\n\nThe formal TOA uncertainties are scaled in both analyses by EFAC parameters, which do not have a straightforward interpretation with respect to physical, excess noise; nominally, EFAC parameters account for misestimation of the system noise level or template matching errors.\nComparing the EFAC parameters is not very enlightening, particularly because the narrowband analysis uses fixed, non-evolving templates and uses a much broader prior on EFAC.\nThe wideband analysis also uses DMEFAC parameters, which can absorb some excess noise that might be modeled by EFAC in the narrowband analysis.\nAs mentioned in Section~\\ref{subsec:timing}, there is a difference between the two analyses in how white noise is modeled in the timing residuals.\nEQUAD and ECORR parameters capture the additional variance in the narrowband analysis, but because ECORR accounts for fluctuations that are completely correlated for simultaneously obtained measurements (i.e., narrowband TOAs), it cannot be differentiated from EQUAD in the wideband analysis and is therefore left out of the noise model.\n\nTo effectively compare the white noise, we plot the MAP EQUAD parameters from our analyses against the quadrature sum of the corresponding maximum-likelihood EQUAD and ECORR parameters in Figure~\\ref{fig:wn_compare}.\nIn the figure, points appear more opaque in proportion to how constrained the posterior distribution of the parameter is.\nThere is a clear correspondence over two orders of magnitude in the white noise parameters, suggesting that both analyses see very similar white noise.\n\nAlthough the integrated pulse profile shapes of MSPs are secularly stable \\citep[][with some exceptions, e.g., \\citet{slk+16}]{Brook18}, they vary minutely (indeed) on short timescales due pulse jitter (see Section~\\ref{subsec:DM_nu}).\nPulse jitter contributes additional uncertainty to the TOA and is expected to manifest in ECORR parameters, though the measured ECORR values exceed the predicted level of jitter \\citep{Lam16}.\nJitter is thought to be weakly or modestly dependent on frequency \\citep{Shannon14,Lam19b}, and its effects can only be reduced by longer integration times or actively accounting for shape change \\citep{Oslowski11}.\nOn the other hand, the various ISM effects that can contribute to EQUAD have a stronger (and mostly pulsar-independent) frequency dependence.\nFrom this perspective, analyzing narrowband TOAs may help to discriminate between sources of excess white noise, although using evolving profile templates would be an improvement to the overall approach.\nIn this way, both forms of analysis may contribute to arriving at the best results for a given pulsar.\nFor example, if some of our MSPs have large white noise because of time-variable scattering, then the pulse broadening can be included as part of the wideband TOA measurement.\n\n\\subsubsection{RMS Timing Residual}\n\\label{subsubsec:rms_compare}\n\n\\input{text\/fig_rms_compare}\n\nA second metric for gauging the overall level of noise is the RMS timing residual (see Table~\\ref{tab:timing_model_summary}).\nIn Figure~\\ref{fig:rms_compare} we compare the RMS values between the two analyses, taking care to use the averaged residuals from the narrowband data set and the whitened set of residuals whenever red noise was detected in either of the analyses.\nAlmost no pulsars differ by more than a factor of 1.5, with a number of exceptions explained as part of Section~\\ref{subsec:pulsar_results}, and half of them agree to within a factor of 1.1.\nHowever, the RMS residual from either analysis can be very sensitive to the exact noise model, which is fixed in the final optimization of the timing model.\nThe noise analyses explore the logarithm of the EQUAD, ECORR, and red noise amplitude parameters, so small statistical deviations in the best-fit parameters arising from the Monte Carlo analysis can lead to rather different RMS values.\nThe RMS should be thought of as a random variable, whose variance is influenced by the posterior distributions of the noise parameters.\nThis is true even when the noise model parameters are constrained, and it underscores the need for advanced noise modeling techniques.\nNevertheless, it is encouraging that 31 of the 47 pulsars show some amount of improvement and that all but five pulsars have RMS residuals no more than $\\sim10\\%$ larger than their narrowband counterparts.\n\n\n\n\n\\subsubsection{Detection of Red Noise}\n\\label{subsubsec:rn_compare}\n\n\\input{text\/fig_rn_compare}\n\nA final, and perhaps most crucial, litmus test for the wideband analyses is the detection of red noise in individual pulsars.\nObviously, the presence of red noise in the wideband data set (or lack thereof), in relation to what is seen in the narrowband data set, guides our expectations of full-scale GW analyses, which heretofore have only been vetted on our narrowband data sets.\nWe introduced the red noise model in Section~\\ref{subsec:timing}; there are additional details in Appendix~\\ref{sec:wb_like}, \\nineyr, \\elevenyr, and \\twelveyr.\nHere we discuss our findings in contrast to those from the narrowband analysis.\n\nIn Figure~\\ref{fig:rn_compare} we show the significantly detected power-law red noise in our analyses compared to those from \\twelveyr.\nWe again find the level of agreement between the data sets reassuring.\nRecall from Section~\\ref{subsec:timing} that a pulsar is deemed to have ``significant red noise'' if the estimated Bayes factor is above one hundred (see Table~\\ref{tab:timing_model_summary} for Bayes factors).\nThirteen pulsars have detected red noise in both analyses, one pulsar has significant red noise detected in just the narrowband analysis (J0613$-$0200), and two black widow pulsars, which are not shown in the plot, are treated differently and not discussed further here (J0023$+$0923 and J2234$+$0944; see Section~\\ref{subsubsec:J2234+0944}).\nTen of these pulsars (plus J1713$+$0747) had detected red noise in \\elevenyr; J1744$-$1134, J1853$+$1303, and J2317$+$1439 are new detections in \\twelveyr, which are all have significant red noise here.\n\nIt is thought that unmitigated ISM effects can manifest as shallow-spectrum red noise \\citep[][]{ShannonCordes2017,CordShan10,FosterCordes90,Rickett90}, which we indicate in Figure~\\ref{fig:rn_compare} for $\\gamma_{\\textrm{\\scriptsize red}} > -3$.\nJ0613$-$0200's DM is in the top third of our sample ($\\sim$ 38.8~$\\textrm{cm}^{-3}~\\textrm{pc}$, respectively), and has fairly shallow red noise in its narrowband analyses.\nRed noise is only marginally favored in its wideband analysis, as indicated by the Bayes factors of $\\sim$~15 in Table~\\ref{tab:timing_model_summary}.\nWhen red noise is included in the wideband analysis (the dashed-dotted lines in Figure~\\ref{fig:rn_compare}), the MAP model has the same index, a slightly smaller amplitude, and similar white noise parameters, than in \\twelveyr.\nWithout red noise, the corresponding wideband white noise EQUAD parameters are statistically unchanged.\nThis suggests that the wideband analysis might be able to mitigate some of the ISM-induced red noise.\n\n\nIntrinsic spin noise in pulsars has been modeled in the literature as a random walk in phase, frequency, or frequency derivative, with corresponding power spectral indices of $-2$, $-4$, and $-6$, respectively, as well as arising from chaotic behavior \\citep[e.g.,][]{Harding90}.\nThe lighter gray region in Figure~\\ref{fig:rn_compare} represents the best fit index ($\\gamma_{\\textrm{\\scriptsize spin}} = -4.46 \\pm 0.16$) for timing noise seen across pulsars of all types from \\citet{Lam17a}, consistent with a mixture of random walks \\citep[e.g.,][]{DAlessandro95,CordesDowns85}.\nThe scatter in this best fit relation, however, is large enough to essentially cover the range of observed spectra.\nIt is therefore difficult to interpret the spread of red noise we have detected, particularly because we suspect that some of the pulsars with shallow red noise are dominated by contributions from the ISM, whereas others may have a mix of contributions.\nCoexisting with the red noise intrinsic to the pulsar and that from the ISM, there is a contribution from the background of stochastic, low-frequency GWs, which is thought to have a steep power-law index ($\\gamma_{\\textrm{\\scriptsize GWB}} = -13\/3$; \\citet{Jaffe2003,Phinney2001}), indicated by a dotted vertical line in the figure.\nFor scale, the dashed vertical line indicates the 95\\% upper limit on the amplitude of the GW background from analyzing the 11-year data set \\citep{Arzoumanian2018b}.\nA more recent search for the stochastic GW background in the 12.5-year narrowband data set is presented in \\citet{Arzoumanian20}.\n\n\n\n\\subsection{Additional Discussion of Individual Pulsars}\n\\label{subsec:pulsar_results}\n\nThe results from a number of pulsars, some of which have been previously mentioned, deserve additional comments, caveats, or emphasis, which we detail here.\nIn addition, for the simple purpose of highlighting one example of generally good, comprehensive agreement with the narrowband results, and one example of where perhaps wideband timing did not prove beneficial, we direct the reader to J0931$-$1902 and J1910$+$1256, respectively.\n\n\\subsubsection{J0931$-$1902}\n\\label{subsubsec:J0931-1902}\n\nJ0931$-$1902 has the distinction of having the largest fractional difference between the number of pulse profiles used in the wideband analysis and the number of TOAs in the narrowband analysis; its wideband data set makes use of 81\\% more profiles (see Table~\\ref{tab:ntoa_nchan}).\nAgain, this difference arises because of the S\/N threshold used in the narrowband analysis; because this pulsar is fairly weak and scintillates, a large number of its low S\/N profiles get individually discarded in the narrowband analysis, even though they combine to yield useful wideband TOAs.\nJ0931$-$1902 is the second worst pulsar in our data set in terms of raw L-band timing precision (see Figure~\\ref{fig:wb_toa_dm_err}), but is somewhere in the middle in terms of RMS ($\\sim$~440~ns in both data sets).\nThere is absolutely nothing else different about its results from the narrowband analysis -- except that its timing model parameters are all $\\sim$~15\\% more precise in the wideband analysis.\nAt least two other pulsars show this level of improvement that is most likely attributable to a similar explanation -- J0340$+$4130 and J0740$+$6620 -- although their differences in data volume are not extreme.\nThese improvements underscore the benefit of using the wideband TOA approach for salvaging all information contained in less bright or scintillating pulsars.\n\n\n\\subsubsection{J1640$+$2224}\n\\label{subsubsec:J1640+2224}\n\nThe difference in J1640$+$2224's ecliptic longitude is the lone culprit referred to earlier for being very different ($\\sim$~6$\\sigma$) from its counterpart in the narrowband analysis.\nHowever, this is a known anomaly to us, albeit of unknown origin; we have previously compared timing results from different timing software using the exact same data sets, and J1640$+$2224's ecliptic longitude was the single outlier to be significantly different (see also the comparison between \\texttt{Tempo} and \\texttt{PINT} \\citep{PINT} in \\twelveyr).\nThe published position from Very Long Baseline Interferometry \\citep{Vigeland18} is not precise enough to discern between the two measurements.\nHowever, it should be noted that the value from \\twelveyr\\ is better than 1$\\sigma$ consistent with the extrapolated value from \\elevenyr, whereas the value from the wideband analysis is $\\sim$~2$\\sigma$ consistent with the extrapolated value from \\nineyr.\n\\citet{Fonseca2016} followed up on \\nineyr\\ and suspected that J1640$+$2224 is a massive neutron star \\citep[see also][]{Deng20}; the improvements on the mass measurements will be presented elsewhere.\n\n\\subsubsection{J1643$-$1224}\n\\label{subsubsec:J1643-1224}\n\nWe have already discussed J1643$-$1224 at some length in Section~\\ref{subsec:DM_nu}.\nIt is worth emphasizing, though, that some of the complexity and chromatic dependence seen in the DM measurements and timing residuals of this pulsar almost certainly arise from the fact that it lies directly behind the HII region Sh 2-27 associated with $\\zeta$-Ophiuchi \\citep{Ocker20}.\nThis association may also be responsible for a protracted decrease in its flux density \\citep{Maitia03}.\nIn addition to the confounding factors of the ISM, at least one intrinsic profile shape change event is thought to have occurred in this pulsar around February 2015 \\citep{slk+16}.\nAlthough we see the corresponding discrete perturbation in J1643$-$1224's timing residuals at this time, the follow-up analysis by \\citet{Brook18} on our 11-year data set argues that ISM effects cannot be ruled out.\nThe $\\sim$~45\\% improvement in its RMS timing residual seen in Figure~\\ref{fig:rms_compare} is almost certainly a result of the mitigation of the chromatic structure in its residuals; see the discussion at the end of Section~\\ref{subsec:DM_nu}.\n\n\\subsubsection{J1747$-$4036}\n\\label{subsubsec:J1747-4036}\n\nSimilarly, we have already discussed J1747$-$4036 in Section~\\ref{subsec:DM_nu}.\nJ1747$-$4036 also stands out in Figure~\\ref{fig:rms_compare}, with the same level of improvement in RMS residual as J1643$-$2224 due to the mitigation of chromatic structure in the timing residuals.\n\n\\subsubsection{J1910$+$1256}\n\\label{subsubsec:J1910+1256}\n\nJ1910$+$1256 has a significantly worse RMS wideband timing residual in Figure~\\ref{fig:rms_compare}, by just over a factor of two.\nWe find a significant L-band EQUAD detected in the wideband analysis, whereas the posterior distributions for the L-band EQUAD and ECORR are consistent with upper-limits.\nInterestingly, the white noise parameters for S-band are significantly measured in both analysis and are of similar amplitude.\nJ1910$+$1256 is in the top ten pulsars by raw L-band timing precision (Figure~\\ref{fig:wb_toa_dm_err}), with the median L-band PUPPI TOA having a precision just above 100~ns; the MAP PUPPI L-band EQUAD is more than three times larger.\nThe source of this discrepancy has not been determined but despite the difference, the timing model parameters are no more than $\\sim$~10\\% worse than their narrowband counterparts.\n\n\\subsubsection{B1937$+$21}\n\\label{subsubsec:B1937+21}\n\nB1937$+$21 (a.k.a. J1939$+$2134) presents a special set of challenges for the wideband analysis, being the brightest pulsar in the data set with the smallest formal measurement uncertainties by a considerable margin (see Figure~\\ref{fig:wb_toa_dm_err}).\nAs mentioned in Section~\\ref{subsec:prof_evol}, its profile modeling is contaminated by spectral leakage because it is so bright, although we do not believe this meaningfully affects the timing results.\nAs mentioned in Section~\\ref{subsec:DM_nu}, it has a substantial amount of scatter in its wideband DM measurements once the long-term trend is removed; this results in the highest DMEFAC parameters in the data set as well as the worst goodness-of-fit value for its timing model, due to the additional contribution from the DM model.\nThe restrictive Gaussian prior (see Section~\\ref{subsec:timing}) inhibits the DMEFAC parameters from taking even larger values, which would encapsulate more of the variance in the DM time series.\nRelaxing the prior is not physically motivated, and so these results direct us to implement an additional DM model parameter in future analyses, one that is analogous to the standard EQUAD parameter.\nGiven that both pulse jitter and variable diffractive interference in the ISM (i.e., the ``finite scintle effect'') play a roll in the observations of this pulsar \\citep{Lam19b}, it is feasible that both effects serve to bias the wideband DM estimates, resulting in extra variance in the DM time series.\nDespite the additional variance in the wideband DM time series, the astrometric timing model parameters are in very good agreement ($<$~1$\\sigma$) with similar uncertainties.\n\n\\subsubsection{J1946$+$3417}\n\\label{subsubsec:J1946+3417}\n\nJ1946$+$3417 is one of the two new pulsars in this data set, which has already been discussed in Section~\\ref{subsec:DM_nu} due to it having the third largest DM in the data set ($\\sim$110.2~$\\textrm{cm}^{-3}~\\textrm{pc}$).\nIt has the distinction of showing the single largest difference in RMS in either direction, seen in Figure~\\ref{fig:rms_compare}; the wideband RMS timing residual is a factor of three smaller.\nBoth analyses examine the same amount of data and the wideband raw timing precision is $\\sim$~10\\% better (see Table~\\ref{tab:toa_summary} and the equivalent table in \\twelveyr).\nThe Bayes factor for red noise in the narrowband analysis is $\\sim$~59, whereas it is not at all favored in the wideband analysis.\nThe preferred red noise model is large and shallow, and as a result of it not being included, the narrowband white noise parameters are larger than their wideband counterparts.\nThe timing model parameters agree to $\\le 1\\sigma$, but with $\\sim$~10\\% larger uncertainties in the wideband analysis.\nIt should also be noted that J1946$+$3417 is an astrophysically interesting source, as it is one of the few eccentric binary MSPs in the field and also contains a massive neutron star \\citep[][all of which also make note of J2234$+$0611]{Barr17,Jiang15,Antoniadis14,Freire14}.\n\n\n\\subsubsection{J2043$+$1711}\n\\label{subsubsec:J2043+1711}\n\nJ2043$+$1711 has the highest sub-threshold Bayes factor in Table~\\ref{tab:timing_model_summary}, $B\\sim50$.\nIn repeated analyses, the statistic $B$ was noisy enough to sometimes cross our significance threshold.\nThe narrowband analysis also favors red noise, but with a lower Bayes factor~$\\sim$~26.\nThe difference between the analyses may arise from the amount of data examined.\nAs can be seen in Table~\\ref{tab:ntoa_nchan}, J2043$+$1711's wideband data set is $\\sim$~70\\% larger than its narrowband counterpart, the third largest difference, which is due to its scintillation characteristics combined with the S\/N ratio cut off in the narrowband analysis.\nJ2043$+$1711 has a fairly low DM ($\\sim$20.8~$\\textrm{cm}^{-3}~\\textrm{pc}$), a timing baseline of six years and, importantly, it has been included in our high-cadence observations at Arecibo since 2015, which has increased its data volume by $\\sim$~70\\% since the 11-year data set.\nThe narrow features in its profile enable this pulsar to be timed very precisely when it is detected (see Tables~\\ref{tab:toa_summary}~\\&~\\ref{tab:timing_model_summary}, and Figure~\\ref{fig:wb_toa_dm_err}), and so we expect the emerging red noise in this pulsar to be significantly detected in the near future.\n\n\\subsubsection{J2234$+$0611}\n\\label{subsubsec:J2234+0611}\n\nAt face value, J2234$+$0611 is the best timed pulsar in the data set: the narrowband and wideband timing RMS values are $\\sim$~60 and 35~ns, respectively (Figure~\\ref{fig:rms_compare}), with no preference for red noise in either analysis.\nThis is partially due to it only having a timing baseline 3.4~years in length, although it is in the top ten pulsars by raw L-band timing precision.\nBoth analyses detect excess white noise, although the wideband analysis measures a significantly larger EQUAD in the 430~MHz band; this results in an overweighting of the wideband L-band data, which may explain the significantly smaller RMS value.\nWe make special mention of this pulsar also because it stands out for its level of disagreement in its timing model parameters with \\twelveyr, as mentioned in Section~\\ref{subsubsec:param_compare}.\nAll of its wideband timing model parameters have larger uncertainties by $\\sim$~30$-$40\\%, but no other pulsar shows quite this level of disagreement.\nIts ecliptic latitude and parallax measurements are $\\sim$~3.5$\\sigma$ different from their narrowband analysis counterparts.\nThe fact that J2234$+$0611 has a relatively short timing baseline but has significantly measured secular binary parameters is a testament to its timing precision.\nIn fact, additional modeling of its binary orbit is necessary, which was carried out in \\citet{Stovall2019} with an additional 1.5~years of data, most of which were NANOGrav observations collected beyond the cutoff of the present data set.\nAlong with the Shapiro delay and annual orbital parallax, \\citet{Stovall2019} were able to determine the 3-D orbital geometry of the binary.\nWe are confident that the discrepancies seen here will be resolved with the implementation of the \\citet{Stovall2019} timing solution in future data sets.\n\n\\subsubsection{J2234$+$0944}\n\\label{subsubsec:J2234+0944}\n\nJ2234$+$0944 was previously mentioned in Sections~\\ref{subsubsec:param_compare}, \\ref{subsubsec:dmx_compare}, and \\ref{subsubsec:rn_compare}.\nThis pulsar is one of four black widow pulsars in the data set (along with J0023$+$0923, J0636$+$5128, and J2214$+$3000), and one of two (along with J2214$+$3000) that do not show orbital or secular variability according to \\citet{BakNielsen20}, who studied three of these systems over $\\sim~8$~year baselines (J0636$+$5128 was not part of their study).\nThis pulsar has very significantly detected ``red noise'' in the wideband analysis, but no indication of it in the narrowband analysis, according to the Bayes factors in Table~\\ref{tab:timing_model_summary} and its analog in \\twelveyr.\nHowever, this ``red noise'' is specious; the preferred model is extremely shallow and the power-law fit to the function in Equation~\\ref{eqn:red_pl} is dominated by the frequencies higher than 1\\,yr$^{-1}$, reflecting the short timing baseline ($T_\\textrm{span} \\sim 4.0$~yr).\nBased on this reasoning, we exclude the red noise component from J2234$+$0944's analysis; we similarly excluded the ``red noise'' seen in J0023$+$0923 for the same reason.\nThe origin of the excess noise seen in the wideband data set is not known; it could be a sign of variability \\citep{Torres17}, but the findings of \\citet{BakNielsen20} refute this.\nInterestingly, the black widow J2214$+$3000 had a very similar issue in \\elevenyr, when it had a similar timing baseline, which was resolved with the additional data in this data set and a reparameterization of its timing model.\nFinally, \\twelveyr\\ reports that the parallax measurement is no longer significant, in contrast to \\elevenyr; this loss of significance in \\twelveyr\\ is marginal and not the case in the wideband analysis.\nDespite being $\\sim$~3$\\sigma$ different from \\twelveyr, J2234$+$0944's parallax is significantly measured and is $< 1\\sigma$ consistent with the value from \\elevenyr.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nInteracting particle systems have attracted a lot of attention\nbecause of their versatile modelling power (see for instance\n\\cite{Durrett99,Liggett97}). However, most available results deal\nwith their asymptotic behavior, and relatively few theorems\ndescribe their transient regime. In particular, central limit\ntheorems for random fields have been available for a long time\n\\cite{Malyshev75,Neaderhouser78,Bolthausen82,Newman80,\nTakahata83,CometsJanzura,ChenShao04}, diffusion approximations and\ninvariance principles have an even longer history\n(\\cite{EthierKurtz} and references therein), but those\nfunctional central limit theorems that describe the\ntransient behavior of an interacting particle system are usually much\nless general than their fixed-time counterparts.\nExisting results (see\n\\cite{HolleyStrook78,HolleyStrook79,KipnisVaradhan,SethuramanXu})\nrequire rather stringent hypotheses:\nspin flip dynamics on $\\mathbb{Z}$, reversibility,\nexponential ergodicity, stationarity\\ldots (see Holley and Strook's\ndiscussion in the introduction of \\cite{HolleyStrook79}).\nThe main objective of this article is to prove a functional central\nlimit theorem for interacting particle systems,\nunder very mild hypotheses, using some new\ntechniques of weakly dependent random fields.\n\nOur basic reference on interacting particle systems\nis the textbook by Liggett \\cite{Liggett}, and\nwe shall try to keep our notations as close to his as possible:\n$S$ denotes the (countable) set of sites, $W$ the (finite) set of\nstates, ${\\cal X}=W^S$ the set of configurations, and\n$\\{\\eta_t\\,,\\;t\\geq 0\\}$ an interacting particle system, i.e. a\nFeller process with values in ${\\cal X}$. If $R$ is a finite\nsubset of $S$, an empirical process is defined by counting how\nmany sites of $R$ are in each possible state at time $t$. This\nempirical process will be denoted by $N^R = \\{N^R_t\\,,\\;t\\geq\n0\\}$, and defined as follows.\n$$\nN^R_t = (N^R_t(w))_{w\\in W}\\;,\\quad N^R_t(w) = \\sum_{x\\in R}\n\\mathbb{I}_w(\\eta_t(x))\\;,\n$$\nwhere $\\mathbb{I}_w$ denotes the indicator function of state $w$. Thus\n$N^R_t$ is a $\\mathbb{N}^W$-valued stochastic process, which is not\nMarkovian in general. Our goal is to show that, under suitable\nhypotheses, a properly scaled version of $N^R$ converges to a\nGaussian process as $R$ increases to $S$. The hypotheses will be\nprecised in sections \\ref{basic} and \\ref{covar} and the main result\n(Theorem \\ref{cltips2}) will be stated and proved in section\n\\ref{clt}. Here is a loose description of our assumptions. Dealing\nwith a sum of random variables, two hypotheses can be made for a\ncentral limit theorem: weak dependence and identical distributions.\n\\begin{enumerate}\n\\item {\\it Weak dependence:} In order to give it a sense, one has\nto\n  define a distance between sites, and therefore a graph\n  structure. We shall first suppose that this (undirected) graph\n  structure has bounded degree. We shall assume also finite range\n  interactions: the configuration can\n  simultaneously change only on a bounded set of sites, and its value\n  at one site can influence transition rates only up to a fixed\n  distance (Definition \\ref{defetenduefinie}).\n  Then if $f$ and $g$ are two functions whose dependence on\n  the coordinates decreases exponentially fast with the distance from\n  two distant finite sets $R_1$ and $R_2$,\n  we shall prove that the covariance between\n  $f(\\eta_s)$ and $g(\\zeta_t)$ decays exponentially fast in the\n  distance between $R_1$ and $R_2$ (Proposition\n  \\ref{propinegcovfini}). The central limit theorem\n  \\ref{cltips2} will actually be proved in a much narrower setting,\n  that of group invariant dynamics on a transitive graph (Definition\n  \\ref{definvariant}). However we believe that a covariance inequality\n  for general finite range interacting particle systems is of\n  independent interest. Of course the bound of Proposition\n  \\ref{propinegcovfini} is not uniform in time, without further\n  assumptions.\n \n \n \n\\item {\\it Identical distributions:} In order to ensure that the\nindicator processes $\\{\\mathbb{I}_w(\\eta_t(x))\\,,\\;t\\geq 0\\}$ are\nidentically distributed, we shall assume that the set of sites $S$\nis endowed with a transitive graph structure (see\n\\cite{GodsilRoyle} as a general reference), and that both the\ntransition rates and the initial distribution are invariant by the\nautomorphism group action. This generalizes the notion of\ntranslation invariance, usually considered in $\\mathbb{Z}^d$\n(\\cite{Liggett} p.~36), and can be applied to non-lattice graphs\nsuch as trees. Several recent articles have shown the interest of\nstudying random processes on graph structures more general than\n$\\mathbb{Z}^d$ lattices: see e.g.\n\\cite{Haggstrometal02,HaggstromPeres99,Haggstrometal00}, and\n for general references \\cite{Peres99,Woess00}.\n\\end{enumerate}\nAmong the potential applications of our result, we chose to focus on\nthe hitting time of a prescribed level by a linear combination of\nthe empirical process. In \\cite{ParoissinYcart04}, such hitting\ntimes were considered in the application context of reliability.\nIndeed the sites in $R$ can be viewed as components of a coherent\nsystem and their states as degradation levels. Then a linear\ncombination of the empirical process is interpreted as the global\ndegradation of the system, and by Theorem \\ref{cltips2}, it is\nasymptotically distributed as a diffusion process if the number of\ncomponents is large. An upper bound for the degradation level can be\nprescribed: the system is working as soon as the degradation is\nlower, and fails at the hitting time. More precisely, let $f~:\nw\\mapsto f(w)$ be a mapping from $W$ to $\\mathbb{R}$. The total degradation\nis the real-valued process $D^R=\\{D^R_t\\,,\\;t\\geq 0\\}$, defined by:\n$$\nD^R_t = \\sum_{w\\in W} f(w)N^R_t(w).\n$$\nIf $a$ is the prescribed level, the failure time of the system\nwill be defined as the random variable\n$$\nT^R_a = \\inf\\{t\\,\\geq 0\\,,\\; D^R_t\\geq a\\,\\}.\n$$\nUnder suitable hypotheses, we shall  prove that $T^R_a$ converges\nweakly to a normal distribution, thus extending Theorem 1.1 of\n\\cite{ParoissinYcart04} to systems with dependent components.\nIn reliability (see \\cite{BarlowProschan} for a general reference),\ncomponents of a coherent system are usually considered as\nindependent. The reason seems to be mathematical convenience rather\nthan realistic modelling. Models with dependent components have been\nproposed in the setting of stochastic Petri nets\n\\cite{LiuChiou01,Volovoi04}. Observing that a Markovian Petri net can\nalso be interpreted as an interacting particle system, we believe that\nthe model studied here is versatile enough to be used in practical\napplications.\n\nThe paper is organized as follows. Some basic facts about\ninteracting particle systems are first recalled in section\n\\ref{basic}. They are essentially those of sections I.3 and I.4 of\n\\cite{Liggett}, summarized here for sake of completeness, and in\norder to fix notations. The covariance inequality for finite range\ninteractions and local functions will be given in section\n\\ref{covar}. Our main result, Theorem \\ref{cltips2}, will be stated\nin section \\ref{clt}. Some examples of transitive graphs are\nproposed in section \\ref{graphs}. The application to hitting times\nand their reliability interpretation is the object of section\n\\ref{hitting}. In the proof of Theorem \\ref{cltips2}, we need a\nspatial CLT for an interacting particle system at fixed time, i.e. a\nrandom field. We thought interesting to state it independently in\nsection \\ref{cltfields}: Proposition \\ref{pro2r} is in the same vein\nas the one proved by Bolthausen \\cite{Bolthausen82} on\n$\\mathbb{Z}^d$, but it uses a somewhat different technique. All\nproofs are postponed to section \\ref{proofs}.\n\\section{Main notations and assumptions}\n\\label{basic}\nIn order to fix notations, we briefly recall the basic\nconstruction of general interacting particle systems, described in\nsections I.3 and I.4 of Liggett's book \\cite{Liggett}.\n\nLet $S$ be a countable set of sites, $W$ a finite set of states,\nand ${\\cal X}=W^S$ the set of configurations, endowed with its\nproduct topology, that makes it a compact set. One defines a\nFeller process on ${\\cal X}$ by specifying the local transition\nrates: to a configuration $\\eta$ and a finite set of sites $T$ is\nassociated a nonnegative measure $c_T(\\eta,\\cdot)$ on $W^T$.\nLoosely speaking, we want the configuration to change on $T$ after\nan exponential time with parameter\n$$\nc_{T,\\eta} = \\sum_{\\zeta\\in W^T} c_T(\\eta,\\zeta).\n$$\nAfter that time, the configuration becomes equal to $\\zeta$ on\n$T$, with probability $c_T(\\eta,\\zeta)\/c_{T,\\eta} $. Let\n$\\eta^\\zeta$ denote the new configuration, which is equal to\n$\\zeta$ on $T$, and to $\\eta$ outside $T$. The infinitesimal\ngenerator should be:\n\\begin{equation}\n\\label{defgenerateur} \\Omega f(\\eta) = \\sum_{T\\subset S}\n\\sum_{\\zeta\\in W^T} c_T(\\eta,\\zeta)(f(\\eta^\\zeta)-f(\\eta)).\n\\end{equation}\nFor $\\Omega$ to generate a Feller semigroup acting on continuous\nfunctions from $X$ into $\\mathbb{R}$, some hypotheses have to be imposed\non the transition rates $c_T(\\eta,\\cdot)$.\n\nThe first condition is that the mapping $\\eta\\mapsto\nc_T(\\eta,\\cdot)$ should be continuous (and thus bounded, since\n${\\cal X}$ is compact). Let us denote by $c_T$ its supremum norm.\n$$\nc_T = \\sup_{\\eta\\in X} \\,  c_{T,\\eta}.\n$$\nIt is the maximal rate of change of a configuration on $T$. One\nessential hypothesis is that the maximal rate of change of a\nconfiguration at one given site is bounded.\n\\begin{equation}\n\\label{hyp0} B = \\sup_{x\\in \\,S}\\ \\sum_{T\\ni\\, x} c_T <\\infty.\n\\end{equation}\nIf $f$ is a continuous function on ${\\cal X}$, one defines\n$\\Delta_f(x)$ as the degree of dependence of $f$ on $x$:\n$$\n\\Delta_f(x) = \\sup\\{\\,|f(\\eta)-f(\\zeta)|\\,,\\;\\eta,\\zeta\\in X\n\\mbox{ and } \\eta(y) = \\zeta(y)\\; \\forall\\, y\\neq x\\,\\}.\n$$\nSince $f$ is continuous, $\\Delta_f(x)$ tends to $0$ as $x$ tends to\ninfinity, and $f$ is said to be {\\it smooth} if $\\Delta_f$ is\nsummable:\n$$\n\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  f\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  = \\sum_{x\\,\\in\\, S} \\Delta_f(x) <\\infty .\n$$\nIt can be proved that if $f$ is smooth, then $\\Omega f$ defined by\n(\\ref{defgenerateur}) is indeed a continuous function on ${\\cal\nX}$ and moreover:\n$$\n\\|\\Omega f\\| \\leq B \\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  f\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  .\n$$\nWe also need to control the dependence of the transition rates on\nthe configuration at other sites. If $y\\in S$ is a site, and\n$T\\subset S$ is a finite set of sites, one defines\n\n$$\nc_T(y) = \\sup \\{ \\, \\| c_T(\\eta_1,\\,\\cdot\\,) -\nc_T(\\eta_2,\\,\\cdot\\,)\\|_{tv}\\,,\\; \\eta_1(z)=\\eta_2(z)\\;\\forall\\,\nz\\neq y\\,\\} ,\n$$\nwhere $\\|\\,\\cdot\\,\\|_{tv}$ is the total variation norm:\n$$\n\\| c_T(\\eta_1,\\,\\cdot\\,) - c_T(\\eta_2,\\,\\cdot\\,)\\|_{tv} =\n\\frac{1}{2} \\sum_{\\zeta \\in W^T} | c_T(\\eta_1,\\zeta) -\nc_T(\\eta_2,\\zeta)|.\n$$\nIf $x$ and $y$ are two sites such that $x\\neq y$, the {\\it influence} of\n$y$ on $x$ is defined as:\n$$\n\\gamma(x,y) = \\sum_{T\\,\\ni\\, x} c_T(y).\n$$\nWe will set  $\\gamma(x,x)=0$ for all $x$. The influences\n$\\gamma(x,y)$ are assumed to be summable:\n\\begin{equation}\n\\label{hypl1} M = \\sup_{x\\in\\, S}\\ \\sum_{y\\in\\, S} \\gamma(x,y) <\n\\infty.\n\\end{equation}\nUnder both hypotheses (\\ref{hyp0}) and (\\ref{hypl1}), it can be\nproved that the closure of $\\Omega$ generates a Feller semigroup\n$\\{ S_t\\,,\\,t\\geq 0\\}$ (Theorem 3.9 p.~27 of \\cite{Liggett}). A\ngeneric process with semigroup $\\{S_t\\,,\\,t\\geq 0\\}$ will be\ndenoted by $\\{ \\eta_t\\,,\\,t\\geq 0\\}$. Expectations relative to its\ndistribution, starting from $\\eta_0=\\eta$ will be denoted by\n$\\mathbb{E}_\\eta$. For each continuous function $f$, one has:\n$$\nS_t f(\\eta) = \\mathbb{E}_\\eta[ f(\\eta_t) ] =\n\\mathbb{E}[f(\\eta_t)\\,|\\,\\eta_0=\\eta].\n$$\nAssume now that $W$ is ordered, (say $W=\\{1,\\ldots,n\\}$). Let\n${\\cal M}$ denote the class of all continuous functions on $X$\nwhich are monotone in the sense that $f(\\eta)\\leq f(\\xi)$ whenever\n$\\eta\\leq \\xi$. As it was noticed by Liggett (1985) it is\nessential to take advantage of monotonicity in order to prove\nlimit theorems for particle systems.  The following theorems\ndiscuss a number of ideas related to monotonicity.\n\\begin{theo}[Theorem 2.2 Liggett, (1985)]\\label{thm1}\nSuppose $\\eta_t$ is a Feller process on $X$ with semigroup $S(t)$.\nThe following statement are equivalent :\n\\begin{enumerate}\n\\item[(a)] $f\\in {\\cal M}$ implies $S(t)f\\in {\\cal M}$, for all\n$t\\geq 0$\n\\item[(b)] $\\mu_1\\leq \\mu_2$ implies $\\mu_1S(t)\\leq \\mu_2S(t)$ for\nall $t\\geq 0$.\n\\end{enumerate}\nRecall that $\\mu_1\\leq \\mu_2$ provided that $\\int fd\\mu_1\\leq \\int\nfd\\mu_2$ for any $f\\in {\\cal M}$.\n\\end{theo}\n\\begin{defi}\nA Feller process is said to be monotone (or attractive) if the\nequivalent conditions of Theorem \\ref{thm1} are satisfied.\n\\end{defi}\n\\begin{theo}[Theorem 2.14 Liggett, (1985)]\\label{thm2}\nSuppose that $S(t)$ and $\\Omega$ are respectively the semigroup\nand the generator of a {\\bf monotone} Feller process on $X$.\nAssume further that $\\Omega$ is a {\\bf{bounded}} operator. Then\nthe following two statements are equivalent:\n\\begin{enumerate}\n\\item[(a)] $\\Omega fg\\geq f\\Omega g+ g\\Omega f$, for all $f$,\n$g\\in {\\cal M}$\n\\item[(b)] $\\mu S(t)$ has positive correlations whenever $\\mu$\ndoes.\n\\end{enumerate}\nRecall that $\\mu$ has positive correlation if $\\int fgd\\mu\\geq\n\\left(\\int fd\\mu\\right)\\left(\\int gd\\mu\\right)$ for any $f,g\\in\n\\,{\\cal M}$.\n\\end{theo}\nThe following corollary gives conditions under which the positive\ncorrelation property continue to hold at later times if it holds\ninitially.\n\\begin{coro}\\label{corligg}[Corollary 2.21 Liggett, (1985)]\nSuppose that the assumptions of Theorem \\ref{thm2} are satisfied\nand that the equivalent conditions of Theorem \\ref{thm2} hold. Let\n$\\eta_t$ be the corresponding process, where the distribution of\n$\\eta_0$ has positive correlations. Then for $t_1<t_2<\\cdots<t_n$\nthe joint distribution of $(\\eta_{t_1},\\cdots,\\eta_{t_n})$, which\nis a probability measure on $X^n$, has positive correlations.\n\\end{coro}\n\\section{Covariance inequality}\n\\label{covar}\nThis section is devoted to the covariance of $f(\\eta_s)$\nand $g(\\eta_t)$\nfor a finite\nrange interacting particle system when the underlying graph\nstructure has bounded degree.\nProposition \\ref{propinegcovfini} shows that if $f$ and $g$\nare mainly located on two finite sets $R_1$ and $R_2$, then the\ncovariance of $f$ and $g$ decays exponentially in the distance between\n$R_1$ and $R_2$.\n\nFrom now on, we assume that the set of sites $S$ is endowed with\nan undirected graph structure, and we denote by $d$ the natural\ndistance on the graph. We will assume not only that the graph is\nlocally finite, but also that the degree of each vertex is\nuniformly bounded.\n$$\n\\forall x\\in S\\;,\\quad |\\{y\\in S\\,,\\; d(x,y)=1\\}|\\leq r\\;,\n$$\nwhere $|\\,\\cdot\\,|$ denotes the cardinality of a finite set. Thus\nthe size of the sphere or ball with center $x$ and radius $n$ is\nuniformly bounded in $x$, and increases at most geometrically in\n$n$.\n$$\n|\\{y\\in S\\,,\\; d(x,y)=n\\}|\\leq \\frac{r}{r-1}(r\\!-\\!1)^n\n\\quad\\mbox{and}\\quad |\\{y\\in S\\,,\\; d(x,y)\\leq n\\}|\\leq\n\\frac{r}{r-2}(r\\!-\\!1)^n.\n$$\nLet $R$ be a finite subset of $S$. We shall use the following\nupper bounds for the number of vertices at distance $n$, or at\nmost $n$ from $R$.\n\\begin{equation}\n\\label{borneboule} |\\{x\\in S\\,,\\; d(x,R)=n\\}|\\leq |\\{y\\in S\\,,\\;\nd(x,R)\\leq n\\}|\\leq 2|R|e^{n\\rho}\\;,\n\\end{equation}\nwith $\\rho=\\log(r-1)$.\n\nIn the case of an amenable graph (e.g. a lattice on $\\mathbb{Z}^d$), the\nball sizes have a subexponential growth. Therefore, for all\n$\\varepsilon>0$, there exists $c$ such that~:\n$$\n|\\{x\\in S\\,,\\; d(x,R)=n\\}|\\leq |\\{y\\in S\\,,\\; d(x,R)\\leq n\\}|\\leq\nce^{n\\varepsilon}.\n$$\nWhat follows is written in the general case, using\n(\\ref{borneboule}). It applies to the amenable case replacing\n$\\rho$ by $\\varepsilon$, for any $\\varepsilon>0$.\n\nWe are going to deal with smooth functions, depending weakly on\ncoordinates away from a fixed finite set $R$. Indeed, it is not\nsufficient to consider functions depending only on coordinates in\n$R$, because if $f$ is such a function, then for any $t>0$, $S_tf$\nmay depend on all coordinates.\n\\begin{defi}\n\\label{defmainlylocated} Let $f$ be a function from $S$ into\n$\\mathbb{R}$, and $R$ be a finite subset of $S$. The function $f$ is said\nto be {\\rm mainly located on} $R$ if there exists two constants\n$\\alpha$ and $\\beta>\\rho$ such that $\\alpha>0$, $\\beta>\\rho$ and\nfor all $x\\in \\mathbb{R}$:\n\\begin{equation}\n\\label{mainlylocated} \\Delta_f(x) \\leq \\alpha e^{-\\beta d(x,R)}.\n\\end{equation}\n\\end{defi}\nSince $\\beta>\\rho$, the sum  $\\sum_x \\Delta_f(x)$ is finite. Therefore a\nfunction mainly located on a finite set is necessarily smooth.\n\nThe system we are considering will be supposed to have finite\nrange interactions in the following sense (cf. Definition 4.17,\np.~39 of \\cite{Liggett}).\n\\begin{defi}\n\\label{defetenduefinie} A particle system defined by the rates\n$c_T(\\eta,\\cdot)$ is said to have {\\it finite range interactions}\nif there exists $k>0$ such that if $d(x,y)>k$:\n\\begin{enumerate}\n\\item $c_T = 0$ for all $T$ containing both $x$ and $y$\\;, \\item\n$\\gamma(x,y)=0$.\n\\end{enumerate}\n\\end{defi}\nThe first condition imposes that two coordinates cannot\nsimultaneously change if their distance is larger than $k$. The\nsecond one says that the influence of a site on the transition\nrates of another site cannot be felt beyond distance $k$.\n\nUnder these conditions, we prove the following covariance\ninequality.\n\n\\begin{prop}\n\\label{propinegcovfini} Assume (\\ref{hyp0}) and (\\ref{hypl1}).\nAssume moreover that the process is of finite range. Let $R_1$ and\n$R_2$ be two finite subsets of $S$.\n Let $\\beta$\nbe a constant such that $\\beta>\\rho$. Let $f$ and $g$ be two\nfunctions mainly located on $R_1$ and $R_2$, in the sense that\nthere exist positive constants $\\kappa_f,\\kappa_g$ such that,\n$$\n\\Delta_f(x) \\leq \\kappa_f e^{-\\beta d(x,R_1)} \\quad\\mbox{and}\\quad\n\\Delta_g(x) \\leq \\kappa_g e^{-\\beta d(x,R_2)}.\n$$\nThen for all positive reals $s,t$,\n\\begin{equation}\n\\label{inegcovariancefini} \\sup_{\\eta\\in X}\n\\Big|\\cov_\\eta(f(\\eta_s),g(\\eta_t))\\Big| \\leq\nC\\kappa_f\\kappa_g(|R_1|\\wedge|R_2|)e^{D(t+s)}e^{-({\\beta-\\rho})d(R_1,R_2)}\\;,\n\\end{equation}\nwhere\n$$\nD = 2Me^{(\\beta+\\rho) k} \\quad\\mbox{and}\\quad C = \\frac{2\nBe^{\\beta k}}{D}\\left(1+\\frac{e^{\\rho\nk}}{1-e^{-\\beta+\\rho}}\\right).\n$$\n\\end{prop}\n\\paragraph{Remark.} Shashkin \\cite{Shashkin} obtains a similar\ninequality for random fields indexed by $\\mathbb{Z}^d$.\n\n\\\n\nWe now consider a {\\it transitive graph}, such that the group of\nautomorphism acts transitively on $S$ (see chapter 3 of\n\\cite{GodsilRoyle}). Namely we need that\n\\begin{itemize}\n    \\item for any $x$ and $y$ in $S$ there exists $a$ in $Aut(S)$, such that\n    $a(x)=y$.\n    \\item for any $x$ and $y$ in $S$ and any radius $n$, there exists $a$ in $Aut(S)$, such that\n    $a(B(x,n))=B(y,n)$.\n\\end{itemize}\nAny element $a$ of the automorphism group acts on configurations,\nfunctions and measures on ${\\cal X}$ as follows:\n\\begin{itemize}\n\\item {\\it configurations:} $a\\cdot \\eta(x) = \\eta(a^{-1}(x))$, \\item\n{\\it functions:} $a\\cdot f(\\eta) = f(a\\cdot \\eta)$, \\item {\\it\nmeasures:} $\\int f\\, d(a\\cdot \\mu) = \\int(a\\cdot f)\\,d\\mu$.\n\\end{itemize}\nA probability measure $\\mu$ on ${\\cal X}$ is invariant through the\ngroup action if $a\\cdot \\mu = \\mu$ for any automorphism $a$, and\nwe want this to hold for the probability distribution of $\\eta_t$\nat all times $t$. It will be the case if the transition rates are\nalso invariant through the group action. In order to avoid\nconfusions with invariance in the sense of the semigroup\n(Definition 1.7, p.~10 of \\cite{Liggett}), invariance through the\naction of the automorphism group of the graph will be\nsystematically referred to as ``group invariance'' in the sequel.\n\\begin{defi}\n\\label{definvariant} Let $G$ be the automorphism group of the\ngraph. The transition rates $c_T(\\eta,\\cdot)$ are said to be {\\rm\ngroup invariant} if for any $a\\in G$,\n$$\nc_{a(T)}(a\\cdot \\eta,a\\cdot \\zeta) = c_T(\\eta,\\zeta).\n$$\n\\end{defi}\nThis definition extends in an obvious way that of translation\ninvariance on $\\mathbb{Z}^d$-lattices (\\cite{Liggett}, p.~36).\n\\\\\n\\\\\n{\\bf{Remark.}}\n Observe\nthat for rates which are both  finite range and group invariant,\nthe hypotheses (\\ref{hyp0}) and (\\ref{hypl1})\nare trivially satisfied. In that case, it is easy to check that\nthe semi-group $\\{S_t\\,,\\;t\\geq 0\\}$ commutes with the\nautomorphism group. Thus if $\\mu$ is a group invariant measure,\nthen so is $\\mu S_t$ for any $t$ (see \\cite{Liggett}, p.~38). In\nother terms, if the distribution of $\\eta_0$ is group invariant,\nthen that of $\\eta_t$ will remain group invariant at all times.\n\n \\section{Functional CLT}\n\\label{clt} Our functional central limit theorem requires that all\ncoordinates of the interacting particle system $\\{\\eta_t\\,,\\;t\\geq\n0\\}$ are identically distributed.\n\n\\vskip 2mm\\noindent\nLet $(B_n)_{n\\geq 1}$ be an increasing sequence of finite subsets of $S$\nsuch that\n\\begin{equation}\\label{slc}\nS=\\bigcup_{n=1}^{\\infty}B_n,\\qquad \\lim_{n\\rightarrow\n+\\infty}\\frac{|\\partial B_n|}{|B_n|}=0\\;,\n\\end{equation}\nrecall that $|\\;\\cdot\\,|$ denotes the cardinality and $\n\\partial B_n=\\{x\\in B_n\\;,\\,\\exists\\,y\\not\\in B_n,\\, d(x,y)=1\\}\n$.\n\\begin{theo}\n\\label{cltips2} Let $\\mu=\\delta_{\\eta}$ be  a  Dirac measure where\n$\\eta\\in {\\cal X}$ fulfills $\\eta(x)=\\eta(y)$ for any $x,y\\in S$.\nSuppose that the transition rates are group invariant. Suppose\nmoreover that the process is of finite range, monotone and\nfulfilling the requirements of Corollary \\ref{corligg}. Let\n$(B_n)_{n\\geq 1}$ be an increasing sequence of finite subsets of\n$S$ fulfilling (\\ref{slc}). Then the sequence of processes\n$$\n\\left\\{\\frac{N_t^{B_n}-\\mathbb{E}_{\\mu}N_t^{B_n}}{\\sqrt{|B_n|}}\\,,\\;t\\geq\n0\\right\\},\\qquad \\mbox{ for } n=1,2,\\ldots$$ converges in\n${D([0,T])}$ as $n$ tends to infinity, to a centered Gaussian,\nvector valued process\\\\ $(B(t,w))_{t\\ge\\,0,\\,w\\in\\, W}$ with\ncovariance function $\\Gamma$ defined, for $w,\\, w'\\in\\, W$, by\n$$\n\\Gamma_\\mu(s,t)(w,w')=\\sum_{x\\in\\, S}\\cov_{\\mu}\\left(\n\\mathbb{I}_{w}(\\eta_s(x)),\\mathbb{I}_{w'}(\\eta_t(x))\\right).\n$$\n\\end{theo}\n\\paragraph{Remark.} One may wonder wether such results can extend under\nmore general initial distributions. The point is that the covariance\ninequality do not extend simply by integration with respect to\ndeterministic configurations. We are thankful to Pr. Penrose for\nstressing our attention on this important restriction. Monotonicity\nallows to get ride of this restriction.\n\n\\section{Examples of graphs}\n\\label{graphs}\nBesides the classical lattice graphs in $\\mathbb{Z}^d$ and their groups of\ntranslations, which are considered by most authors (see\n\\cite{DurrettLevin94,Liggett,Liggett97}), our setting applies to a\nbroad range of graphs. We propose some simple examples of\nautomorphisms on trees, which\ngive rise to a large variety of non classical situations.\n\\vskip 2mm\nThe simplest example\n corresponds to regular trees defined as\nfollows. Consider the non-commutative free group $S$ with finite\ngenerator set $G$. Impose that each generator $g$ is its own\ninverse ($g^2=1$). Now consider $S$ as a graph, such that $x$ and\n$y$ are connected if and only if there exists $g \\in G$ such that\n$x=yg$. Note that $S$ is a regular tree of degree equal to the\ncardinality $r$ of $G$. The size of spheres is exponential:\n$\\left|\\{y\\,,\\, d(x,y)=n\\}\\right|=r^n$. Now consider the group\naction of $S$ on itself: $x\\cdot y=xy$: this action is transitive\non $S$ (take $a=yx$). \\vskip 2mm From this basic example it is\npossible to get a large class of graphs by adding relations\nbetween generators; for example take the tree of degree $4$,\ndenote by $a$, $b$, $c$, and $d$ the generators, and add the\nrelation $ab=c$. Then, the corresponding graph is a regular tree\nof degree $4$ were nodes are replaced by tetrahedrons. The spheres\ndo not grow at rate $4^n$: $\\left|\\{y\\,,\\,\nd(x,y)=n\\}\\right|=4\\cdot3^{n\/2}$ if $n$ is even and $\\left|\n\\{y\\,,\\, d(x,y)=n\\}\\right|=6\\cdot3^{(n-1)\/2}$ if $n$ is odd.\n\n\\begin{figure}[ht]\n\\vskip 7truecm \\vbox{ \\centerline{ \\psset{unit=1cm}\n\\pspicture(2.5,2.5)(5,-0.5) \\psline(1,7)(2,7.5)(3,7)(2,6.5)(1,7)\n\\psline(1,7)(9,7) \\psline(2,6.5)(2,5.5)\n\\psline(1,5)(2,5.5)(3,5)(2,4.5)(1,5) \\psline(2,7.5)(2,6.5)\n\\psline(3,7)(4,7)\n\\psline[showpoints=true](4,7)(5,7.5)(6,7)(5,6.5)(4,7)\n\\psline(6,7)(7,7)\n\\psline[showpoints=true](7,7)(8,7.5)(9,7)(8,6.5)(7,7)\n\\psline(2,4.5)(2,6) \\psline(1,5)(3,5) \\psline(5,7.5)(5,4.5)\n\\psline[showpoints=true](4,5)(5,5.5)(6,5)(5,4.5)(4,5)\n\\psline(4,5)(6,5) \\psline(8,7.5)(8,6.5) \\rput(1,7){$\\bullet$}\n\\rput(0.75,7.3){$db$} \\psline[linewidth=3pt](1,7)(0,7)\n\\psline[linewidth=3pt](1.09,5)(0,5)\n \\rput(2,7.5){$\\bullet$} \\rput(1.75,7.8){$da$}\n\\psline[linewidth=3pt](2,7.5)(2,8.5)\n \\rput(3,7){$\\bullet$} \\rput(3.2,7.3){$d$}\n\\rput(4.2,7.3){$1$} \\psline[linewidth=3pt](3,7)(4,7)\n\\psline[linewidth=3pt](6,7)(7,7)\n\\psline[linewidth=3pt](2,6.5)(2,5.5)\n\\psline[linewidth=3pt](5,6.5)(5,5.5)\n \\rput(5.2,7.8){$a$}\n\\psline[linewidth=3pt](5,7.5)(5,8.5) \\rput(6.2,7.3){$b$}\n\\rput(2,6.5){$\\bullet$} \\rput(4.75,6.3){$c$} \\rput(1.75,6.3){$dc$}\n\\rput(2,5.5){$\\bullet$} \\rput(2.35,5.8){$dcd$}\n\\rput(2,4.5){$\\bullet$} \\rput(2.41,4.3){$dcdc$}\n\\rput(2,4.5){$\\bullet$} \\rput(1,5){$\\bullet$} \\rput(1,5.3){$dcda$}\n\\rput(3,5){$\\bullet$} \\rput(3.1,5.3){$dcdb$}\n \\psline[linewidth=3pt](2,4.5)(2,3.5)\n \\rput(4.7,5.8){$cd$}\\rput(4.6,4.3){$cdc$}\n\\psline[linewidth=3pt](5,4.5)(5,3.5)\n \\rput(5.999,5.3){$cdb$}\n \\psline[linewidth=3pt](6,5)(7,5)\n\\rput(3.999,5.3){$cda$} \\psline[linewidth=3pt](4,5)(3.75,5)\n\\psline[linewidth=3pt](2.9,5)(3.25,5)\n \\rput(4,7){$\\bullet$} \\rput(7.1,7.3){$bd$}\n\\rput(8.4,7.8){$bda$} \\psline[linewidth=3pt](8,7.5)(8,8.5)\n \\rput(9.1,7.3){$bdb$}\n\\psline[linewidth=3pt](9,7)(10,7)\n \\rput(8.4,6.3){$bdc$}\n \\psline[linewidth=3pt](8,6.5)(8,5.5)\n\\endpspicture\n} } \\vskip -3truecm\n \\caption{Graph structure of the tree with tetrahedron\ncells. The graph consists in a regular tree  of degree 4 (bold\nlines), where nodes have been replaced by tetrahedrons.\nAutomorphisms in this graph correspond to composition of\nautomorphisms exchanging couples of branches of the tree (action\nof generator $a$ for example) and displacements in the subjacent\nregular tree.}\n\\end{figure}\n\n\\section{CLT for hitting times}\\label{hitting}\nIn this section we consider the case where $W$ is ordered, the\nprocess is monotone and satisfies the assumptions in Theorem\n\\ref{cltips2}, the initial condition is fixed and $f$ is an\nincreasing function from $W$ to $\\mathbb{R}$. In the reliability\ninterpretation, $f(w)$ measures a level of degradation for a\ncomponent in state $w$. The total degradation of the system in state\n$\\eta$ will be measured by the sum $\\sum_{x \\in B_n} f(\\eta(x))$. So\nwe shall focus on the process $D^{(n)} = \\{D_t^{(n)}\\,,\\,t\\geq 0\\}$,\nwhere $D_t^{(n)}=D_t^{B_n}$ is the total degradation of the system\nat time $t$ on the set $R=B_n$:\n$$\nD_t^{(n)} = \\sum_{x\\, \\in\\, B_n} f(\\eta_t(x)).\n$$\nIt is natural to consider the instants at which $D_t^{(n)}$\nreaches a prescribed level of degradation. Let $k=(k(n))$ be a\nsequence of real numbers. Our main object is the {\\it failure\ntime} $T_n$, defined as:\n$$\nT_n = \\inf\\{t\\geq 0\\,,\\; D_t^{(n)}\\geq k(n)\\}.\n$$\nIn the particular case where\n$W=\\{\\mbox{working},\\,\\mbox{failed}\\}$ (binary components), and\n$f$ is the indicator of a failed component, then $D_t^{(n)}$\nsimply counts the number of failed components at time $t$, and our\nsystem is a so-called ``$k$-out-of-$n$'' system\n\\cite{BarlowProschan}.\n\nLet $w_0$ be a particular state (in the reliability $w_0$ could be\nthe ``perfect state'' of an undergrade component). Let $\\eta$ be the\nconstant configuration where all components are in the perfect state\n$w_0$, for all $ x\\in S$. Our process starts from that configuration\n$\\eta$, which is obviously group invariant. We shall denote by\n$m(t)$ (respectively, $v(t)$) the expectation (resp., the variance)\nof the degradation at time $t$ for one component.\n$$\nm(t) = \\mathbb{E}[f(\\eta_t(x))\\,|\\,\\eta_0=\\eta]\\;,\\qquad\nv(t)=\\lim_{n\\to\\infty}\\frac{\\Var D_t^{(n)}}{|B_n|}.\n$$\nThese expressions do not depend on $x\\in S$, due to group\ninvariance.\n\nThe average degradation $D_t^{(n)}\/|B_n|$ converges in probability\nto its expectation $m(t)$. We shall assume that $m(t)$ is strictly\nincreasing on the interval $[0,\\tau]$, with $0<\\tau\\leq+\\infty$ (the\ndegradation starting from the perfect state increases on average).\nMathematically, one can assume that the states are ranked in\nincreasing order, the perfect state being the lowest. This yields a\npartial order on configurations. If the rates are such that the\ninteracting particle system is monotone (see \\cite{Liggett}), then\nthe average degradation increases. In the reliability\ninterpretation, assuming monotonicity is quite natural: it amounts\nto saying that the rate at which a given component jumps to a more\ndegraded state is higher if its surroundings are more degraded.\n\nWe consider a ``mean degradation level'' $\\alpha$, such\nthat $m(0)<\\alpha<m(\\tau)$. Assume the threshold $k(n)$ is such\nthat:\n$$\nk(n) = \\alpha |B_n| + o(\\sqrt{ |B_n|}).\n$$\nTheorem \\ref{cltips2} shows that the degradation process $D^{(n)}$\nshould remain at distance $O(\\sqrt{ |B_n|})$ from the deterministic\nfunction $\\left|B_n\\right| m$. Therefore it is natural to expect\nthat $T_n$ is at distance $O(1\/\\sqrt{|B_n|})$ from the instant\n$t_\\alpha$ at which $m(t)$ crosses $\\alpha$:\n$$\nt_\\alpha = \\inf\\{t,\\ m(t)=\\alpha\\}.\n$$\n\\begin{theo}\n\\label{clthitting} Under the above hypotheses,\n$$\n\\sqrt{ |B_n|}\\left(T_n-t_\\alpha\\right) \\xrightarrow[n\\rightarrow\n+\\infty]{{\\cal{L}}} {\\cal N}(0,\\sigma^2_\\alpha),\n$$\nwith:\n$$\n\\sigma_\\alpha^2 = \\frac{v(t_\\alpha)}{(m'(t_\\alpha))^2}.\n$$\n\\end{theo}\n\\section{CLT for weakly dependent random fields}\n\\label{cltfields}\nAs in section \\ref{clt}, we consider a transitive graph ${\\cal\nG}=(S,E)$, where $S$ is the set of vertices and $E\\subset\n\\Big\\{\\{x,y\\},\\, x,y\\in\\,S,\\, x\\neq y\\Big\\}$ the set of edges. For\na transitive graph, the degree $r$ of each vertex is constant (cf.\nLemma 1.3.1 in Godsil and Royle \\cite{GodsilRoyle}).\n\nFor any $x$ in $S$ and for any positive integer $n$, we denote by\n$B(x,n)$ the open ball of $S$ centered at $x$, with radius $n$:\n$$\nB(x,n)=\\{y\\in S,\\,\\,\\, d(x,y)< n\\}.\n$$\nThe cardinality of the ball $B(x,n)$ is constant in $x$ and bounded as follows.\n\\begin{equation}\\label{car}\n\\sup_{x\\in S}|B(x,n)|\\leq 2 r^n=2e^{n\\rho}=: \\kappa_n,\n\\end{equation}\nwhere $\\rho= \\ln(\\max(r,4)-1)$: compare with\nformula (\\ref{borneboule}).\n\nLet $Y=(Y_x)_{x\\in S}$ be a real valued random field. We will\nmeasure covariances between coordinates of $Y$ on two distant sets\n$R_1$ and $R_2$ through Lipschitz functions (see\n\\cite{DoukhanLouhichi}). A Lipschitz function is a real valued\nfunctions $f$ defined on $\\mathbb{R}^n$ for some positive integer $n$,\nfor which\n$$\n\\Lip f :=\\sup_{x\\neq y}\\frac{\\textstyle\n\\left|f(x)-f(y)\\right|}{\\textstyle\\sum_{i=1}^n|x_i-y_i|}<\\infty.\n$$\nWe will assume the the random field $Y$ satisfies the following covariance inequality:\nfor any positive real $\\delta$, for any\ndisjoint finite subsets  $R_1$ and $R_2$  of $S$ and\n for any Lipschitz functions\n$f$ and $g$\n  defined respectively on\n$\\mathbb{R}^{|R_1|}$ and $\\mathbb{R}^{|R_2|}$, there exists a positive constant\n$C_\\delta$ (not depending on $f$ $g$, $R_1$ and $R_2$)\n such that\n\\begin{equation}\\label{slcov}\n\\left|\\Cov\\left(f(Y_x,\\, x\\in R_1), g(Y_x,\\, x\\in\nR_2\\right)\\right|\\leq C_\\delta \\,\\Lip f\\,\\Lip\ng\\,\\left(|R_1|\\wedge|R_2|\\right)\\exp\\left(-\\delta\nd(R_1,R_2)\\right).\n\\end{equation}\n\\vskip 3mm\nFor any finite subset $R$ of $S$, let $Z(R)=\\sum_{x\\in R}Y_x$. Let\n$(B_n)_{n\\in\\mathbb{N}}$ be an increasing sequence of finite subsets of\n$S$ such that $|B_n|$ goes to infinity with $n$. Our purpose in\nthis section is to establish a central limit theorem for $Z(B_n)$,\nsuitably normalized. We suppose that\n $(Y_x)_{x\\in S}$ is a  weakly dependent\nrandom field according to the covariance inequality (\\ref{slcov}).\n\\\\\nIn Proposition \\ref{pro2r} below we prove that, as in the\nindependent setting, a central limit theorem holds as soon as\n$\\Var Z(B_n)$ behaves, as $n$ goes to infinity, like $|B_n|$ (cf.\nCondition (\\ref{limff}) below). So the purpose of Proposition\n\\ref{slpro2} is to study the behavior of $\\Var Z(B_n)$. We prove\nthat the limit (\\ref{limff}) holds under two additional\nconditions. The first one supposes that the cardinality of\n$\\partial B_n$ is asymptotically negligible compared to $|B_n|$\n(cf. Condition (\\ref{slc}) in section \\ref{clt}); the second\ncondition supposes an invariance by the automorphisms of the group\n${\\cal G}$, of the joint distribution $(Y_x,Y_y)$ for any two\nvertices $x$ and $y$. More precisely we need to have Condition\n(\\ref{slstationarity}) below,\n\\begin{equation}\\label{slstationarity}\n\\Cov(Y_x,Y_y)=\\Cov(Y_{a(x)},Y_{a(y)}),\n\\end{equation}\nfor any automorphism $a$ of ${\\cal G}$.\n\\vskip 2mm\nIn order to prove Proposition \\ref{pro2r}, we shall use some\nestimations of Bolthausen \\cite{Bolthausen82} that yield a central\nlimit theorem for stationary random fields on $\\mathbb{Z}^d$ under mixing\nconditions. Recall that the mixing coefficients used there are\ndefined as follows, noting by ${\\cal A}_{R}$ the\n$\\sigma$-algebra generated by $(Y_x,\\, x\\in R)$,\n$$\n\\alpha_{k,l}(n)=\\sup\\{|\\mathbb{P}(A_1\\cap A_2)-\\mathbb{P}(A_1)\\mathbb{P}(A_2)|, \\,\nA_i\\in {\\cal A}_{R_i},\\, |R_1|\\leq k,\\,\n|R_2|\\leq l,\\, d(R_1,R_2)\\geq n\\},\n$$\nfor $n\\in \\mathbb{N} $ and $k,l\\in \\mathbb{N} \\cup {\\infty}$,\n$$\n\\rho(n)=\\sup\\{|\\cov(Z_1,Z_2)|,\\, Z_i\\in L_2({\\cal\nA}_{\\{\\rho_i\\}}),\\, \\|Z_i\\|_2\\leq 1,\\, d(\\rho_1,\\rho_2)\\geq n\\}.\n$$\nUnder suitable decay of $(\\alpha_{k,l}(n))_n$ or of $(\\rho(n))_n$,\nBolthausen \\cite{Bolthausen82} proved a central limit theorem for stationary\nrandom fields on $\\mathbb{Z}^d$, using an idea of Stein. In our case,\ninstead of using those mixing coefficients, we describe the\ndependence structure of the random fields $(Y_x)_{x\\in S}$  in\nterms of the gap between two Lipschitz transformations of two\ndisjoint blocks (the covariance inequality (\\ref{slcov})\nabove). Those manners of describing the dependence of random\nfields are quite different. As one may expect, the techniques of proof\nwill be different as well (see section \\ref{proofs}).\n\\begin{prop}\\label{pro2r}\n Let ${\\cal G}=(S,E)$ be a transitive graph. Let\n$(B_n)_{n\\in\\mathbb{N}}$ be an increasing sequence of finite subsets of\n$S$ such that $|B_n|$ goes to infinity with $n$. Let $(Y_x)_{x\\in\nS}$ be a real valued random field, satisfying (\\ref{slcov}).\nSuppose that, for any $x\\in S$, $\\mathbb{E} Y_x=0$ and $\\sup_{x\\in\nS}\\|Y_x\\|_{\\infty}< \\infty$.  If, there exists a finite real\nnumber $\\sigma^2$ such that\n\\begin{equation}\\label{limff}\n\\lim_{n\\rightarrow \\infty}\\frac{\\Var Z(B_n)}{|B_n|}=\\sigma^2,\n\\end{equation}\n then the quantity\n$\\frac{\\textstyle Z(B_n)}{\\textstyle \\sqrt{|B_n|}}$ converges in\ndistribution to a centered normal law with variance $\\sigma^2$.\n\\end{prop}\n\\begin{prop}\\label{slpro2}\nLet ${\\cal G}=(S,E)$ be a transitive graph. Let $(Y_x)_{x\\in S}$\nbe a centered real valued random field, with finite variance.\nSuppose that the conditions (\\ref{slcov}) and\n(\\ref{slstationarity}) are satisfied.\n Let $(B_n)_n$ be a sequence of finite and\nincreasing sets of $S$ fulfilling (\\ref{slc}). Then $$\\sum_{z\\in\nS}|\\Cov(Y_0,Y_{z})|<\\infty\\ \\ \\ {\\mbox{and}}\\ \\ \\\n\\lim_{n\\rightarrow \\infty} \\frac{1}{|B_n|}\\Var Z(B_n)=\\sum_{z\\in\nS}\\Cov(Y_0,Y_{z}).$$\n\\end{prop}\n\n\\section{Proofs}\n\\label{proofs}\n\\subsection{Proof of Proposition \\ref{propinegcovfini}}\nLet $\\Gamma$ denote the matrix $(\\gamma(x,y))_{x,y\\in S}$, and let\nit operate on the right on the space of summable series\n$\\ell_1(S)$ indexed by the denumerable set $S$:\n$$\nu = (u(x))_{x\\in S} \\mapsto \\Gamma u= (\\Gamma u(y))_{y\\in S}\\,,\n$$\nwith~:\n$$\n\\Gamma u(y) = \\sum_{x\\in S} u(x)\\, \\gamma(x,y).\n$$\n(We have followed Liggett's \\cite{Liggett} choice of denoting by $\\Gamma\nu$ the product of $u$ by $\\Gamma$ on the right.)\nThanks to hypothesis (\\ref{hypl1}), this defines a bounded\noperator of $\\ell_1(S)$, with norm $M$. Thus for all $t\\geq 0$,\nthe exponential of $t\\Gamma$, is well defined, and gives another\nbounded operator of $\\ell_1(S)$:\n$$\n\\exp(t\\Gamma)u = \\sum_{n=0}^\\infty \\frac{t^n \\Gamma^n u}{n!}.\n$$\nIf $f$ is a smooth function, then $\\Delta_f =(\\Delta_f(x))_{x\\in\nS}$, is an element of $\\ell_1(S)$. Applying $\\exp(t\\Gamma)$ to\n$\\Delta f$ provides a control on $S_tf$ as shows the following\nproposition (cf. Theorem 3.9 of  \\cite{Liggett}).\n\\begin{prop}\n\\label{propsmooth} Assume {\\rm (\\ref{hyp0})} and {\\rm (\\ref{hypl1})}. Let $f$\nbe a smooth function. Then,\n\\begin{equation}\n\\label{ineg1} \\Delta_{S_tf} \\leq \\exp(t\\Gamma)\\Delta_f .\n\\end{equation}\n\\end{prop}\nIt follows immediately that if $f$ is a smooth function then\n$S_tf$ is also smooth and:\n$$\n\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  S_t f\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  \\leq e^{tM}\\,\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert  f\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert \\;,\n$$\nbecause the norm of $\\exp(t \\Gamma)$ operating on $\\ell_1(S)$ is\n$e^{tM}$. \\\\\nA similar bound for covariances will be our starting point (cf.\nProposition 4.4, p.~34 of \\cite{Liggett}).\n\\begin{prop}\n\\label{inegcovbasique} Assume {\\rm (\\ref{hyp0})} and {\\rm (\\ref{hypl1})}. Then\nfor any smooth functions $f$ and $g$ and for all $t\\geq 0$, one\nhas,\n\\begin{equation}\n\\label{inegcovariance} \\| S_tfg - (S_tf)(S_tg)\\| \\leq \\sum_{y,z\\in\nS} \\left(\\sum_{T\\ni y,z} c_T\\right) \\int_0^t (\\exp(\\tau\n\\Gamma)\\Delta_f)(y) (\\exp(\\tau \\Gamma)\\Delta_g)(z)\\,d\\tau.\n\\end{equation}\n\\end{prop}\nIn terms of the process $\\{\\eta_t\\,,\\,t\\geq 0\\}$, the left member\nof (\\ref{inegcovariance}) is the uniform bound for the covariance\nbetween $f(\\eta_t)$ and $g(\\eta_t)$.\n$$\n\\| S_tfg - (S_tf)(S_tg)\\| = \\sup_{\\eta\\in X} \\Big|\n\\mathbb{E}_\\eta[f(\\eta_t)g(\\eta_t)] -\n\\mathbb{E}_\\eta[f(\\eta_t)]\\mathbb{E}_\\eta[g(\\eta_t)] \\Big|.\n$$\nA slight modification of (\\ref{inegcovariance}) gives a bound on\nthe covariance of $f(\\eta_s)$ with $g(\\eta_t)$, for $0\\leq s\\leq\nt$. From now on, we shall denote by $\\cov_\\eta$ covariances\nrelative to the distribution of $\\{\\eta_t\\,,\\;t\\geq 0\\}$, starting\nat $\\eta_0=\\eta$:\n$$\n\\cov_\\eta(f(\\eta_s),g(\\eta_t)) = \\mathbb{E}_\\eta[f(\\eta_s)g(\\eta_t)] -\n\\mathbb{E}_\\eta[f(\\eta_s)]\\mathbb{E}_\\eta[g(\\eta_t)].\n$$\n\\begin{coro}\n\\label{corocovariancest} Assume {\\rm (\\ref{hyp0})} and {\\rm (\\ref{hypl1})}.\nLet $f$ and $g$ be two smooth functions. Then for all $s$ and $t$\nsuch that $0\\leq s\\leq t$,\n\\begin{equation}\n\\label{inegcovariancest} \\sup_{\\eta\\in X}\n\\Big|\\cov_\\eta(f(\\eta_s),g(\\eta_t))\\Big| \\leq \\sum_{y,z\\in S}\n\\left(\\sum_{T\\ni y,z} c_T\\right) \\int_0^s\n(\\exp(\\tau\\Gamma)\\Delta_f)(y)(\\exp(\\tau\\Gamma)\\Delta_{S_{t-s}g})(z)\\,d\\tau.\n\\end{equation}\n\\end{coro}\n{\\bf{Proof of Corollary \\ref{corocovariancest}.}}  We have, using\nthe semigroup property,\n$$\n\\mathbb{E}_\\eta[ f(\\eta_s) g(\\eta_t)] =\n\\mathbb{E}_\\eta[f(\\eta_s)\\mathbb{E}[g(\\eta_t)\\,|\\,\\eta_s]] = \\mathbb{E}_\\eta[f(\\eta_s)\nS_{t-s}g(\\eta_s)] = S_s(fS_{t-s}g)(\\eta).\n$$\nAlso,\n$$\n\\mathbb{E}_\\eta[g(\\eta_t)] = S_tg(\\eta) = S_s(S_{t-s}g)(\\eta).\n$$\nApplying (\\ref{inegcovariance}) at time $s$ to $f$ and $S_{t-s}g$,\nyields the result.\\,\\, $\\Box$\n\\vskip 3mm\n In order to apply\n(\\ref{inegcovariancest}) to functions mainly located on finite\nsets, we shall need to control the effect of $\\exp(t\\Gamma)$ on a\nsequence $(\\Delta_f(x))$ satisfying (\\ref{mainlylocated}). This\nwill be done through the following technical lemma.\n\\begin{lemm}\n\\label{lemmetechniquegamma} Suppose that the process is of finite\nrange. Let $R$ be a finite\nset of sites. Let $u=(u(x))_{x\\in S}$ be an element of\n$\\ell_1(S)$. If for all $x\\in S$, $u(x)\\leq \\alpha e^{-\\beta\nd(x,R)}$, with $\\alpha>0$ and $\\beta>\\rho$, then for all $y\\in S$,\n$$\n|(\\exp(t\\Gamma)u)(y)| \\leq \\alpha\\exp(2tMe^{(\\beta+\\rho)\nk})\\,e^{-\\beta d(y,R)}.\n$$\n\\end{lemm}\nThis lemma, together with Proposition \\ref{propsmooth}, justifies\nDefinition \\ref{defmainlylocated}. Indeed, if $f$ is mainly\nlocated on $R$, then by (\\ref{ineg1}) and Lemma\n\\ref{lemmetechniquegamma}, $S_tf$ is also mainly located on $R$,\nand the rate of exponential decay $\\beta$ is the same for both\nfunctions.\n\\\\\n{\\bf{Proof of Lemma \\ref{lemmetechniquegamma}.}} Recall that\n$$\n\\Gamma u(y) = \\sum_{x\\in S} u(x)\\gamma(x,y).\n$$\nObserve that if $\\gamma(x,y)>0$, then the distance from $x$ to $y$\nmust be at most $k$ and thus the distance from $x$ to $R$ is at\nleast $d(y,R)-k$. If $u(x)\\leq \\alpha e^{-\\beta d(x,R)}$ then:\n$$\n\\Gamma u(y) \\leq 2\\alpha e^{\\rho k}e^{-\\beta(d(y,R)-k)}M = 2\\alpha\ne^{(\\beta +\\rho)k} M e^{-\\beta d(y,R)}.\n$$\nHence by induction,\n$$\n\\Gamma^n u(y) \\leq \\alpha 2^ne^{(\\beta+\\rho) kn} M^{n} e^{-\\beta\nd(y,R)}.\n$$\nThe result follows immediately.\\,\\, $\\Box$\n\\vskip 3mm\nTogether with (\\ref{inegcovariancest}), Lemma\n\\ref{lemmetechniquegamma} will be the key ingredient in the proof\nof our covariance inequality.\n\\\\\n{\\bf{End of the proof of Proposition \\ref{propinegcovfini}.}}\nBeing mainly located on finite sets, the functions $f$ and $g$ are\nsmooth. By (\\ref{inegcovariancest}), the covariance of $f(\\eta_s)$\nand $g(\\eta_t)$ is bounded by $M(s,t)$ with:\n$$\nM(s,t)= \\sum_{y,z\\in\\, S} \\left(\\sum_{T\\ni\\, y,z} c_T\\right)\n\\int_0^s\n(\\exp(\\tau\\Gamma)\\Delta_f)(y)(\\exp(\\tau\\Gamma)\\Delta_{S_{t-s}g})(z)\\,d\\tau.\n$$\nLet us apply Lemma \\ref{lemmetechniquegamma} to $\\Delta_f$ and\n$\\Delta_{S_{t-s}g}$.\n\\begin{equation}\n\\label{maj1} (\\exp(\\tau\\Gamma)\\Delta_f)(y) \\leq \\kappa_f\\exp(\\tau\nMe^{(\\beta+\\rho) k})\n  e^{-\\beta d(y,R_1)}\n= \\kappa_f e^{D\\tau}e^{-\\beta d(y,R_1)}.\n\\end{equation}\nThe last bound, together with (\\ref{ineg1}), gives\n$$\n\\Delta_{S_{t-s}g}(x)\\leq (\\exp((t-s)\\Gamma)\\Delta_g)(x) \\leq\n\\kappa_g e^{D(t-s)}e^{-\\beta d(x,R_2)}.\n$$\nTherefore~:\n\\begin{equation}\n\\label{maj2} (\\exp(\\tau\\Gamma)\\Delta_{S_{t-s}g})(z)\\leq \\kappa_g\ne^{D(\\tau+t-s)} e^{-\\beta d(z,R_2)}.\n\\end{equation}\nInserting the new bounds (\\ref{maj1}) and (\\ref{maj2}) into\n$M(s,t)$, we obtain\n$$\nM(s,t)\\leq \\sum_{y,z\\in\\, S} \\left(\\sum_{T\\ni\\, y,z} c_T\\right)\n\\kappa_f\\kappa_g e^{-\\beta(d(y,R_1)+d(z,R_2))} \\int_0^s\ne^{D(2\\tau+t-s)}\\,d\\tau.\n$$\nNow if $d(y,z)>k$ and $y,z\\in T$, then $c_T$ is null by Definition\n\\ref{defetenduefinie}. Remember moreover that by hypothesis\n(\\ref{hyp0}):\n$$\nB=\\sup_{u\\in S} \\sum_{T\\ni u} c_T  <\\infty.\n$$\nTherefore~:\n\\begin{equation}\\label{avantderniere}\nM(s,t) \\leq \\kappa_f\\kappa_g\\frac{Be^{D(s+t)}}{2D}\\sum_{y\\in S}\n\\sum_{d(y,z)\\leq k} e^{-\\beta(d(y,R_1)+d(z,R_2))}.\n\\end{equation}\nIn order to evaluate the last quantity, we have to distinguish two\ncases. \\\\\n\\\\\n$\\bullet$ If $d(R_1,R_2)\\leq k$, then\n\\begin{eqnarray*}\n\\sum_{y\\in S} \\sum_{d(y,z)\\leq k}\ne^{-\\beta(d(y,R_1)+d(z,R_2))}&\\leq& 2e^{\\rho k}\\sum_{y\\in S}\ne^{-\\beta d(y,R_1)}\n\\\\\n&\\leq & 2e^{\\rho k}\\sum_{n\\in\\mathbb{N}}\\sum_{y\\in S} e^{-\\beta\nd(y,R_1)}\\mathbb{I}_{d(y,R_1)=n}\n\\\\\n&\\leq& 4|R_1|e^{\\rho k}\\sum_{n=0}^\\infty e^{(\\rho-\\beta )n}\n\\\\\n&\\leq &  \\frac{4|R_1| e^{\\rho k}}{1-e^{-(\\beta-\\rho)}}\\\\\n& \\leq & |R_1|\\frac{4 e^{(\\rho+\\beta)\nk}}{1-e^{-(\\beta-\\rho)}}e^{-\\beta d(R_1,R_2)}\\\\\n& \\leq & |R_1|\\frac{4 e^{(\\rho+\\beta)\nk}}{1-e^{-(\\beta-\\rho)}}e^{-(\\beta - \\rho) d(R_1,R_2)}\n\\end{eqnarray*}\n$\\bullet$ If $d(R_1,R_2)>k$, then we have, noting that\n$d(y,R_1)+d(z,R_2)\\geq d(R_1,R_2)-d(y,z)$ and that $d(y,z)\\leq k$,\n\n\\begin{eqnarray*}\n{\\lefteqn{\\sum_{y\\in S} \\sum_{d(y,z)\\leq k}\ne^{-\\beta(d(y,R_1)+d(z,R_2))}}}\\\\\n&&\\leq \\sum_{d(y,R_1)\\leq d(R_1,R_2)-k} \\sum_{d(y,z)\\leq\nk}e^{-\\beta (d(R_1,R_2)-k)} +\\sum_{d(y,R_1)\\geq d(R_1,R_2)-k}\n\\sum_{d(y,z)\\leq k}e^{-\\beta d(y,R_1)}\n\\\\[2ex]\n&&\\leq 4 |R_1|\\,e^{\\rho (d(R_1,R_2)-k)}e^{\\rho k}e^{-\\beta\n(d(R_1,R_2)-k)} +4|R_1|e^{\\rho k}\\sum_{n\\geq d(R_1,R_2)-k}\ne^{(\\rho-\\beta )n}\n\\\\[2ex]\n&&\\leq 4|R_1|\\, e^{\\beta\nk}\\left(1+\\frac{1}{1-e^{-(\\beta-\\rho)}}\\right)e^{-(\\beta-\\rho)d(R_1,R_2)}.\n\\end{eqnarray*}\n\nBy inserting the latter bound into (\\ref{avantderniere}), one\nobtains,\n$$\nM(s,t)\\leq\nC\\kappa_f\\kappa_g|R_1|e^{D(t+s)}e^{-({\\beta-\\rho})d(R_1,R_2)}\\;,\n$$\nwith~:\n$$\nC = \\frac{2B}{D}e^{\\beta k}\\left(1+\\frac{e^{\\rho\nk}}{1-e^{-\\beta+\\rho}}\\right).\\qquad \\Box\n$$\nThe covariance inequality (\\ref{inegcovariancefini}) implies that\nthe covariance between two functions essentially located on two\ndistant sets decays exponentially with the distance of those two\nsets, whatever the instants at which it is evaluated. However the\nupper bound increases exponentially fast with $s$ and $t$. In the\ncase where the process $\\{\\eta_t\\,,\\;t\\geq 0\\}$ converges at\nexponential speed to its equilibrium, it is possible to give a\nbound that increases only in $t-s$, thus being uniform in $t$ for\nthe covariance at a given instant $t$.\n\n\\subsection{Proof of Theorem  \\ref{cltips2}}\n\\subsubsection{Finite dimensional laws}\nLet ${\\cal G}=(S,E)$ be a transitive graph and $Aut({\\cal G})$ be\nthe automorphism group of ${\\cal G}$. Let $\\mu$ be a probability\nmeasure on ${\\cal X}$ invariant through the automorphism group\naction. Let $(\\eta_t)_{t\\geq 0}$ be an interacting particle system\nfulfilling the requirements of Theorem \\ref{cltips2}. Recall that\n$\\{S_t\\,,\\;t\\geq 0\\}$ denotes the semigroup and $\\mu S_t$ the\ndistribution of $\\eta_t$, if the distribution of $\\eta_0$ is $\\mu$.\n\\begin{prop}\\label{fidi0}\nLet $(B_n)_n$ be an increasing sequence of finite subsets of $S$\nfulfilling {\\rm (\\ref{slc})}. Let assumptions of   Theorem\n\\ref{cltips2} hold.\n Then for any fixed positive real numbers $t_1\\leq\nt_2\\leq\\cdots\\leq t_k$, the random vector\n$$\\frac{1}{\\sqrt{|B_n|}}\\left(N_{t_1}^{B_n}-\\mathbb{E}_{\\mu} N_{t_1}^{B_n},\nN_{t_2}^{B_n}-\\mathbb{E}_{\\mu} N_{t_2}^{B_n}\n,\\ldots,N_{t_k}^{B_n}-\\mathbb{E}_{\\mu} N_{t_k}^{B_n}\\right)$$ converges\nin distribution, as $n$ tends to infinity, to a centered Gaussian\nvector with covariance matrix $(\\Gamma_{\\mu}(t_i,t_j))_{1\\leq\ni,j\\leq k}$.\n\\end{prop}\n{\\bf{Proof of Proposition \\ref{fidi0}.}} We will only study the\nconvergence in distribution of the vector\n$$\\frac{1}{\\sqrt{|B_n|}}\\left(N_{t_1}^{B_n}-\\mathbb{E}_{\\mu} N_{t_1}^{B_n},\nN_{t_2}^{B_n}-\\mathbb{E}_{\\mu} N_{t_2}^{B_n}\\right)\\;,$$ the general\ncase being similar.\n For $i=1,2$, we denote by $\\alpha_i=(\\alpha_i(w))_{w\\in W}$ two\n fixed vectors of $\\mathbb{R}^{|W|}$. We have, denoting by $\\cdot$ the usual\n scalar product,\n \\begin{eqnarray*}\n{\\lefteqn{\\frac{1}{\\sqrt{|B_n|}}\\sum_{i=1}^2 \\alpha_i \\cdot\n\\left(N_{t_i}^{B_n}-\\mathbb{E}_{\\mu}\nN_{t_i}^{B_n}\\right)}}\\\\\n&& = \\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in\nB_n}\\left(\\sum_{i=1}^2\\left(\\sum_{w\\in W}\n\\alpha_i(w)(\\mathbb{I}_w(\\eta_{t_i}(x))-\\mathbb{P}_{\\mu}(\\eta_{t_i}(x)=w))\\right)\\right)\\\\\n&&= \\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in B_n}Y_x,\n\\end{eqnarray*}\n where $(Y_x)_{x\\in S}$ is the random field defined by\n\\begin{equation}\\label{Ydef}\nY_x = \\sum_{i=1}^2\\left(\\sum_{w\\in W}\n\\alpha_i(w)(\\mathbb{I}_w(\\eta_{t_i}(x))-\\mathbb{P}_{\\mu}(\\eta_{t_i}(x)=w))\\right)=:F_1(\\eta_{t_1}(x))\n+ F_2(\\eta_{t_2}(x)).\n\\end{equation}\nThe purpose is then to prove a central limit theorem for the sum\n$\\sum_{x\\in B_n}Y_x$. For this, we shall study the nature of the\ndependence of $(Y_x)_{x\\in S}$.\n\nLet $R_1$ and $R_2$ be two finite and disjoints subsets of $S$.\nLet $k_1$ and $k_2$ be two\n real valued functions defined respectively on\n$\\mathbb{R}^{|R_1|}$ and $\\mathbb{R}^{|R_2|}$. Let\n  $K_1$, $K_2$ be two real valued functions,\ndefined respectively on $W^{R_1}$ and $W^{R_2}$, by\n$$K_j(\\nu,\\eta)=k_j(F_1(\\nu(x))+ F_2(\\eta(x)),\\ x\\in R_j),\\ \\\n\\ j=1,2. $$\nLet ${\\cal L}$ be the class of real valued Lipschitz functions $f$ defined\non $\\mathbb{R}^n$, for some positive integer $n$, for which\n$$\n\\Lip f :=\\sup_{x\\neq y}\\frac{\\textstyle\n\\left|f(x)-f(y)\\right|}{\\textstyle\\sum_{i=1}^n|x_i-y_i|}<\\infty.\n$$\nWe assume that $k_1$ and $k_2$ belong to ${\\cal L}$.\n  Recall that\n\\begin{eqnarray*}\n\\Cov_{\\eta}(k_1(Y_x,\\, x\\in R_1), k_2(Y_x,\\,x \\in\nR_2))&=&\\Cov_{\\eta}\\left(K_1(\\eta_{t_1},\\eta_{t_2}),\nK_2(\\eta_{t_1},\\eta_{t_2})\\right) \\\\\n\\end{eqnarray*}\nBut\n$$|K_1(\\eta_{t_1},\\eta_{t_2})-K_1(\\eta'_{t_1},\\eta_{t_2})|\\leq\n4\\Lip k_1\\sum_{w\\in W}|\\alpha_1(w)|\\sum_{x\\in\nR_1}|\\eta_{t_1}(x)-\\eta'_{t_1}(x)|\n$$\nDenote $ A_1(W)=4\\Lip k_1\\sum_{w\\in W}|\\alpha_1(w)|$. Then, the\nfunctions\n$$\n\\eta_{t_1}\\longrightarrow (\\Lip k_1)A_1(W) \\sum_{x\\in\nR_1}\\eta_{t_1}(x) \\pm\\,\\,K_1(\\eta_{t_1},\\eta_{t_2})\n$$\nare increasing. Hence, the functions\n$$\nG_1^{\\pm}:\\,(\\eta_{t_1},\\eta_{t_2})\\longrightarrow \\Lip k_1\n\\sum_{x\\in R_1}\\left(A_1(W)\\eta_{t_1}(x)+A_2(W)\\eta_{t_2}(x)\\right)\n\\pm\\,\\,K_1(\\eta_{t_1},\\eta_{t_2})\n$$\nare increasing  coordinate by coordinate. This also holds for,\n$$\nG_2^{\\pm}:\\,(\\eta_{t_1},\\eta_{t_2})\\longrightarrow \\Lip k_2\n\\sum_{x\\in R_2}(A_1(W)\\eta_{t_1}(x)+A_2(W)\\eta_{t_2}(x))\n\\pm\\,\\,K_2(\\eta_{t_1},\\eta_{t_2}).\n$$\nUnder assumptions of Theorem \\ref{thm2} and of its Corollary\n\\ref{corligg}, the vector $(\\eta_{t_1},\\eta_{t_2})$ has positive\ncorrelation so that\n$$\n\\Cov_{\\eta}(G_1^{\\pm}(\\eta_{t_1},\\eta_{t_2}),\nG_2^{\\pm}(\\eta_{t_1},\\eta_{t_2}))\\geq 0.\n$$\nThis gives\n\\begin{eqnarray*}\n&&\\left|\\Cov_{\\eta}(k_1(Y_x,\\, x\\in R_1), k_2(Y_x,\\,x \\in\nR_2))\\right|\\\\\n&& \\leq \\Lip k_1\\Lip k_2 \\sum_{x\\in R_1}\\sum_{y\\in\nR_2}\\Cov_{\\eta}(A_1(W)\\eta_{t_1}(x)+A_2(W)\\eta_{t_2}(x),A_1(W)\\eta_{t_1}(y)+A_2(W)\\eta_{t_2}(y)).\n\\end{eqnarray*}\n From this bilinear formula, we now apply Proposition\n\\ref{propinegcovfini} and  obtain the following covariance\ninequality: for finite subsets $R_1$ and $R_2$ of $S$, we have\nletting $\\delta=\\beta-\\rho$,\n$$\\left|\\Cov_{\\eta}\\left(K_1(\\eta_{t_1},\\eta_{t_2}),K_2(\\eta_{t_1},\\eta_{t_2})\\right)\\right|\\leq  C_{\\delta} \\Lip k_1\\Lip\nk_2\\left(|R_1|\\wedge|R_2|\\right)\\exp\\left(-\\delta\nd(R_1,R_2)\\right),$$ where $C_{\\delta}$ is a positive constant\ndepending on $\\beta$ and not depending on $R_1$, $R_2$, $k_1$ and\n$k_2$.\n\\\\\nWe then deduce from Proposition \\ref{pro2r} that\n$\\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in B_n}Y_x$ converges in distribution\nto a centered normal law as soon as the quantity\n$\\Var_{\\mu}(\\sum_{x\\in B_n}Y_x)\/|B_n|$ converges as $n$ tends to\ninfinity to a finite number $\\sigma^2$. This variance converges if\nthe requirements of Proposition \\ref{slpro2} are satisfied. For\nthis, we first check the condition of invariance\n(\\ref{slstationarity}):\n$$\n\\cov_{\\mu}(Y_x,Y_y)=\\cov_{\\mu}(Y_{a(x)},Y_{a(y)}),\n$$\nfor any automorphism $a$ of ${\\cal G}$ and for $Y_x$ as defined by\n(\\ref{Ydef}). We recall that the initial distribution  is a Dirac\ndistribution on the configuration $\\eta$. Then it has positive\ncorrelations. We have supposed that $\\eta(x)=\\eta(y)$ for all\n$x,y\\in S$, hence $a\\cdot \\mu=\\mu$ and the group invariance\nproperty of the transition rates proves that $\\mu=\\delta_{\\eta}$\nfulfills (\\ref{essai}) below and then (\\ref{slstationarity}) will\nhold. Condition (\\ref{essai}) is true thanks to the following\nestimations valid for any suitable real valued functions $f$ and\n$g$,\n\\begin{eqnarray}\\label{essai}\n{\\lefteqn{\\mathbb{E}_{\\mu}(f(\\eta_{t_1})g(\\eta_{t_2}))}}\n{\\nonumber}\\\\\n&& =\\int d\\mu(\\eta)S_{t_1}\\left(fS_{t_2-t_1}g\\right)(\\eta)\n{\\nonumber}\n\\\\\n&&= \\int d\\mu(\\eta)\\, a\\cdot\nS_{t_1}\\left(fS_{t_2-t_1}g\\right)(\\eta)\\ \\ \\ {\\mbox{since}}\\ \\ \\ \\\na\\cdot\\mu=\\mu {\\nonumber}\n\\\\\n&&=\\int d\\mu(\\eta) S_{t_1}\\left((a\\cdot f)S_{t_2-t_1}(a\\cdot\ng)\\right)(\\eta)\\\n\\ \\ {\\mbox{since}}\\ \\ \\ \\ a\\cdot (S_sf)=S_s(a\\cdot f) {\\nonumber}\\\\\n&& = \\mathbb{E}_{\\mu}((a\\cdot f)(\\eta_{t_1})(a\\cdot\ng)(\\eta_{t_2}))=\\mathbb{E}_{\\mu}(f(a\\cdot\\eta_{t_1})g(a\\cdot\\eta_{t_2})).\n\\end{eqnarray}\n Hence Proposition \\ref{slpro2} applies and gives\n\\begin{eqnarray*}\n{\\lefteqn{\\sigma^2=\\sum_{z\\in S}\\cov_{\\mu}(Y_0,Y_z)}} \\\\\n&& = \\sum_{i,j=1}^2\\sum_{w,w'\\in W}\\alpha_i(w)\\alpha_j(w')\\sum_{z\\in\nS}\\cov_{\\mu}\\left(\\mathbb{I}_w(\\eta_{t_i}(0)),\\mathbb{I}_{w'}(\\eta_{t_i}(z))\\right)\n\\\\\n&&= \\sum_{i,j=1}^2 \\alpha_i^t \\Gamma_{\\mu}(t_i,t_j) \\alpha_j,\n\\end{eqnarray*}\nwhere $\\Gamma_{\\mu}(t_i,t_j)$ is the covariance matrix as defined in\nTheorem \\ref{cltips2}; with this we complete the proof of\nProposition \\ref{fidi0}.\n\n\\subsubsection{Tightness}\\label{secwdep}\n\n\n\nFirst we establish covariance inequalities for the counting\nprocess. Denote $g_{s,t,w}(\\eta,y)=\\mathbb{I}_w(\\eta_{\nt}(y))-\\mathbb{I}_w(\\eta_{s}(y))$ and for any multi-index ${\\bf\ny}=(y_1,\\ldots,y_u)\\in S^u$, for any  state vector ${\\bf\nw}=(w_1,\\ldots, w_u)\\in W^u$, $\\Pi_{{\\bf y},{\\bf w}}=\\prod_{\n\\ell=1}^ug_{s,t,w_\\ell}(\\eta,y_\\ell)$.\n Following (\\ref{inegcovariancefini}), for $\\beta>\\rho$, for any\n$r$-distant finite multi-indices ${\\bf y}\\in S^u$ and ${\\bf z}\\in\nS^v$ ,\n for any times $0\\le s\\le t\\leq T$ and\n for any  state vectors ${\\bf w}\\in W^u$\nand $ {\\bf w'}\\in W^v$\n\\begin{equation}\\label{inegcovsimp}\n\\left|\\cov_\\eta\\left(\\Pi_{{\\bf y},{\\bf w}},\\Pi_{{\\bf z},{\\bf w'}}\n\\right)\\right|\\leq 4C (u\\wedge v)e^{2DT }e^{-(\\beta-\\rho) r}\\equiv\nc_0  (u\\wedge v)e^{-c r} ,\n\\end{equation}\nfor $c=\\beta-\\rho$ and $c_0=\\frac{\\textstyle\n4Be^{2DT}e^{-(\\beta-\\rho)r}(2-e^{-c})}{\\textstyle Me^{\\rho\nk}(1-e^{-c})}$.\n\\begin{lemm}  \\label{cor:covx}There exist $\\delta_0>0$ and $K_\\Omega>0$ such that for\n$|s-t|<\\delta_0$:\n\\begin{equation}\\label{eqn:covx1}\n\\left|\\cov_\\eta\\left(\\Pi_{{\\bf x},{\\bf w}},\\Pi_{{\\bf y},{\\bf w'}}\n\\right)\\right|\\leq K_\\Omega |t-s|.\n\\end{equation}\n\\end{lemm}\n\n\\begin{proof} Denote $f(\\eta)=\\mathbb{I}_w(\\eta(x))$ then\n$g_{t+h,t,w}(\\eta,x)=S_hf(\\eta_{t})-f(\\eta_t)$; the properties of\nthe generator $\\Omega$ imply that\n$$\n\\lim_{h\\rightarrow 0} \\frac{S_{h}f(\\eta)-f(\\eta)}h= \\Omega f(\\eta)\n$$\nBut\n\\begin{eqnarray*}\n|\\Omega f(\\eta)|&\\leq& \\sum_{T\\subset S}\\sum_{\\zeta \\in\nW^T}c_T(\\eta,\\zeta)|f(\\eta^\\zeta)-f(\\eta)|\n\\\\ &\\leq&\\sum_{T\\subset S, x \\in T}c_T(\\eta)\\leq \\sum_{T\\subset S, x \\in T}c_T\\leq  C_\\Omega\n\\end{eqnarray*}\nso that for $h>0$ tending to zero\n$$\n|g_{s,s+h,w}(\\eta,x)|\\leq C_\\Omega h+o(h)\n$$\nBecause $\\Omega$ is group invariant, the remainder term is uniform\nwith respect to index $x$, so that we find convenient $\\delta_0$\nand $K_\\Omega$ uniformly with respect to location.\n\\end{proof}\n\\noindent From inequality (\\ref{inegcovsimp}) and lemma\n\\ref{cor:covx}, we deduce the following moment inequality:\n\\begin{prop}\\label{pro:moment}\nChoose $l$ and $c$ such that $\\rho(2l-1)<c$. For $(s,t)$ such that\n$|t-s|<\\delta_0\\wedge c_0e^c\/K_\\Omega$:\n\\begin{multline}\n\\mathbb{E}  (N_t^{B_n}-N_s^{B_n})^{2l}\\leq\\frac{(4l-2)!(c_0e^\n{2c})^{\\frac{\\rho l}c}}{(2l)!(2l-1)!}\n\\\\\\left(\\frac{2^{2l}(2l)!(c_0e^{ 2c})^{\\frac{\\rho (l-1)}c}}{c_1}\n|B_n|^{1-l}(K_\\Omega|t-s|)^{1-\\frac{\\rho (2l-1)}c}\n +\\left(\\frac8 {c_1}\\right)^{l}(K_\\Omega|t-s|)^{l-\\frac {\\rho l} c}\n\\right),\n\\end{multline}\nwhere $c_1=\\rho \\wedge (c-\\rho(2l-1))$.\n\\end{prop}\n\n\\begin{proof} Recall  that $\nN_t^{B_n}-N_s^{B_n}=\\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in B_n }\ng_{s,t,w}(\\eta,x)$.  Note that  the value of $\\Pi_{\\bf x}$  does\nnot depend on the order of the elements $x_1,\\ldots,x_L.$ The\nindex $\\bf x$ is said to split into ${\\bf y}=(y_1,\\ldots, y_M)$\nand ${\\bf z}=(z_1,\\ldots,z_{L-M})$ if one can write\n$y_1=x_{\\sigma(1)},\\ldots,y_M=x_{\\sigma(M)}$ and\n$z_1=x_{\\sigma(M+1)},\\ldots,z_{L-M}=x_{\\sigma(L)}$ for some\nbijection $\\sigma:\\{1,\\ldots, L\\}\\to\\{1,\\ldots, L\\}$. We adapt\nlemma 14 in Doukhan \\& Louhichi  \\cite{DoukhanLouhichi} to the\nseries $(g_{t,s,w}(\\eta,x))_{x\\in B_n}$. For any integer $q\\geq\n1$, set :\n\\begin{equation}\\label{eqn:defaq}\nA_q(n)=\\sum_{{\\bf x}\\in B_n^q} \\left|\\mathbb{E} \\Pi_{\\bf x,w}\\right|,\n\\end{equation}\nthen,\n\\begin{equation}\\label{eqn:201}\n\\mathbb{E} (N_s^{B_n}-N_t^{B_n})^{2l}\\leq |B_n|^{-l}A_{2l}(n).\n\\end{equation}\nIf $q\\geq 2$, for a multi-index ${\\bf x}=(x_1,\\ldots,x_q)$ of\nelements of $S$, the gap is defined by the maximum of the integers\n$r$ such that the index may split into two non-empty sub-indices\n${\\bf y}=( y_1,\\ldots,y_h)$ and ${\\bf z}=(z_1,\\ldots,z_{q-h})$\nwhose mutual distance equals $r$: $d({\\bf y(x)},{\\bf\nz(x)})=\\min\\{d(y_a,z_b);1\\le a\\le h,1\\le b\\le q-h \\}=r$. If the\nsequence is constant, its gap is $0$. Define the set $G_r(q,n) =\\{\n{\\bf x}\\in B_n^q$ and the gap of ${\\bf x}$ is $r$\\}. Sorting the\nsequences of indices by their gap:\n\\begin{eqnarray}\nA_q(n)&\\leq & \\sum_{x_1\\in B_n}\\mathbb{E}  | g_{s,t,w}(\\eta,x_1)|^q\n+\\sum_{r=1}^{n}\\sum_{{\\bf x}\\in G_r(q,n)} \\left|\\cov \\left(\n\\Pi_{{\\bf y(x),w}}, \\Pi_{{\\bf\nz(x),w}}\\right)\\right|\\label{eqn:majaq}\\\\\n&&+\\sum_{r=1}^{n}\\sum_{{\\bf x}\\in G_r(q,n)} \\left|\\mathbb{E}\n\\left(\\Pi_{{\\bf y(x),w}}\\right) \\mathbb{E} \\left(\\Pi_{{\\bf\nz(x),w}}\\right)\\right|.\\label{eqn:majq}\n\\end{eqnarray}\nDenote $$ V_q(n)=\\sum_{x_1\\in B_n}\\mathbb{E}  | g_{s,t,w}(\\eta,x_1)|^q\n+\\sum_{r=1}^{n}\\sum_{{\\bf x}\\in G_r(q,n)} \\left|\\cov \\left(\n\\Pi_{{\\bf y(x),w}}, \\Pi_{{\\bf z(x),w}}\\right)\\right|.$$  In order\nto prove that the expression (\\ref{eqn:majq}) is bounded by the\nproduct $\\sum_h A_h(n) A_{q-h}(n)$ we make a first summation over\nthe $\\bf x$'s such that ${\\bf y(x)}\\in B_n^h$. Hence: $$\nA_q(n)\\leq V_q(n)+\\sum_{h=1}^{q-1} A_h(n) A_{q-h}(n). $$ To build\na multi-index ${\\bf x}=(x_1,\\ldots,x_q)$ belonging to $G_r(q,n)$,\nwe first fix one of the $|B_n|$ points of $B_n$, say $x_1$.  We\nchoose a second point $x_2$ with $d(x_1,x_2)= r$. The third point\n$x_3$ is in one of the ball with radius $r$ centered in one of the\nprevious points, and so on\\ldots Thus, because the maximal\ncardinality of a ball with radius $r$ writes\n $b(r)\\le e^{\\rho r}$\n$$\n|G_r(q,n)|\\leq |B_n| b(r)2b(r)\\cdots (q-1)b(r)\n \\leq |B_n|\n (q-1)!2^{q-1}e^{\\rho(q-1) r}.$$\n We use lemma\n\\ref{cor:covx} to deduce:\n\n$$ V_q(n)\\leq |B_n| \\left(K_\\Omega|t-s|+ (q-1)!2^{q-1} \\sum_{r=1}^{\\infty}\ne^{\\rho(q-1)r} (c_0qe^{-cr}\\wedge K_\\Omega |t-s|)\\right).\n$$\nLet $R$ be an integer to be specified, then\n$$ V_q(n)\\leq |B_n| q!2^{q-1}\\left(K_\\Omega|t-s|\\sum_{r=0}^{R-1} e^{\\rho(q-1)r}+\n c_0\\sum_{r=R}^{\\infty}e^{(\\rho (q-1)-c)r}\n \\right). $$ Comparing those summations with integrals:\n\\begin{eqnarray*}\n V_q(n)&\\leq & |B_n|q!2^{q-1} \\left( \\frac{K_\\Omega|t-s|}{\\rho (q-1)}e^{\\rho (q-1)R}\n  + \\frac{c_0}{c-\\rho (q-1)}e^{(\\rho (q-1)-c)(R-1)}\\right)\\\\\n & \\leq & |B_n| q!2^{q-1}\\frac{K_\\Omega|t-s|}{c_1}e^{\\rho (q-1)R}\n \\left( 1  + \\frac{c_0}{K_\\Omega|t-s|}e^{c-cR}\\right),\n\\end{eqnarray*}\nwhere $c_1=\\rho \\wedge (c-\\rho(2l-1))$. Assume that $(s,t)\\in T$\nare such that $|t-s|<{c_0e^{c}}\/{K_\\Omega}$ and choose $R\\geq 1$\nas the integer such that $ e^{c(R-1)}\n\\leq\\frac{c_0e^{c}}{K_\\Omega|t-s|}\\leq e^{cR}$.\n\\begin{equation}\\label{eq:deuxsom}\nV_q(n)\\leq   |B_n|q! \\frac{2^qK_\\Omega|t-s|e^{2\\rho\n(q-1)}}{c_1}\\left(\\frac{c_0}{K_\\Omega|t-s|}\\right)^{\\frac{\\rho\n(q-1)}c},\n\\end{equation}\nso that $ V_q(n)$ is a function of $q$ that satisfies condition\n$({\\cal H}_0)$ of Doukhan \\& Louhichi \\cite{DoukhanLouhichi}. Then\n\\begin{eqnarray*}\nA_{2l}(n) &\\leq &\\frac{(4l-2)!}{(2l)!(2l-1)!}\\left(V_{2l}(n)+V_2(n)^{l}\\right)\\\\\n &\\leq &\\frac{(4l-2)!(c_0e^{2c})^{\\frac{\\rho l}c}}{(2l)!(2l-1)!}\n\\left(\\frac{2^{2l}(2l)!(c_0e^{ 2c})^{\\frac{\\rho (l-1)}c}}{c_1} |B_n|(K_\\Omega|t-s|)^{1-\\frac{\\rho (2l-1)}c}\n \\right.\\\\&&\\left.+\\left(\\frac8 {c_1}\\right)^{l}|B_n| ^{l}(K_\\Omega|t-s|)^{l-\\frac {\\rho l} c}\n\\right),\n\\end{eqnarray*}\nand Proposition \\ref{pro:moment} is proved.\n\\end{proof}\nTo prove\nthe tightness of the sequence of processes $N^{B_n}$, we study\nits oscillations:\n\\begin{equation*}\nw(\\delta,N^{B_n})=\\sup_{\\|t-s\\|_1<\\delta}|N_t^{B_n}-N_s^{B_n}|\n\\end{equation*}\nFix $\\varepsilon$ and $\\eta$. We have to find $\\delta$ and $n_0$\nsuch that for all $n>n_0$ :\n\\begin{equation*}\n\\mathbb{P} (w(\\delta,N^{B_n})\\geq \\varepsilon)\\leq \\eta\n\\end{equation*}\nDefine $n_0$ as the smallest integer such that\n$|B_{n_0}|>\\delta^{-1-\\rho\/c}$, then for $n>n_0$, $|t-s|<\\delta$,\n$l=2$ and $c>3\\rho$, Proposition \\ref{pro:moment} yields:\n\\begin{equation*}\n\\mathbb{E} (N_t^{B_n}-N_s^{B_n})^{4}\\leq C \\delta^{2(1-\\frac{\\rho}{c})}\n\\end{equation*}\nand we now follow the proof in Billingsley \\cite{Billingsley} to\nconclude.\n\\subsection{Proof of Theorem \\ref{clthitting}}\n\nThe proof is close to that of the analogous result in\n\\cite{ParoissinYcart04}. The convergence in distribution of\n$Z_n=(Z_n(t))_{t\\ge0}$, where $Z_n(t)=(D_t^{(n)}-|B_n|\\cdot\nm(t))\/\\sqrt{ |B_n|}$, does not directly imply the CLT for $T_n$. The\nSkorohod-Dudley-Wichura representation theorem is a much stronger\nresult (see Pollard \\cite{Pollard}, section IV.3). It implies that\nthere exist versions $Z^{*}_n$ of $Z_n$  and non-decreasing\nfunctions $\\phi_n$ such that for any fixed $s$ such that for $Z^*$,\nlimit in distribution of $Z_n$:\n\\begin{equation*}\n\\lim_{n \\rightarrow \\infty} \\sup_{0 \\leq t \\leq s} \\left|\nZ_n^{*}(t) - Z^*(\\phi_n(t)) \\right| = 0 \\quad a.s.\n\\end{equation*}\nand:\n\\begin{equation*}\n\\lim_{n \\rightarrow \\infty} \\sup_{0 \\leq t \\leq s} \\left|\n\\phi_n(t) - t \\right| = 0 \\quad a.s.\n\\end{equation*}\nSince $Z^*$ has continuous paths, it is uniformly continuous on\n$[0,s]$, and hence:\n\\begin{equation}\\label{unif}\n\\lim_{n \\rightarrow \\infty} \\sup_{0 \\leq t \\leq s} \\left| Z^*_n(t)\n- Z^*(t) \\right| = 0 \\quad a.s.\\;,\n\\end{equation}\nWe shall first use (\\ref{unif}) to prove that the distributions of\n$\\sqrt{ |B_n|}(T_n-t_\\alpha)$ are a tight sequence. Let $c$ be a\npositive constant. On the one hand, if\n$D^{(n)}_{t_\\alpha+c\/\\sqrt{ |B_n|}} \\geq k(n)$, then $T_n \\leq\nt_\\alpha+c\/\\sqrt{ |B_n|}$. Thus:\n\\begin{eqnarray*}\n{\\mathbb{P} [ \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq c]}\n &\\geq&{\\mathbb{P} [D^{(n)}_{t_\\alpha+c\/\\sqrt{ |B_n|}}\\geq k(n)]} \\\\[1ex]\n &=&{\\mathbb{P} [Z^*_n(t_\\alpha+c\/\\sqrt{ |B_n|})\\geq\n \\sqrt{ |B_n|}(\\alpha-m(t_\\alpha+c\/\\sqrt{ |B_n|}))+o(1)]} \\\\[1ex]\n &=&{\\mathbb{P} [Z^*_n(t_\\alpha+c\/\\sqrt{ |B_n|})\\geq\n -cm'(t_\\alpha)+o(1)]} \\\\[1ex]\n &=&{\\mathbb{P} [Z^*(t_\\alpha)\\geq\n -cm'(t_\\alpha)]+o(1)}\\;,\n\\end{eqnarray*}\nusing (\\ref{unif}) and the continuity of $Z^*$. Since\n$m'(t_\\alpha)>0$, we obtain that:\n\\begin{equation}\n\\label{liminf} \\lim_{c \\rightarrow \\infty} \\liminf_{n \\rightarrow\n\\infty} \\mathbb{P} [\n  \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq c] = 1 .\n\\end{equation}\nOn the other hand, we have:\n$$\n\\mathbb{P} [ \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq -c]\n =\\mathbb{P} [\\exists t \\leq t_\\alpha-c\/\\sqrt{ |B_n|}\\;,\\, Z^*_n(t)\\geq\n \\sqrt{ |B_n|}(\\alpha-m(t))+o(1)] .\n$$\nBut since the function $m$ is increasing, for all $t\\leq\nt_\\alpha-c\/\\sqrt{ |B_n|}$ we have:\n\\begin{equation*}\n\\sqrt{ |B_n|}(\\alpha-m(t)) \\geq \\sqrt{\n|B_n|}(\\alpha-m(t_\\alpha-c\/\\sqrt{ |B_n|})) = cm'(t_\\alpha)+o(1) .\n\\end{equation*}\nHence:\n\\begin{eqnarray*}\n\\displaystyle{\\mathbb{P} [ \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq -c]}\n&\\leq&\\displaystyle{\\mathbb{P} [\\exists t \\leq t_\\alpha-c\/\\sqrt{\n|B_n|}\\,, Z^*_n(t)\\geq\n cm'(t_\\alpha)+o(1)]} \\\\[1ex]\n&\\leq&\\displaystyle{ \\mathbb{P} [\\exists t \\leq t_\\alpha\\,, Z^*_n(t)\\geq\n cm'(t_\\alpha)+o(1)]} \\\\[1ex]\n&=&\\displaystyle{ \\mathbb{P} [\\exists t \\leq t_\\alpha\\,, Z^*(t)\\geq\n cm'(t_\\alpha)+o(1)]+o(1)} .\n\\end{eqnarray*}\nThe process $Z$ being \\textsl{a.s.} bounded on any compact set\nand $m'(t)$ being positive on $[0,\\tau]$, we deduce that:\n\\begin{equation}\n\\label{limsup} \\lim_{c \\rightarrow \\infty} \\limsup_{n \\rightarrow\n\\infty} \\mathbb{P} [\n  \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq -c] = 0 .\n\\end{equation}\nNow (\\ref{liminf}) and (\\ref{limsup}) mean that the sequence of\ndistributions of $(\\sqrt{ |B_n|}(T_n-t_\\alpha))$ is tight. Hence\nto conclude it is enough to check the limit. \\vskip 1mm Using\nagain (\\ref{unif}), together with the almost sure continuity of\n$Z$ yields:\n\\begin{eqnarray*}\nD^{(n)}_{t_\\alpha+c\/\\sqrt{ |B_n|}}\n & = &  |B_n|m(t_\\alpha+u\/\\sqrt{ |B_n|})\n +\\sqrt{ |B_n|}Z^*(t_\\alpha+u\/\\sqrt{ |B_n|})+o(\\sqrt{ |B_n|}) \\quad a.s. \\\\[1ex]\n & = &  |B_n|\\alpha +u\\sqrt{ |B_n|}m'(t_\\alpha)\n +\\sqrt{ |B_n|}Z^*(t_\\alpha)+o(\\sqrt{ |B_n|}) \\quad a.s.\n\\end{eqnarray*}\nTherefore:\n\\begin{eqnarray*}\n\\inf \\left\\{ u \\;; D^{(n)}_{t_\\alpha+u\/\\sqrt{ |B_n|}} \\geq k(n) \\right\\}\n  & = & \\inf \\Big\\{ u \\;; u\\sqrt{ |B_n|}m'(t_\\alpha)\n +\\sqrt{ |B_n|}Z^*(t_\\alpha)+o(\\sqrt{ |B_n|}) \\geq 0\\Big\\}\\\\[1ex]\n & = & - \\frac{Z^*(t_\\alpha)}{m'(t_\\alpha)} +o(1) .\n\\end{eqnarray*}\nThe distribution of $-Z^*(t_\\alpha)\/m'(t_\\alpha)$ is normal with\nmean $0$ and variance $\\sigma^2_\\alpha$, hence the result.\n\\subsection{Proof of Proposition \\ref{pro2r}}\nLet ${\\cal F}_{2,3}$ be\nthe set of real valued functions $h$\n  defined on $\\mathbb{R}$, three times differentiable, such\n  that $h(0)=0$, $\\|h''\\|_{\\infty}<+\\infty$, and\n  $\\|h^{(3)}\\|_{\\infty}<+\\infty$. For a function $h\\in {\\cal\n    F}_{2,3}$, we will denote by $b_2$ and $b_3$ the supremum norm of\n  its second and third derivatives.\nWe first need the following lemma.\n\\begin{lemm}\\label{lem1}\nLet $h$ be a fixed function of the set ${\\cal F}_{2,3}$. Let\n$R$ be a fixed and finite subset of $S$. Let $r$ be a fixed\npositive real. For any $x\\in R$, let $V_x=B(x,r)\\cap R$.\nLet $(Y_x)_{x\\in S}$ be a real valued random field. Suppose that,\nfor any $x\\in S$, $\\mathbb{E} Y_x=0$ and $\\mathbb{E} Y_x^2<+\\infty$. Let\n$Z(R)=\\sum_{x\\in R}Y_x$. Then\n\\begin{eqnarray}\\label{cov1}\n{\\lefteqn{\\left|\\mathbb{E}(h(Z(R))) -\\Var Z(R)\\int_0^1t\n\\mathbb{E}(h''\\left(tZ(R)\\right))dt\\right|}} \\nonumber\\\\ &&\\leq\n\\int_0^1\\hspace{-1.5mm}\\sum_{x\\in\nR}\\left|\\Cov\\left(Y_x,h'(tZ(V_x^c))\\right)\\right|dt + 2\\sum_{x\\in\nR}\\mathbb{E} |Y_x||Z(V_x)|\\left[b_2\\wedge b_3|Z(V_x)|\\right] \\nonumber\\\\\n&& + b_2 \\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right| + b_2 \\sum_{x\\in\nR}\\left|\\Cov(Y_x, Z(V^c_x))\\right|,\n\\end{eqnarray}\nwhere $V^c_x=R\\setminus V_x$.\n\\end{lemm}\n{\\bf{Remark.}} For an independent random field $(Y_x)_{x\\in S}$,\nfulfilling $\\sup_{x\\in S}\\mathbb{E} Y_x^4<+\\infty$, Lemma \\ref{lem1}\napplied with $V_x=\\{x\\}$, ensures $$ \\left|\\mathbb{E}(h(Z(R))) -\\Var\nZ(R)\\int_0^1t \\mathbb{E}(h''\\left(tZ(R)\\right))dt\\right|\\leq 2 \\sum_{x\\in\nR}\\mathbb{E} |Y_x|^2\\left(b_2\\wedge b_3|Y_x|\\right) +\nb_2\\sqrt{|R|}\\sup_{x\\in S}\\|Y_x^2\\|_2 . $$\n{\\bf{Proof of Lemma \\ref{lem1}.}} We have,\n\\begin{eqnarray*}\\label{d1}\n{\\lefteqn{h(Z(R)) = Z(R)\\int_0^1 h'(tZ(R))dt  =\n\\int_0^1\\left(\\sum_{x\\in R}Y_xh'(tZ(R))\\right)dt}} {\\nonumber}\\\\\n&& = \\int_0^1\\left(\\sum_{x\\in R}Y_xh'(tZ(V_x^c))\\right)dt +\n\\int_0^1\\left(\\sum_{x\\in\nR}Y_x\\left(h'(tZ(R))-h'(tZ(V_x^c))-tZ(V_x)h''(tZ(R))\\right)\\right)dt{\\nonumber}\\\\\n&& + \\sum_{x\\in R}Y_xZ(V_x)\\int_0^1th''(tZ(R))dt - \\sum_{x\\in\nR}\\mathbb{E}\\left(Y_xZ(V_x)\\right)\\int_0^1th''(tZ(R))dt\\\\ && + \\sum_{x\\in\nR}\\mathbb{E}\\left(Y_xZ(V_x)\\right)\\int_0^1th''(tZ(R))dt\n-\\sum_{x\\in R}\\mathbb{E}\\left(Y_xZ(R)\\right)\\int_0^1th''(tZ(R))dt {\\nonumber}\\\\\n&& + \\sum_{x\\in\nR}\\mathbb{E}\\left(Y_xZ(R)\\right)\\int_0^1th''(tZ(R))dt{\\nonumber}.\n\\end{eqnarray*}\nWe take expectation in the last equality. The obtained formula,\ntogether with the following estimations, proves Lemma \\ref{lem1}.\n\\begin{eqnarray*}\n{\\lefteqn{\\left|h'(tZ(R))-h'(tZ(V_x^c))-tZ(V_x)h''(tZ(R))\\right|}}\n\\\\ && \\leq\n\\left|h'(tZ(R))-h'(tZ(V_x^c))-tZ(V_x)h''(tZ(V_x^c))\\right|+\n|Z(V_x)||h''(tZ(R))-h''(tZ(V_x^c))| \\\\ && \\leq 2\n|Z(V_x)|\\left(b_2\\wedge b_3|Z(V_x)|\\right).\\qquad\\Box\n\\end{eqnarray*}\n\n\\vskip 3mm\\noindent\nOur purpose now is to control the right hand side of the bound\n(\\ref{cov1}) for a random field $(Y_x)_{x\\in S}$ fulfilling the\ncovariance inequality (\\ref{slcov}) and the requirements of\nProposition \\ref{pro2r}.\n\\begin{coro}\\label{cor1}\nLet $h$ be a fixed function of the set ${\\cal F}_{2,3}$. Let\n$R$ be a finite subset of $S$.  For any $x\\in R$ and for any\npositive real $r$, let $V_x=B(x,r)\\cap R$. Let $(Y_x)_{x\\in\nS}$ be a real valued random field, fulfilling the covariance\ninequality (\\ref{slcov}). Suppose that, for any $x\\in S$, $\\mathbb{E}\nY_x=0$ and $\\sup_{x\\in S}\\|Y_x\\|_{\\infty}< M$, for some positive\nreal $M$. Recall that $Z(R)=\\sum_{x\\in R}Y_x$. Then, for\nany $\\delta>0$, there exists a positive constant $C({\\delta},M)$\nindependent of $R$, such that\n\\begin{eqnarray*}\n{\\lefteqn{\\sup_{h\\in {\\cal F}_{2,3}}\\left|\\mathbb{E}(h(Z(R))) -\\Var\nZ(R)\\int_0^1t \\mathbb{E}(h''\\left(tZ(R)\\right))dt\\right|}} \\\\ && \\leq\nC({\\delta},M)\\left\\{ b_2 |R|e^{-\\delta r}+ b_3 |R| \\kappa_r\n+ b_2|R|^{1\/2}\\kappa_r\\left(\\sum_{k=[3r]}^{\\infty}\\kappa_k e^{-\\delta(k-2r)}\\right)^{1\/2}\\right.\\\\\n&& \\left. +\nb_2|R|^{1\/2}\\kappa_{3r}\\left(\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k\\right)^{1\/2}\\right\\},\n\\end{eqnarray*}\nrecall that $\\sup_{x\\in S}|B(x,n)|\\leq \\kappa_n$.\n\\end{coro}\n{\\bf{Proof of Corollary \\ref{cor1}}}\n\\\\\nWe have $$V_x^c=\\{y\\in S,\\ \\ d(x,y)\\geq r\\}\\cap R. $$ Hence\n$$d(\\{x\\}, V^c_x)\\geq r. $$ The last bound together with\n(\\ref{slcov}), proves that\n\\begin{eqnarray}\\label{t1}\n\\sum_{x\\in R}\\left|\\Cov\\left(Y_x,h'(tZ(V_x^c))\\right)\\right|&\\leq\n& C_{\\delta} b_2 \\sum_{x\\in R}(|V_x^c|\\wedge 1)e^{-\\delta d(\\{x\\},\nV^c_x)} \\nonumber\n\\\\ & \\leq &  C_{\\delta} b_2 |R|e^{-\\delta r}.\n\\end{eqnarray}\nIn the same way, we prove that\n\\begin{eqnarray}\\label{t2}\nb_2 \\sum_{x\\in R}\\left|\\Cov(Y_x, Z(V^c_x))\\right| & \\leq &\nC_{\\delta} b_2 |R|e^{-\\delta r}.\n\\end{eqnarray}\nNow\n\\begin{eqnarray}\\label{t3}\n\\sum_{x\\in R}\\mathbb{E} |Y_x||Z(V_x)|\\left(b_2\\wedge b_3|Z(V_x)|\\right)\n&\\leq & b_3 M |R| \\sup_{x\\in S} \\mathbb{E}|Z(V_x)|^2 {\\nonumber}\\\\\n&\\leq &\n b_3 M |R|\\kappa_r \\sup_{y\\in S}\n\\sum_{z\\in S}|\\Cov(Y_y,Y_z)|\n\\end{eqnarray}\nThe last bound is obtained since $|V_x|\\leq \\kappa_r$ and\n$\\sup_{y\\in S} \\sum_{z\\in S}|\\Cov(Y_y,Y_z)|<\\infty$ (the proof of\nthe last inequality is done along the same lines as that of\nProposition \\ref{slpro2}) .\n\\\\\nIt remains to control $$\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|.$$ For this, we\nargue as Bolthausen \\cite{Bolthausen82}. We have\n\\begin{eqnarray*}\n\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|^2 &=&\n\\Var(\\sum_{x\\in R}Y_xZ(V_x)) \\\\ & =& \\sum_{x\\in R}\\sum_{y\\in\nR}\\Cov(Y_xZ(V_x), Y_yZ(V_y)).\n\\end{eqnarray*}\nHence, since $V_x\\subset B(x,r)$,\n\\begin{eqnarray}\\label{i1}\n\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|^2\\leq \\sum_{x\\in\nR}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|.\n\\end{eqnarray}\nWe have,\n\\begin{eqnarray}\\label{second}\n \\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\n \\leq  \\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\geq 3r}+\n\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}.\n\\end{eqnarray}\nWe begin by controlling the first term. The covariance inequality\n(\\ref{slcov}) together with some elementary estimations, ensures\n\\begin{eqnarray*}\n\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\geq 3r} & \\leq\n& \\sum_{k=[3r]}^{\\infty}\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{k\\leq d(x,y)< k+1} \\\\ & \\leq & 2 M^2\nC_{\\delta}\\sum_{k=[3r]}^{\\infty}e^{-\\delta\nd(\\{x,x'\\},\\{y,y'\\})}\\mathbb{I}_{k\\leq d(x,y)< k+1}\\\\ & \\leq & 2 M^2\nC_{\\delta}\\sum_{k=[3r]}^{\\infty}e^{-\\delta(k-2r)}\\mathbb{I}_{ d(x,y)<\nk+1},\n\\end{eqnarray*}\nthe last bound is obtained since, for any $x'\\in B(x,r)$ and\n$y'\\in B(y,r)$, we have, $$ d(\\{x,x'\\},\\{y,y'\\})+2r \\geq\nd(\\{x,x'\\},\\{y,y'\\})+ d(x,x')+ d(y,y')\\geq d(x,y).$$ Hence,\n\\begin{eqnarray}\\label{i2}\n{\\lefteqn{\\sum_{x\\in R}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\geq\n3r}}}\\nonumber \\\\ &&\\leq 2 M^2\nC_{\\delta}\\kappa_r^2\\sum_{k=[3r]}^{\\infty}\\sum_{x\\in R}\\sum_{y\\in\nR}e^{-\\delta(k-2r)} \\mathbb{I}_{y\\in B(x,k+1)}\\nonumber\n\\\\\n&&\\leq 2 M^2\nC_{\\delta}|R|\\kappa_r^2\\sum_{k=[3r]}^{\\infty}\\kappa_{k+1}e^{-\\delta(k-2r)}.\n\\end{eqnarray}\nWe now control the second term in (\\ref{second}). Inequality\n(\\ref{slcov}) and the fact that\\\\ $d(\\{x\\},\\{x',y,y'\\})\\leq\nd(\\{x\\},\\{x'\\})$, ensure\n\\begin{eqnarray*}\n{\\lefteqn{\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}}}\\\\\n &&\\leq \\left|\\Cov(Y_x,\nY_{x'}Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}+\n\\left|\\Cov(Y_x,Y_{x'})\\right|\\left|\\Cov(Y_y,Y_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq\n3r}\\\\ && \\leq 2M^2C_{\\delta}e^{-\\delta\nd(\\{x\\},\\{x',y,y'\\})}\\mathbb{I}_{d(x,y)\\leq 3r}.\n\\end{eqnarray*}\nWe deduce, using the last bound, that\n\\begin{eqnarray}\\label{bff}\n{\\lefteqn{\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}}}\\nonumber\\\\\n &&\\leq \\sum_{k=1}^{[3r]+1}\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}\\mathbb{I}_{k-1\\leq\nd(\\{x\\},\\{x',y,y'\\})< k}\\nonumber\\\\ && \\leq\n2M^2C_{\\delta}\\sum_{k=1}^{[3r]+1}e^{-\\delta (k-1)}\\mathbb{I}_{d(x,y)\\leq\n3r}\\mathbb{I}_{d(\\{x\\},\\{x',y,y'\\})< k}.\n\\end{eqnarray}\nWe have $$\\mathbb{I}_{d(\\{x\\},\\{x',y,y'\\})\\leq k}\\leq\n\\mathbb{I}_{d(\\{x\\},\\{x'\\})\\leq k}+ \\mathbb{I}_{d(\\{x\\},\\{y\\})\\leq\nk}+\\mathbb{I}_{d(\\{x\\},\\{y'\\})\\leq k}.$$ Hence, we check that,\n\\begin{eqnarray}\\label{bf}\n\\sum_{x\\in R}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\mathbb{I}_{d(x,y)\\leq 3r}\\mathbb{I}_{d(\\{x\\},\\{x',y,y'\\})\\leq k} \\leq\n3|R|\\kappa_{3r}^2\\kappa_k.\n\\end{eqnarray}\nWe obtain combining (\\ref{bff}) and (\\ref{bf}),\n\\begin{multline}\n \\sum_{x\\in R}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq\n3r}\\\\\n\\label{i3} \\leq\n6e^{\\delta}M^2C_{\\delta}|R|\\kappa_{3r}^2\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k.\n\\end{multline}\nWe collect the bounds (\\ref{i1}), (\\ref{i2}) and (\\ref{i3}), we\nobtain,\n\\begin{eqnarray}\\label{t4}\n{\\lefteqn{\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|}}\\nonumber\\\\\n&&\\leq\nC(\\delta,M)|R|^{1\/2}\\left\\{\\kappa_r\\left(\\sum_{k=[3r]}^{\\infty}\\kappa_{k+1}e^{-\\delta(k-2r)}\\right)^{1\/2}\n+ \\kappa_{3r} \\left(\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k\\right)^{1\/2}\\right\\}.\n\\end{eqnarray}\nFinally, the bounds (\\ref{t1}), (\\ref{t2}), (\\ref{t3}),\n(\\ref{t4}), together with Lemma \\ref{lem1} prove Corollary\n\\ref{cor1}. \\ \\ \\ \\  $\\Box$\n\\vskip 3mm\\noindent\n {\\bf{End of the proof of Proposition \\ref{pro2r}.}}\nWe apply Corollary \\ref{cor1} to the real and imaginary parts of\nthe function $x\\rightarrow \\exp(iux\/{\\sqrt{|B_n|}})-1$. Those\nfunctions belong to the set ${\\cal F}_{2,3}$, with\n$b_2=\\frac{\\textstyle u^2}{\\textstyle |B_n|}$ and\n$b_3=\\frac{\\textstyle |u|^3}{\\textstyle |B_n|^{3\/2}}$.\n\\\\\nWe obtain, noting by $\\phi_n$ the characteristic function of the\nnormalized sum $Z(B_n)\/{\\sqrt{|B_n|}}$,\n\\begin{eqnarray*}\n{\\lefteqn{\\left|\\phi_n(u)-1+\\frac{\\Var Z(B_n)}{|B_n|}u^2\n\\int_0^1t\\phi_n(tu)dt\\right|}}\n\\\\ && \\leq C({\\delta},M,u)\\left\\{\ne^{-\\delta r}+ \\frac{\\kappa_r}{\\sqrt{|B_n|}} +\n\\frac{\\kappa_r}{\\sqrt{|B_n|}}\\left(\\sum_{k=[3r]}^{\\infty}\\kappa_k e^{-\\delta(k-2r)}\\right)^{1\/2}\\right.\\\\\n&& \\left.\n+\\frac{\\kappa_{3r}}{\\sqrt{|B_n|}}\\left(\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k\\right)^{1\/2}\\right\\}.\n\\end{eqnarray*}\nLet $\\delta$ be a fixed positive real such that\n$\\delta>12\\rho$, recall that\n$$\\sup_{x\\in S}|B(x,r)|\\leq 2 e^{r\\rho}=:\\kappa_r.$$ Hence\n\\begin{eqnarray*}\n{\\lefteqn{\\left|\\phi_n(u)-1+\\frac{\\Var Z(B_n)}{|B_n|}u^2\n\\int_0^1t\\phi_n(tu)dt\\right|}}\n\\\\ && \\leq C(\\delta,M,u)\\left\\{\ne^{-\\delta r}+\n\\frac{e^{r\\rho}}{\\sqrt{|B_n|}}+\\frac{e^{(\\rho+\\delta)r}}{\\sqrt{|B_n|}}\\left(\\sum_{k=[3r]}^{\\infty}e^{-(\\delta-\\rho)k}\\right)^{1\/2}\n +\\frac{e^{3\\rho\nr}}{\\sqrt{|B_n|}}\\left(\\sum_{k=1}^{[3r]+1}e^{-(\\delta-\\rho)\nk}\\right)^{1\/2}\\right\\}\\\\ &&\\leq\nC(M,\\rho,\\delta,u)\\left(e^{-\\delta\nr}+\\frac{e^{3r\\rho}}{\\sqrt{|B_n|}}+\n\\frac{e^{-(\\delta-5\\rho)r\/2}}{\\sqrt{|B_n|}}\\right).\n\\end{eqnarray*}\nFor a suitable choice of the sequence $r$ (for example we can take\n$r=\\frac{2}{\\delta}\\ln|B_n|$), the right hand side of the last\nbound tends to $0$ an $n$ tends to infinity:\n\\begin{equation}\\label{limf}\n\\lim_{n\\rightarrow \\infty}\\left|\\phi_n(u)-1+\\frac{\\Var\nZ(B_n)}{|B_n|}u^2 \\int_0^1t\\phi_n(tu)dt\\right|=0.\n\\end{equation}\nWe now need the following lemma.\n\\begin{lemm}\\label{lem2} Let $\\sigma^2$ be a positive real. Let $(X_n)$\nbe a sequence of real valued random variables such that\n$\\sup_{n\\in \\mathbb{N}}\\mathbb{E} X_n^2<+\\infty$. Let $\\phi_n$ be the\ncharacteristic function of $X_n$. Suppose that for any $u\\in \\mathbb{R}$,\n\\begin{equation}\\label{cara}\n\\lim_{n\\rightarrow\n+\\infty}\\left|\\phi_n(u)-1+\\sigma^2\\int_0^ut\\phi_n(t)dt\\right|=0.\n\\end{equation}\nThen, for any $u\\in \\mathbb{R}$, $$\\lim_{n\\rightarrow\n+\\infty}\\phi_n(u)=\\exp(-\\frac{u^2\\sigma^2}{2}). $$\n\\end{lemm}\n{\\bf{Proof of Lemma \\ref{lem2}.}}\n Lemma \\ref{lem2} is  a variant of Lemma 2 in\nBolthausen \\cite{Bolthausen82}. The  Markov inequality and the\ncondition $\\sup_{n\\in \\mathbb{N}}\\mathbb{E} X_n^2<+\\infty$ imply that the\nsequence $(\\mu_n)_{n\\in \\mathbb{N}}$ of the laws of $(X_n)$ is tight.\nTheorem 25.10 in Billingsley \\cite{Billingsley} proves the\nexistence of a subsequence $\\mu_{n_k}$ and a probability measure\n$\\mu$ such that $\\mu_{n_k}$ converges weakly to $\\mu$ as $k$ tends\nto infinity. Let $\\phi$ be the characteristic function of $\\mu$.\nWe deduce from (\\ref{cara}) that, for any $u\\in \\mathbb{R}$, $$\n\\phi(u)-1+\\sigma^2\\int_0^ut\\phi(t)dt=0, $$ or equivalently, for\nany $u\\in \\mathbb{R}$, $$ \\phi'(u)+\\sigma^2u\\phi(u)=0. $$ We obtain,\nintegrating the last equation, that for any $u\\in \\mathbb{R}$, $$\n\\phi(u)= \\exp(-\\frac{\\sigma^2 u^2}{2}). $$ The proof of Lemma\n\\ref{lem2} is completed by using Theorem 25.10 in Billingsley\n\\cite{Billingsley} and its corollary. \\,\\,\\, $\\Box$ \\vskip 1mm\nProposition \\ref{pro2r} follows from (\\ref{limff}), (\\ref{limf})\nand Lemma \\ref{lem2}.\\ \\ \\ $\\Box$ \\vskip 3mm\n\\subsection{Proof of Proposition \\ref{slpro2}.}\nWe deduce from\n(\\ref{slcov}) that for any positive real $\\delta$ there\nexists a positive constant $C_{\\delta}$ such that for different\nsites $x$ and $y$ of $S$,\n\\begin{equation}\\label{slc1}\n\\left|\\Cov(Y_x,Y_y)\\right|\\leq C_{\\delta} e^{-\\delta d(x,y)}.\n\\end{equation}\nHence, the first conclusion of Proposition \\ref{slpro2} follows\nfrom the bound (\\ref{slc1}), together with the following\nelementary calculations, for $\\rho<\\delta$,\n\\begin{eqnarray}\\label{slcf}\n\\sum_{z\\in S}|\\Cov(Y_0,Y_{z})| &\\leq & C_{\\delta}  \\sum_{z\\in\nS}\\exp(-\\delta d(0,z)){\\nonumber}\\\\\n& \\leq & C_{\\delta}\\sum_{z\\in S}\\sum_{r=0}^{\\infty}\\exp(-\\delta\nd(0,z))\\mathbb{I}_{r \\leq d(0,z)< r+1} {\\nonumber}\\\\ & \\leq &\nC_{\\delta}\\sum_{r= 0}^{\\infty}\\exp(-\\delta\nr)\\sum_{z\\in S}\\mathbb{I}_{d(0,z)< r+1}{\\nonumber}\\\\\n& \\leq & C_{\\delta}\\sum_{r=0}^{\\infty}\\exp(-\\delta\nr)|B(0,r+1)|{\\nonumber}\\\\ & \\leq & C(\\delta,\\rho)\\sum_{r=\n0}^{\\infty}\\exp(-(\\delta -\\rho)r),\n\\end{eqnarray}\nwhere $C(\\delta,\\rho)$ is a positive constant depending  on\n$\\delta$ and $\\rho$.\n\\\\\nWe now prove the second part of Proposition \\ref{slpro2}. Thanks\nto (\\ref{slc}), we can find a sequence $u=(u_n)$ of positive real\nnumbers such that\n\\begin{equation}\\label{sllimd}\n\\lim_{n\\rightarrow +\\infty}u_n=+\\infty, \\, \\lim_{n\\rightarrow\n+\\infty} \\frac{|\\partial B_n|}{|B_n|}\\exp(\\rho u_n)=0.\n\\end{equation}\nLet $(\\partial _{u}B_n)_n$ be the sequence of subsets of $S$\ndefined by\n$$\\partial _{u}B_n=\\{s\\in B_n\\,:\\, d(s,\\partial B_n)< u_n\\}. $$\nThe bound (\\ref{borneboule}) gives\n$$ |\\partial _{u}B_n|\\leq  2 |\\partial B_n| e^{u_n\\rho},\n$$\nwhich together with the suitable choice of the sequence  $(u_n)$\nensures\n\\begin{equation}\\label{sllimt3}\n\\lim_{n\\rightarrow +\\infty}\\frac{|\\partial _{u}B_n|}{|B_n|}=0,\n\\end{equation}\nwe shall use this fact below without further comments. Let\n$B^{u}_n=B_n\\setminus \\partial _{u}B_n$. We decompose the quantity\n$\\Var\\,S_n$ as in Newman \\cite{Newman80}:\n\\begin{eqnarray*}\\label{sldeco}\n\\frac{1}{|B_n|}\\Var\\, S_n & = & \\frac{1}{|B_n|}\\sum_{x\\in\nB_n}\\sum_{y\\in B_n}\\Cov\\left(Y_x,Y_y\\right) = T_{1,n} + T_{2,n}\n+T_{3,n},\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\nT_{1,n} & = & \\frac{1}{|B_n|}\\sum_{x\\in B^{u}_n}\\,\\,\\sum_{y\\in\nB_n\\setminus B(x,u_n)}\\Cov\\left(Y_x,Y_y\\right),\\\\\nT_{2,n} & = & \\frac{1}{|B_n|}\\sum_{x\\in B^{u}_n}\\,\\,\\sum_{y\\in\nB_n\\cap B(x,u_n)}\\Cov\\left(Y_x,Y_y\\right),\\\\ T_{3,n} & = &\n\\frac{1}{|B_n|}\\sum_{x\\in \\partial_u B_n}\\sum_{y\\in\nB_n}\\Cov\\left(Y_x,Y_y\\right).\n\\end{eqnarray*}\n{\\bf{Control of $T_{1,n}$.}} We have, since $|B^{u}_n| \\leq |B_n|$\nand applying (\\ref{slc1})\n\\begin{eqnarray}\\label{slt1bisbis}\n|T_{1,n}| &\\leq & \\sup_{x\\in S} \\sum_{y\\in S\\setminus B(x,u_n)\n}\\left|\\Cov(Y_x,Y_y)\\right| \\leq  C_\\delta \\sup_{x\\in S}\n\\sum_{y\\in S\\setminus B(x,n)}\\exp(-\\delta d(x,y)).\n\\end{eqnarray}\nFor any fixed $x\\in S$, we argue as for (\\ref{slcf}) and we obtain\nfor $\\rho<\\delta$,\n\\begin{equation}\\label{slt1bis}\n\\sum_{y\\in S\\setminus B(x,n)}\\exp(-\\delta d(x,y)) \\leq\nC(\\delta)\\sum_{r= [u_n]}^{\\infty}\\exp(-(\\delta-\\rho) r) \\leq\nC(\\delta,\\rho)\\exp(-(\\delta-\\rho)u_n)\n\\end{equation}\n We obtain, collecting (\\ref{slt1bisbis}), (\\ref{slt1bis}) together with\nthe first limit in (\\ref{sllimd}) :\n\\begin{equation}\\label{sllimT1}\n\\lim_{n\\rightarrow +\\infty} T_{1,n}=0.\n\\end{equation}\n\\\\\n{\\bf{Control of $T_{3,n}$.}} We obtain using (\\ref{slc1}) :\n\\begin{eqnarray}\\label{slt3bis}\n|T_{3,n}| &\\leq & \\frac{|\\partial_u B_n|}{|B_n|} \\sup_{x\\in S}\n\\sum_{y\\in S}\\left|\\Cov(Y_x,Y_y)\\right|.\n\\end{eqnarray}\nThe last bound, together with the limit (\\ref{sllimt3})\ngives\n\\begin{equation}\\label{sllimT3}\n\\lim_{n\\rightarrow +\\infty}T_{3,n}=0.\n\\end{equation}\n \\\\ {\\bf{Control of $T_{2,n}$.}}\n We deduce using the following implication, if $x\\in\nB^{u}_n$ and $y$ is not belonging to $B_n$ then $d(x,y)\\geq u_n$,\nthat\n\\begin{eqnarray*}\nT_{2,n} & = & \\frac{1}{|B_n|}\\sum_{x\\in B^{u}_n}\\,\\,\\sum_{y\\in\nB(x,u_n)}\\Cov(Y_x,Y_y)\n\\end{eqnarray*}\nWe claim that,\n\\begin{equation}\\label{ccc}\n\\sum_{y\\in B(x,u_n)}\\Cov(Y_x,Y_y) = \\sum_{z\\in\nB(0,u_n)}\\Cov\\left(Y_0,Y_z\\right),\n\\end{equation}\nin fact, since the graph ${\\cal G}$ is transitive, there exits an\nautomorphism $a_x$, such that $a_x(x)=0$ ($0$ is a fixed vertex in\n$S$). Equality (\\ref{slstationarity}) gives\n$$\n\\sum_{y\\in B(x,u_n)}\\Cov(Y_x,Y_y)=\\sum_{y\\in\nB(x,u_n)}\\Cov(Y_0,Y_{a_x(y)}).\n$$\nNow, Lemma 1.3.2 in Godsil and Royle \\cite{GodsilRoyle} yields that $d(x,y)=\nd(a_x(x), a_x(y))=d(0,a_x(y))$. From this we deduce that $y\\in\nB(x,u_n)$ if and only if $a_x(y)\\in B(0,u_n)$. From above, we\nconclude that,\n$$\n\\sum_{y\\in B(x,u_n)}\\Cov(Y_x,Y_y)=\\sum_{a_x(y)\\in\nB(0,u_n)}\\Cov(Y_0,Y_{a_x(y)})=\\sum_{z\\in B(0,u_n)}\\Cov(Y_0,Y_{z}),\n$$\nwhich proves (\\ref{ccc}). Consequently,\n\\begin{eqnarray*}\nT_{2,n} & = &  \\frac{|B^{u}_n|}{|B_n|}\\sum_{z\\in B(0,u_n)\n}\\Cov(Y_0,Y_z).\n\\end{eqnarray*}\nThe last equality together with the first limit in (\\ref{sllimd})\nand (\\ref{sllimt3}), ensures\n\\begin{equation}\\label{sllimT2}\n\\lim_{n\\rightarrow +\\infty} T_{2,n}= \\sum_{z\\in S}\\Cov(Y_0,Y_z).\n\\end{equation}\nThe second conclusion of Proposition \\ref{slpro2} is proved by\ncollecting the limits (\\ref{sllimT1}), (\\ref{sllimT3}) and\n(\\ref{sllimT2}). $\\Box$\n\\paragraph{Acknowledgements.} We wish to thank Professor Mathew Penrose for\nhis important remarks which helped us to derive the present version\nof this work. He mentioned an error in a previous draft for this\nwork, see the Remark following Theorem \\ref{cltips2}. We also thank\nDavid Coupier for his precious comments.\n\\bibliographystyle{plain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDepth First Search (DFS) is a very well-known method for visiting the vertices and edges of a directed or undirected graph. DFS differs from other ways of traversing the graph such as Breadth First Search (BFS) by the following DFS protocol: Whenever two or more vertices were discovered by the search method and have unexplored incident (out)edges, an (out)edge incident on the most recently discovered such vertex is explored first. This DFS traversal produces a rooted spanning tree (forest), called DFS tree (forest) along with assigning an index to every vertex $v$ i.e., the time vertex $v$ is discovered for the first time during DFS. We call it depth-first-index (DFI($v$)). Let $G=(V,E)$ be a graph on $n=|V|$ vertices and $m=|E|$ edges where $V =\\{v_1, v_2, \\cdots, v_n\\}$. It takes $O(m+n)$ time to perform a DFS traversal of $G$ and to generate its DFS tree (forest) with DFIs of all the vertices. The DFS rule confers a number of structural properties on the resulting graph traversal that cause DFS to have a large number of applications. These properties are captured in the DFS tree (forest), and can be used crucially to design efficient algorithms for many basic and fundamental algorithmic graph problems, namely, biconnectivity~\\cite{Tarjan72}, $2$-edge connectivity~\\cite{Tarjan74}, strongly connected components~\\cite{Tarjan72},  topological sorting~\\cite{Tarjan72}, dominators~\\cite{Tarjan741}, {\\it st}-numbering~\\cite{EvenT76} and planarity testing~\\cite{HopcroftT74} among many others.\n\nThere are two versions of DFS studied in the literature. In the lexicographically smallest DFS or lex-DFS problem, when DFS looks for an unvisited vertex to visit in an adjacency list, it picks the \"first\" unvisited vertex where the \"first\" is with respect to the appearance order in the adjacency list. The resulting DFS tree will be unique. In contrast to lex-DFS, an algorithm that outputs some DFS numbering of a given graph, treats an adjacency list as a set, ignoring the order of appearance of vertices in it, and outputs a vertex ordering $Q$ such that there exists some adjacency ordering $R$ such that $Q$ is the DFS numbering with respect to $R$. We say that such a DFS algorithm performs general-DFS. In this work, we focus only on lex-DFS, thus, given a source vertex, the DFS tree is always unique. Given the lex-DFS tree, the non-tree edges of a given directed graph can be classified into four categories as follows. An edge directed from a vertex to its ancestor in the tree is called a back edge. Similarly, an edge directed from a vertex to its descendant in the tree is called a forward edge. Further, an edge directed from right to left in the DFS tree is called a cross edge. The remaining edges directed from left to right in the tree are called anti-cross edges. In the undirected graphs, there are no cross edges. Note that, we can store the complete DFS tree explicitly using $O(n \\lg n)$ bits by storing pointers between nodes. In what follows, we formally define the problem which we call the {\\sf{DFS-Indexing}}\\ problem.\n\n\\begin{center}\n\\fbox{\\begin{minipage}{11cm}\n\\noindent{{\\sf{DFS-Indexing}}\\ problem}\\\\\n\\noindent {\\em Input}: A directed or undirected graph $G=(V,E)$ where $|V|=n$, $|E|=m$, and a source vertex $v_s$, preprocess $G$ and answer the following queries with respect to the DFS tree $T$ rooted at $v_s$:\n\\begin{enumerate}\n\\item Given any pair of vertices $v_i$ and $v_j$,\n\\begin{enumerate}\n\\item Who is visited first in the DFS traversal of $G$?\n\\item Is $v_i$ an ancestor of $v_j$ in $T$?\n\\end{enumerate}\n\\item Given $v_i$, \n\\begin{enumerate}\n\\item Return the parent of $v_i$ in $T$.\n\\item Return the number of children (if any) of $v_i$ in $T$.\n\\item Enumerate all the children (if any) of $v_i$ in $T$.\n\\item Return the DFI of $v_i$.\n\\end{enumerate}\n\\item Enumerate the order in which vertices of $G$ are visited in the DFS.\n\\item Given $1\\leq i \\leq n$, return the vertex with DFI $i$.\n\\end{enumerate}\n\\end{minipage}}\n\\end{center} \n\nWe study the {\\sf{DFS-Indexing}}\\ problem in two well-known models: the {\\it indexing} and {\\it encoding} models~\\cite{Navarro}. In the indexing model, we wish to build an index {\\it ind} after preprocessing the input graph $G$ such that queries can be answered using both {\\it ind} and $G$ whereas in the encoding model, we seek to build a data structure {\\it encod} after preprocessing the input graph $G$ such that queries have to be answered using {\\it encod} only. Typically the parameters of interest are (i) query time, (ii) space consumed (in bits) by {\\it ind} and {\\it encod} resp. and (iii) the preprocessing time and space. We address all these issues in our paper for the {\\sf{DFS-Indexing}}\\ problem, assuming our computational model is a Random-Access-Machine with constant time operations on $O(\\lg n)$-bit words. In both models, it is not hard to see that using $O(n \\lg n)$ bits, we can answer all the queries of the {\\sf{DFS-Indexing}}\\ problem in the optimal $O(1)$ time except the query of 3 which takes $O(n)$ time. \nOur main objective here is to beat this trivial $O(n \\lg n)$ bit space bound without compromising too much on the query time. \n\nThe motivation for studying this question mainly stems from the rise of the ``big data'' phenomenon and its implications. To illustrate, the rate at which we store data is increasing even faster than the speed and capacity of computing hardware. Thus, if we want to use the stored data efficiently, we need to represent it in sophisticated ways. Many applications dealing with huge data structures can benefit from keeping them in compressed form. Compression has many advantages: it can allow a representation to fit in main memory rather than swapping out to disk, and it improves cache performance since it allows more data to fit into the cache. However, such a data structure is only handy if it allows the application to perform fast queries to the data, and this is the direction we want to explore for the DFS tree. More specifically, we are interested in representing the DFS tree of a given graph compactly while supporting all the queries mentioned above efficiently.\n\n\\subsection{Representation of the Input Graph}\nWe assume that the input graphs $G=(V,E)$ are represented using the {\\it adjacency array} format, i.e., $G$ is given by an array of length $|V|$ where the $i$-th entry stores a pointer to an array that stores all the neighbors of the $i$-th vertex. For the directed graphs, we assume that the input representation has both in\/out adjacency array for all the vertices i.e., for directed graphs, every vertex $v$ has access to two arrays, one array is for all the in-neighbors of $v$ and the other array is for all the out-neighbors of $v$. This form of input graph representation has now become somewhat standard and was recently used in plenty of other works~\\cite{Banerjee2018,Cha_thesis,CTW,Chakraborty00S18,ChakrabortyRS17,ChakrabortyS19}. Throughout this paper, we call a graph sparse when $m=O(n)$, and dense otherwise (i.e., $m=\\omega(n)$).\n\n\\subsection{Our Main Results and Organization of the Paper}\nWe start by mentioning some preliminary results that will be used throughout the paper in Section~\\ref{prelim}. Section~\\ref{main_algo} contains the description of our main index for solving the {\\sf{DFS-Indexing}}\\ problem in the indexing model.  Our main results here can be summarized as follows,\n\n\n\\begin{theorem}\\label{sparsecase}\nIn the indexing model, given any sparse (dense resp.) undirected or directed graph $G$, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing algorithm which outputs a data structure of size $O(n)$ ($O(n \\lg (m\/n))$ resp.) bits, using which the queries 1(a), 1(b), 2(d) and 4 can be reported in $O(\\lg n)$ time, 2(a) and 2(b) \nin $O(1)$ time, 2(c) in time proportional to the number of solutions, and finally 3 can be solved in $O(n)$ time resp. for the {\\sf{DFS-Indexing}}\\ problem.\n\\end{theorem} \n\nWe want to emphasize that obtaining better results for sparse graphs is not only interesting from theoretical perspective\nbut also from practical point of view as these graphs do appear very frequently in most of the realistic network scenario in real world applications, e.g., Road networks and the Internet.\n\nIn Section~\\ref{encoding}, we provide the detailed proof of our index in the encoding model. This contains a space lower bound for any index for the {\\sf{DFS-Indexing}}\\ problem, followed by an index whose size asymptotically matches the lower bound and has efficient query time. We summarize our main results~below.\n\n\\begin{theorem}\\label{encoding_lower_bound}\nIn the encoding model, the size of any data structure for the {\\sf{DFS-Indexing}}\\ problem must be $\\Omega(n \\lg n)$ bits. On the other hand, given any (un)directed graph, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing scheme that outputs an index of size $(1+\\epsilon)n \\lg n+2n+o(n)$ bits (for any constant $\\epsilon > 0$), using which the queries 1(a), 1(b), 2(a), 2(b), 2(d) can be reported in $O(1)$ time, 2(c) in time proportional to the number of solutions, 3 in $O(n\/\\epsilon)$ time, and finally 4 in $O(1\/\\epsilon)$ time resp. for the {\\sf{DFS-Indexing}}\\ problem in this setting.\n\\end{theorem}\n\n\nBuilding on all these aforementioned results, we also show a host of applications of our techniques in designing indices for other fundamental graph problems in Appendix~\\ref{appendix}. \nFinally, we conclude in Section~\\ref{conclude} with some open problems and possible future directions to explore further.\n\n{\\bf Remark.} At this point we want to emphasize that our results are more general, i.e., they can be extended to store any arbitrary labeled tree (arising from some underlying graph) along with the mechanism for fast querying. This method is very useful as many graph algorithms (like shortest path, minimum spanning tree, biconnectivity etc) induce a tree structure which is used subsequently during the execution of the algorithm. Hence, we can use our technique to store and query those trees compactly as well as efficiently. Thus, we also believe that our algorithm may find many other potential interesting applications. However, we chose to provide all the details in terms of DFS as DFS is very widely popular graph traversal technique and is used as the backbone for multiple fundamental algorithms, yet there is no explicit indexing scheme for storing DFS tree compactly. In Appendix~\\ref{app}, \nwe show how one can extend these techniques to design indexing schemes for a variety of other classical and fundamental graph problems.\n\n\\subsection{Related Works}\nThere already exists a large body of work concerning compactly representing various specific classes of graphs, for example planar, constant genus graphs etc~\\cite{Acan,BlandfordBK03,FerresSG0N17,MunroN16,MunroR01,Navarro,YamanakaN10}. All of these works are able to store an $n$-vertex unlabeled planar graph in $O(n)$ bits, and some of them even allow for $O(1)$-time neighbor queries. Generally what is meant by unlabeled is that the algorithm is free to choose an ordering on the vertices (integer labels from $1$ to $n$). Our setting here is slightly different as we work with graphs whose vertices are labeled, and matches closely with~\\cite{BarbayAHM07}. Also we want to support more complex queries whereas the previous works only focused on adjacency queries mostly. Even though DFS being such a widely known method, and having many applications, to the best of our knowledge, we are not aware of any previous work focusing on compactly representing the DFS tree with efficient query support.\n\n\n\\section{Preliminaries} \\label{prelim}\n{\\bf Rank-Select.} We make use of the following theorem:\n\\begin{theorem}\\cite{Clark96}\n \\label{staticrs}\nWe can store a bitstring $B$ of length $n$ with additional $o(n)$ bits such that rank and select operations (defined below) can be supported in $O(1)$ time. Such a structure can also be constructed from the given bitstring in $O(n)$ time and space.\n\\end{theorem}\n\nFor any $ a\\in \\{0,1\\}$, the rank and select operations are defined as follows :\n\\begin{itemize}\n \\item $rank_a(B,i)$ = the number of occurrences of $a$\n\n in $B[1,i]$, for $1\\leq i\\leq n$;\n \\item $select_a(B,i)$ = the position in $B$ of the $i$-th occurrence of $a$, for $1\\leq i\\leq n$.\n\\end{itemize}\n\nWhen the bitvector $B$ is sparse, the space overhead of $o(n)$ bits can be avoided by using the following theorem, which will also be used later in our paper. \n\\begin{theorem}\\cite{Navarro}\n\\label{sparse}\nWe can store a bitstring $B$ of length $n$ with $m$ $1$s using $m\\lg(n\/m)+O(m)$ bits such that $select_1(B,1)$ can be supported in $O(1)$ time, $select_0(B,1)$ in $O(\\lg m)$ time, and both the rank queries ($rank_1(B,i)$ and $rank_0(B,i)$) can be supported in $O(\\text{min}(\\lg m,\\lg n\/m))$ time. Such a structure can also be constructed from $B$ in $O(n)$ time and space.\n\\end{theorem}\n\n{\\bf Permutation.} We also use the following theorem:\n\\begin{theorem}\\cite{MunroRRR12}\n\\label{perm}\nA permutation $\\pi$ of length $n$ can be represented using $(1+\\epsilon)n \\lg n$ bits so that $\\pi(i)$ is answered in $O(1)$ time and $\\pi^{-1}$ in time $O(1\/\\epsilon)$ for any constant $\\epsilon > 0$. Such a representation can be constructed using $O(n)$ time and space.\n\\end{theorem}\n\n{\\bf Succinct Tree Representation.} We need following result from~\\cite{FarzanM14}.\n\\begin{theorem}~\\cite{FarzanM14}\n\\label{succ_tree}\nThere exists a data structure to succinctly encode an ordered tree with $n$ nodes using $2n+o(n)$ bits such that, given a node $v$, (a) child($v$,$i$): $i$-th child of $v$, (b) degree($v$): number of children of $v$, (c) depth($v$): depth of $v$, (d) $select_{pre}$($v$): position of $v$ in preorder, (e) $LA(v,i)$: ancestor of $v$ at level $i$ can be supported in $O(1)$ time among many others. Such a structure can also be constructed in $O(n)$ time and space.\n\\end{theorem}\n\n\n\n\n\n\\section{Algorithms in the Indexing Model}\\label{main_algo}\nIn this section, we provide the main algorithmic ideas needed for the solution of the {\\sf{DFS-Indexing}}\\ problem in the indexing model. We start by describing the preprocessing procedure which is followed by the query algorithms.\n\\subsection{Preprocessing Step}\nWe first describe our algorithms for undirected graphs, and later mention the modifications required for the case of directed graphs. The preprocessing step of the algorithm is divided into two parts. In the first part, we perform a DFS of the input graph $G$ along with storing some necessary data structures. In the second step, we perform a partition of the DFS tree of $G$ using the well-known ``tree covering technique'' of the succinct data structures world~\\cite{FarzanM11}, and also store some auxiliary data structures. Later, in the final step of our algorithm, we show how to use these data structures to answer the required queries. In what follows, we describe each step in detail.\n\n{\\bf Step 1: Creating Parent-Child Array using Unary Degree Sequence Array.} The main idea of this step is to perform a DFS traversal of $G$ and store in a {\\it compact way} the parent-child relationship of the DFS tree $T$. The way we achieve this is by using three bitvectors of length $O(m+n)$ bits.\nRecall that, our input graphs $G=(V,E)$ are represented using the standard adjacency array. Central to our preprocessing algorithm is an encoding of the degrees of the vertices in unary. As usual, let $V =\\{v_1, v_2, \\cdots, v_n\\}$ be the vertex set of $G$. The unary degree sequence encoding $D$ of the undirected graph $G$ has $n$ $1$s to represent the $n$ vertices and each $1$ is followed by a number of $0$s equal to its degree. Moreover, if $d$ is the degree of vertex $v_i$, then $d$ $0$s following the $i$-th $1$ in the $D$ array corresponds to $d$ neighbors of $v_i$ (or equivalently the edges from $v_i$ to the $d$ neighbors of $v_i$) in the same order as in the adjacency array of $v_i$. Clearly $D$ uses $n+2m$ bits and can be obtained from the neighbors of each vertex in $O(m+n)$ time.  Now using {\\it rank\/select} queries of Theorem~\\ref{staticrs} in Section~\\ref{prelim}, the $j$-th outgoing edge of vertex $v_i$ can be identified with the position $p = select_1(D,i)+j$ of $D$ ($1 \\le j \\le degree(v_i)$ where $degree(v_i)$ denotes the degree of the vertex $v_i$). From a position $p$, we can obtain an endpoint of the corresponding edge by $i = rank_1(D, p)$, and the other endpoint is the $j$-th neighbor of $v_i$ where $j = p - select_1(D, i)$.\n\nWe also use two bitvectors $E, P$ of the same length where every bit is initialized to $0$, and the bits in $E, P$ are in one-to-one correspondence with bits in $D$. The bitvector $E$ will be used to mark the tree edges of the DFS tree $T$, and the bitvector $P$ to mark the unique parent of every vertex in $T$.\nThe marking is carried out while performing a DFS of $G$ in the preprocessing step. I.e., if $(v_i, v_j)$ is an edge in the DFS tree where $v_i$ is the parent of $v_j$, and suppose $k$ is the index of the edge $(v_i, v_j)$ in $D$, then the corresponding location in $E$ is marked as $1$ during DFS. At the same time, we scan the adjacency array of $v_j$ to find the position of $v_i$ (as $G$ is undirected, there will be two entries for each edge in the adjacency array), and suppose $t$ is the index of the edge $(v_j,v_i)$ in $D$, then the corresponding location in $P$ is marked as $1$ during DFS. Thus, assuming $G$ is a connected graph, once DFS finishes traversing $G$, the number of ones in $E$ is exactly the number of tree edges (which is $n-1$) and the number of ones in $P$ will be $n-1$ as root does not have any parent. \n\nThe parent of $v_i$ in $T$ is computed in $O(1)$ time\nas follows.\nLet $v_r$ be the root of $T$.  Then if $i > r$\n(resp. $i<r$), the marked bit representing the parent of\n$v_i$ is the $(i-1)$-st (resp. $i$-th) $1$ in $P$.\nLet $p = select_1(P, i-1)$ (resp. $p = select_1(P, i)$)\nand $j = p - select_1(D, i)$.  Then the parent of $v_i$\nis the $j$-th neighbor of $v_i$.\n\nWe use another bitvector $D_T$ of length $2n$,\nwhich encodes the degree of each vertex in $T$\nby unary sequences.  Then the degree of vertex $v_i$ in $T$\nis $select_1(D_T, i+1)-select_1(D_T, i)-1$, and\n$j$-th child of $v_i$ in $T$ is $p$-th neighbor of $v_i$\nin $G$ where $p = select_1(E, select_1(D_T, i-1)+j)-select_1(D, i)$.  These are computed in constant time.\n\n\nNote that, the classical linear time implementation of DFS~\\cite{CLRS} uses a stack (which could grow to $O(n \\lg n)$ bits) and a color array (of size $O(n)$ bits). Thus, the procedure takes $O(m+n)$ time and $O(n \\lg n)$ bits overall. First, we argue that using the same linear time, we can also create bitvectors $D, E$ and $P$ and fill up them correctly. It's easy to see that creating $D$ as well as initializing $E$ and $P$ to all zero takes $O(m+n)$ time. All it remains is to show, how one can fill up $E$ and $P$ while performing DFS. For this purpose, we build the data structures to support the constant time {\\it rank\/select} query (of Theorem~\\ref{staticrs}) on $D$ (and on $E, P$ as well, the reason will be clear in the query step) and use the result of the select query to mark the tree edges on $E$ (as they are in one-to-one correspondence). To illustrate, suppose, while traversing from $v_i$, DFS discovers the edge $(v_i, v_j)$ as a tree edge in $T$ where $v_i$ is the parent of $v_j$, and suppose $v_j$ is the $c$-th neighbor in $v_i$'s adjacency array, then we find the index of the $c$-th zero after $i$-th one in $D$ (using select query), and the corresponding index is marked as $1$ in the $E$ array. This takes $O(1)$ time for each tree edge marking. After this, we mark the index in $P$ as $1$ corresponding to the edge $(v_j,v_i)$ to denote that $v_i$ is the parent of $v_j$. Thus, marking parent takes $O(degree({v_j}))$ time for the vertex $v_j$. Note that, all of this happens along with the classical stack-based DFS implementation. Thus overall it takes $O(m+n)$ time, and space required to store all these arrays is $O(m+n)$ bits. We refer to the bitvector $D$ as the unary degree sequence array, $E$ as the child array, and $P$ the parent array. These three arrays are stored and used for the query step of our algorithm. Thus, we obtain the following lemma.\n\n\\begin{lemma}\nGiven an undirected graph $G$, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing algorithm to construct the unary degree sequence array, parent and child arrays for $G$, each of which takes $O(m+n)$ bits of space.\n\\end{lemma}\n\n\n{\\bf Step 2: Decomposing the DFS tree by the Tree Covering Technique.} The main idea of this step is to perform a decomposition of the DFS tree, and along with storing some crucial informations which will be very useful for navigating the tree during the query step of our algorithm. For this purpose, we use the well-known tree covering technique in the context \nof succinct representation of rooted ordered trees. The high level idea is to decompose the tree into subtrees called \n{\\it minitrees}, and further decompose the minitrees into yet smaller subtrees called {\\it microtrees}. The microtrees are small enough to be stored in a compact table. The root of a \nminitree can be shared by several other minitrees. To represent \nthe tree, we only have to represent the connections and links between the subtrees. One such tree decomposition method was given by Farzan and Munro~\\cite{FarzanM11} where each minitree has at most one node, other than the root of the minitree, that is connected to the root of another minitree. This guarantees that in each minitree, there exists at most one non-root node which is \nconnected to (the root of) another minitree. We use this decomposition in our algorithms, and the main result of Farzan et al.~\\cite{FarzanM11} is summarized in the following theorem:\n\n\\begin{theorem}[\\cite{FarzanM11}]\\label{thm:tree-decomposition}\nFor any parameter $L \\ge 1$, a rooted ordered tree with $n$ nodes can be decomposed into $\\Theta(n\/L)$ minitrees of size at most $2L$ which are pairwise disjoint aside from the minitree \nroots. Furthermore, aside from edges stemming from the minitree root, there is at most one edge \nleaving a node of a minitree to its child in another minitree. The decomposition can be performed \nin linear time using linear words of space.\n\\end{theorem}\n\n\\begin{figure}[bt]\n\\begin{center}\n \\includegraphics[scale=.45, keepaspectratio=true]{TC}\n\\end{center}\n\\caption{An example of Tree Covering technique with $L=5$. \nEach closed region formed by the dotted lines represents a minitree. Here each minitree has at most one `child' minitree (other than the minitrees that share its root).}\\label{fig:1}\n\\end{figure}\n\nSee Figure~\\ref{fig:1} for an illustration. For the purpose of our algorithms, we apply Theorem~\\ref{thm:tree-decomposition} with $L =\\lg n$ on the DFS tree $T$ of $G$. For this parameter $L$, since the number of minitrees is only $O(n\/\\lg n)$, we can represent the structure of the minitrees within the original tree (i.e., how the minitrees are connected with each other) \nusing $O(n)$ bits by simply storing both way pointers (so that we can traverse easily) between the roots of the minitrees. We refer to this as the {\\it skeleton} $S$ of the DFS tree $T$. See Figure~\\ref{fig:2} for a demonstration of Figure~\\ref{fig:1}'s skeleton. The decomposition algorithm of~\\cite{FarzanM11} also ensures that each minitree has at most one `child' minitree (other than the minitrees that share its root) in this structure. We use this property crucially later. \n\n\\begin{figure}[bt]\n\\begin{center}\n \\includegraphics[scale=.7, keepaspectratio=true]{skeleton}\n\\end{center}\n\\caption{(a) A rough sketch of the skeleton of the tree decomposition shown in Figure~\\ref{fig:1}. In this diagram, the triangles represent the minitrees along with the roots of the minitrees are marked inside the circle. For example, the minitrees $M_1$ and $M_2$ share the same root $b$. Also the node $o$ is a minitree on its own. Strictly speaking, the skeleton will not have the traingles, rather it just contains the pointers between the roots of the minitrees (i.e., circles in this diagram). But we put this diagram for better visual description of the compact representation of the previous diagram.}\\label{fig:2}\n\\end{figure}\n\nIn what follows, we explain how we compactly represent the minitree structure, and we refer to this compact representation obtained using this tree covering (TC) approach as the TC representation of the DFS tree. Towards this, first observe that every minitree root has unique first child and last child inside the minitree. In some cases, both are the same (see the minitree rooted at node $d$ of Figure~\\ref{fig:1}), and in some cases, both are absent (see the minitree rooted at node $o$ of Figure~\\ref{fig:1}). Thus, if we specify these two quantities, we can uniquely identify the root of the minitree (along with the exact portion of the nodes which are children of the root of this minitree and also belong to the same minitree as the first and last child of the root) even though the root is shared between multiple minitrees. We use this idea crucially in the design of the TC representation of the DFS tree. \n\nWe mark in a bitvector $R$ of size $n$ all the nodes which are the last child of a minitree root inside a minitree. Note that, there are \n$O(n\/\\lg n)$ such nodes which are marked as $1$ in $R$. In the case of a minitree root not having any children, we mark the minitree root itself as $1$ in $R$. We also build the data structure to support $O(1)$ time {\\it rank\/select} queries on $R$ using Theorem~\\ref{staticrs}. Next, we create an array $C$ where each of the $O(n\/\\lg n)$ entries are $O(\\lg n)$ bits long, thus overall it takes $O(n)$ bits. Basically, each entry of $C$ stores some informations regarding the minitree for which the last child of the minitree root is marked $1$ in $R$. More specifically, For a typical node, say $v_i$, which is the last child of some minitree, we have $R[i]=1$, and $C[j]$ (where $j=rank_1(R,i)$) comprises of the following six informations (some of which could be empty), (i) label of the minitree root, say $v_r$, for which $v_i$ is the last child inside the minitree, (ii) location of the first child, say $v_j$, of $v_r$ inside the minitree in the adjacency array of $v_r$, (iii) DFI of $v_j$, (iv) the edge $(v_c,v_d)$ (if any) that goes out of the minitree, (v) the size of the subtree rooted at $v_c$ in the DFS tree, (vi) depth of $v_r$ in the DFS tree. The tree decomposition method ensures that a minitree has at most one edge $(v_c,v_d)$, where $v_c$ is a non-root node of minitree and $v_d$ is a root of a different minitree, that goes out of the minitree. We also mark in a bitvector $Z$ of size $n$ bits all such vertices like $v_c$ (also note, there could be $O(n\/\\lg n)$ such vertices). We mark in a bitvector $L$ all the vertices which are the rightmost leaves of every minitree. Note that these vertices (there are, again, $O(n\/\\lg n)$ of them) have the highest DFI inside the minitree. In another bitvector $A$, we mark all the roots of the minitrees as $1$, and build rank\/select structure on top of $A$. Correspondingly, the $F$ array will store the DFI of the roots so that we can retrieve them in constant time. More specifically, for a minitree root $v_r$, $A[r]=1$ and $F[j]$ ($j=rank_1(A,r)$) will store the DFI of $v_r$. Next we build the $O(1)$ time level ancestor data sturcture, say $LA$, on the $O(n\/\\lg n)$ minitree roots (i.e., on the skeleton structure) using~\\cite{BenderF04}. Thus, here, $LA$ takes $O(n)$ bits and $O(n)$ time. As a root of the minitree is shared between multiple minitrees, from each node $v_i$ of the skeleton $S$ (where $v_i$ is a root of a minitree), we store pointers to all the minitrees (in $C$ array) which has $v_i$ as their root. Overall these pointers also consume $O(n)$ bits. This completes the description of the TC representation. Note that, the creation of the skeleton and the TC representation for $T$ can be done in $O(n)$ time using $O(n \\lg n)$ bits (using Theorem~\\ref{thm:tree-decomposition}) after the DFS (which takes $O(m+n)$ time and $O(n \\lg n)$ bits). Hence, we obtain the following,\n\n\\begin{lemma}\nGiven an undirected graph $G$, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing algorithm to construct the skeleton $S$ and the TC representation of the DFS tree $T$ of $G$, each of which occupies $O(n)$ bits.\n\\end{lemma}\n\nFirst observe that, the outputs of the previous step are the unary degree sequence array ($D$), parent array ($E$), child  array ($P$), the $D_T$ array, TC representation of $T$ (this includes $R$, $C$, $Z$, $A$, $F$ and $L$) along with the skeleton $S$ with pointers to $C$, and finally the $LA$ structure on $S$. The arrays $D$, $E$, and $P$ take $O(m+n)$ bits, and the others take $O(n)$ bits. Now we show how to efficiently solve the {\\sf{DFS-Indexing}}\\ problem using these structures. \n\n{\\bf Query Algorithms.} Given $v_i$, to answer 2(a) in $O(1)$ time, we do the following. If $v_i$ is the root of the DFS tree, we return null. \nOtherwise, we can compute the answer by using\nonly $select_1$ queries on $P$ and $D$, as described previously.\n\nTo answer 2(b) in $O(1)$ time, \nwe use $select_1$ queries on the bitvector $D_T$.\n\nTo answer 2(c), \nwe first compute the number of children of $v_i$ in $T$\nusing the query 2(b).  Then $j$-th child is obtained\nin constant time as described above.\n\nNote that, the queries 2(a), 2(b) and 2(c) can be answered using only $D$, $E$, $P$ and $D_T$ arrays. Before explaining the algorithms for the rest of the queries, we first prove\nthe following very crucial lemma.\n\\begin{lemma}\\label{reconstruct}\nGiven any query node $v_i$ which is not a root of a minitree, we can reconstruct the minitree $M$ containing $v_i$ in time proportional to the size of $M$ along with the DFIs of all the nodes inside $M$. In the same amount of time, we can also retrieve the root node of $M$.\n\\end{lemma}\n\n\\begin{proof}\nFirst note that if a node $v$ belongs to the minitree $M$,\nits children in $T$ also belong to $M$, except for the\nfollowing two cases.  The first case is that $v$ is the\nroot $v_r$ of $M$ and the second case is that $v$ is $v_c$\nof $M$.  In the first case, as we have stored the location (in the adjacency array) of the first child, say $v_j$, of $v_r$ inside $M$ in the $C$ array, we can enumerate all the children of $v_r$ in $M$\nin constant time for each until we hit the rightmost child\nof $v_r$ in $M$, which is stored in $R$.\nIn the second case, we can also enumerate all the children\nof $v_c$ in $T$ in constant time for each and discard $v_d$.\nFor other vertices in $M$, we can enumerate children\nusing constant time for each output.  Note also that\ngoing to the parent can be performed in constant time.\n\nThe algorithm to achieve the claim can be broken down into three steps. In the first step, given $v_i$, we launch a DFS starting from $v_i$, and continue till we retrieve the rightmost leaf, say $v_j$, of the minitree $M$. In second step, we follow the path in $T$ (by going to $v_j$'s parent, then its parent and so on) till we reach the rightmost child, say $v_k$, of the root, say $v_r$, inside $M$ by using the query algorithm to find the parent repeatedly. In the third and final step, we use the $C$ array, by using $v_k$, to extract all the informations needed to reconstruct the full minitree by performing another round of DFS. We provide the details below.\n\nTo perform the first step, we only need to use the parent and child related queries, whose execution we already showed previously. Note that, as we have stored the information (in $Z$ array) regarding the only edge that goes out of the minitree, we never incorrectly go out of $M$. Also we can verify if we have reached the unique node $v_j$ which is the rightmost leaf of $M$ from the $L$ array. Once we reach $v_j$, it's easy to see that $v_k$ has to be an ancestor of $v_j$ (note that $v_k$ and $v_j$ could be same in some cases). Thus, we can reach $v_k$ from $v_j$ by repeatedly using the parent query algorithm, and this completes the second step. Finally, once we reach $v_k$, we use the informations in $C[j]$ (where $j=rank_1(R,k)$) to retrieve the root $v_r$ of $M$ and other informations. Then we carry out a DFS from $v_r$ by first going to the first child of $v_r$ inside $M$ (retrieved from $C$), then its first child and so on till we fully reconstruct $M$. This step also requires repeated invoking of parent and child query only.\n\nIn order to retrieve the DFIs of the nodes inside $M$, observe that, if $M$ doesn't have any child minitree (i.e., no edge is going out of $M$), then while doing the final DFS from $v_r$, we can easily compute the DFIs of all the nodes inside $M$. Otherwise, assume the edge $(v_c,v_d)$ goes out of $M$ where $v_c$ belongs to $M$, then the DFI of next node inside $M$ can be calculated by adding the size of the subtree rooted at $v_c$ in $T$ (which is stored in $C$) to the DFI of $v_c$. It is clear that all of these procedures can be performed in the time proportional to the size of $M$, which is $O(\\lg n)$ here. This completes the description of the proof.\n\\end{proof}\n\nAs a corollary of the previous lemma, it is easy to see that the query of 2(d) can be reported in $O(\\lg n)$ time for any node $v_i$ which is not a root of the minitree. Otherwise, it can be done in $O(1)$ time by reporting the value stored in $F[j]$ where $j=rank_1(A,i)$. \n\nTo answer 1(a), first we invoke query algorithm of 2(d) for both $v_i$ and $v_j$ to retrieve their DFIs respectively, and then answer accordingly. Thus, this also takes $O(\\lg n)$ time.\n\nAnswering 1(b) involves a few cases. In the first case, if both of them belong to the same minitree then we can figure out the answer by reconstructing the complete minitree. Secondly, suppose $v_i$ and $v_j$ are roots of the two separate minitrees, and their depths in $T$ are $x$ and $y$ respectively (depth can be obtained from $C$ array). Then using these values in $LA$ data structure, we can figure out the required answer. Finally, if both of these nodes belong to two separate minitrees but are not the roots of the minitrees, then first we retrieve the roots of those minitrees using Lemma~\\ref{reconstruct}, then follow almost the same procedure as before to figure out the answer. Note that, in this case, it is enough to reconstruct the path from $v_c$ (of the minitree located near to the root) to the root of that minitree (for the case when one of the minitree root is an ancestor of the other minitree root) to figure out the answer of the query. Thus, overall, it takes $O(\\lg n)$ time.\n\nTo return the query for 3, we do a standard DFS traversal on the skeleton $S$ and each time we visit a new node $v_i$ in $S$, we follow the pointer from $v_i$ in $S$ to the part of the $C$ array where the informations regarding the minitree rooted at $v_i$ is stored. Note that, $v_i$ might be shared between multiple minitrees, hence, we always start following these pointers from left to right. More specifically, if $v_i$ is the root of $p$ minitrees, we have $p$ pointers emanating from $v_i$, and going to $p$ different locations of $C$ array. As these pointers are stored from left to right order, which is the same order in DFS of all the minitrees that share the root $v_i$. Thus we follow the first pointer, and reach the specific portion of $C$, use Lemma~\\ref{reconstruct} to generate the complete minitree along with the DFIs of the nodes. Then if this minitree has any child minitree, we go on to explore that and so on (by following the $(v_c,v_d)$ edge stored in that minitree). Once we finish all the descendant minitree of the first minitree rooted at $v_i$, we come back and start exploring the second minitree (by following the second pointer from $v_i$) and so on. Thus, we need to store these intermediate pointers, in stack, to know how much progress has been made in every node's (in skeleton) list. This procedure is continued until all the nodes of $S$ are exhausted. It is clear that this procedure takes $O(n)$ time as there are $O(n\/\\lg n)$ nodes in $S$ and for each node, we spend $O(\\lg n)$ time. Also, we need $O(n)$ bits (as there could be $O(n\/\\lg n)$ pointers) of intermediate space for the execution of the DFS. \n\nTo answer 4, first note that, in any minitree $M$, if there is no egde going out (i.e., no $(v_c,v_d)$ type edge), then the DFIs inside $M$ are consecutive, i.e., in general, first child of root inside $M$ has the smallest DFI and the rightmost leaf in $M$ has the maximum DFI, and the numbers are consecutive. Otherwise, DFIs are consecutive from the root of $M$ to the DFI of $v_c$, then there is a jump of DFI by the size of the subtree rooted at $v_c$ in the DFS tree, then it's consecutive DFI again until the rightmost leaf (which has the largest DFI inside $M$) of $M$. Thus, the range of DFIs of the vertices inside any arbitrary minitree $M$ can be broken into at most two disjoint consecutive intervals. We store these (at most $O(n\/\\lg n)$) intervals in an interval tree along with augmenting it with the last child of $M$ inside $M$. Now, given $i$, we first find the interval where $i$ belongs to from the tree and simultaneously retrieve the last child, say $v_a$, of the corresponding minitree, all using $O(\\lg n)$ time. Then, we use the information from $R$ and $C$ array corresponding to $v_a$ to invoke Lemma~\\ref{reconstruct}, and retrieve the desired vertex with DFI $i$ using $O(\\lg n)$ overall time. This completes the description of the query algorithms for undirected graphs. \n\nWe can handle directed graphs similarly except a few changes in the data structures. Recall that, for directed graphs, every vertex $v_i$ has access to its in-neighbors array as well as out-neighbors array, and additionally we create two unary degree sequence arrays (each of size $O(m+n)$ bits), $D_1$ for the out-neighbors and $D_2$ for the in-neighbors. We also have two separate arrays, say $E_1$ (having one-to-one map with $D_1$), for marking child of every node and $E_2$ (having one-to-one map with $D_2$) where parents are marked. It is easy to see that almost in a similar fashion as in the undirected case, we can correctly mark, for any node $v_i$, the children of $v_i$ in $E_1$ array and parent of $v_i$ in $E_2$ array using both the $D_1$ and $D_2$ arrays while performing DFS of $G$. The second preprocessing step doesn't require any changes for the  directed graphs. Now reporting queries also can be suitably modified to make use of these changes without affecting the asymptotic running time of the query algorithms. Basically the only change that takes place is as follows, whenever we need to find the parent of a node, now we need to use the in-neighbor array whereas finding children can be handled by consulting out-neighbor array along with the mapping with their respective unary degree sequence array. We omit the details. Thus, we obtain the following,\n\n\\begin{theorem}\\label{main_theorem}\nGiven any undirected or directed graph $G$, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing algorithm which outputs a data structure of size $O(m+n)$ bits, using which the queries 1(a), 1(b), 2(d) and 4 can be reported in $O(\\lg n)$ time, 2(a) and 2(b) in $O(1)$ time, 2(c) can be answered in time proportional to the number of solutions, and finally 3 can be solved in $O(n)$ time respectively for the {\\sf{DFS-Indexing}}\\ problem.\n\\end{theorem}\n\nNote that, if the given input graph is sparse (i.e., $m=O(n)$), then both unary degree sequence array ($D$), and parent and child arrays ($E, P$) take $O(n)$ bits, and every other data structure anyway takes $O(n)$ in total, thus, we obtain the result mentioned in Theorem~\\ref{sparsecase} for the case of sparse graphs. When the input graph is dense (i.e., $m=\\omega(n)$), we compress the $D, E, P$ arrays using Theorem~\\ref{sparse}. Note that we use only $select_1$ queries on compressed\narrays and thus query time complexity on the arrays\nis still constant. Hence, we obtain the result of Theorem~\\ref{sparsecase} for the dense graph case. It is worth mentioning that except the case for very dense graphs, our space bound {\\it always} beats the space bound of the naive algorithm for every edge density in the full spectrum, albeit with super-constant query time. Thus, when the graph is sufficiently dense, it is better to use the standard solution which uses $O(n \\lg n)$ bits with constant query time. This completes the description of our algorithms in the indexing model, and hence, the proof of Theorem~\\ref{sparsecase}.\n\n\n\\section{Algorithms in the Encoding Model}\\label{encoding}\nRecall that in the encoding model, we seek to build a data structure {\\it encod}\nafter preprocessing the input graph $G$ such that queries have to be answered using {\\it encod} only, {\\it without} accessing $G$. To this end, we first provide a lower bound for the space needed for {\\it encod} to answer queries of the {\\sf{DFS-Indexing}}\\ problem.\n\n\\subsection{Space lower-bound}\nObserve that, in order to correctly answer the queries, the data structure {\\it encod} must contain the information regarding the topology of the DFS tree $T$ of the graph $G$ along with the labels of the vertices of $T$ as, unlike the indexing model, we don't have the access to $G$ during the query time. It's easy to see that we need $\\Omega(n \\lg n)$ bits to store the vertex labels mappings. In what follows, we give a proof for the space needed to store the topology of the DFS tree by counting the number of such trees in any arbitrary graph $G$.\n\n\\begin{lemma}\nFor a graph with $n$ vertices and $m$ edges, the size of data structures\nfor storing the topology of the DFS trees is $\\Omega\\left(n \\lg \\frac{m}{n}\\right)$ bits.\n\\end{lemma}\n\n\\begin{proof}\nLet us consider the following graph $G$ with $n$ vertices and $m$ edges ($m < n^2\/2$).\nIt has a vertex $r$, $k = m\/n$ vertices $u_1,\\ldots,u_k$,\nand $n-k-1$ vertices $v_1,\\ldots, v_{n-k-1}$.\nThe vertex $r$ is connected to all $u_i$, and each $v_j$ is also connected to\nall $u_i$.  To construct a spanning tree of $G$, we choose one edge among all\n$k$ edges connected to each $v_j$.  Then the number of different spanning trees of $G$ is at least $k^{n-k-1}$, and for all different spanning trees the set of DFI's are different. Therefore the size of data structure must be\nat least $\\lg k^{n-k-1}$ bits, which is~$\\Omega\\left(n \\lg \\frac{m}{n}\\right)$.\n\\end{proof}\nThus, the space lower bound for {\\it encod} is $\\Omega(\\textit{max}\\{n \\lg \\frac{m}{n}, n \\lg n\\})$ bits, which is $\\Omega(n \\lg n)$ bits as mentioned in Theorem~\\ref{encoding_lower_bound}. In what follows, we complement the above claim by providing a simple indexing structure which asymptotically matches this lower bound.\n\n\\subsection{Upper-bound}\n{\\bf Preprocessing Step.} Our index for the {\\sf{DFS-Indexing}}\\ problem consists of two components which we prepare during the preprocessing step. In the first component, we store, for every vertex $v_i$, DFI($v_i$) as permutation using the structure of Theorem~\\ref{perm} of Section~\\ref{prelim}. Secondly, we encode the DFS tree succinctly using the structure of Theorem~\\ref{succ_tree} of Section~\\ref{prelim}.\n\n{\\bf Query Algorithm.} We answer the queries using the two above mentioned structures as follows. To answer 2(d), we just use $\\pi(i)$. \nSimilarly, 4 can be answered by invoking $\\pi^{-1}(i)$. We report $v_i$ (resp. $v_j$) as the answer for query 1(a) if $\\pi(i) < \\pi(j)$ (resp. $\\pi(i) > \\pi(j)$). We enumerate the vertex ordering as traversed in the DFS order by invoking $\\pi^{-1}(1)$, then $\\pi^{-1}(2)$, and so on till $\\pi^{-1}(n)$. We answer 1(b) in affirmative by checking if $LA(v_j,depth(v_i))$ matches with $v_i$, otherwise no. To answer 2(a), we return $LA(v_i,depth(v_i)-1)$. We return the answer of 2(b) by using the query $degree(v_i)$. Finally, we enumerate the children of a node $v_i$ as requested in query 2(c) by using the query $child(v_i,1)$ till $child(v_i,degree(v_i))$. Hence we obtain the results mentioned in Theorem~\\ref{encoding_lower_bound}.\n\n\n\\section{Conclusion}\\label{conclude}\nIn this paper, we provided procedures for compactly storing the DFS tree for any graph with efficiently supporting various queries in the indexing and encoding models, and showed how to extend these techniques to design indexing schemes for other fundamental and basic graph problems. With some work, our algorithm can be extended for indexing BFS tree (and other graph search tree also) as well while supporting similar types of queries. Also, as mentioned previously, our results are more general, and can be used in other situations as well. \n\n\nThis work opens up many possible future directions to explore. Can we further improve the query time while keeping the space bound same in the indexing model? Can we prove a space lower bound in the indexing model? Can we design compact data structures for indexing problems like maximum flow?\nFinally, we conclude by remarking that using \n\\cite{Banerjee2018,ChakrabortyRS17}, we can improve the preprocessing space of our algorithms\nto $O(n)$ bits (from $O(n \\lg n)$ bits) with marginal\nincrement in the preprocessing time. \n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis is a contribution to the 3rd International Symposium about Quantum Mechanics based on a ``Deeper Level Theory'', but the message I want to present is that there is no need for a deeper theory. In my view bare quantum mechanics provides a good explanation of all that we see around us. In fact, it seems that this series of the Symposia leads in this direction. The  quote from the  objectives of the EmerQuM (2011) reads:\n\\begin{quote}\nThe theme of ``emergent quantum mechanics'' is, we believe, an appropriate present-day topic, which can both serve as ... a means to point towards promising future directions in physics. We intend to bring together many of those physicists who are interested in or work on attempts to understanding quantum mechanics as emerging from a suitable classical (or, more generally: deeper level) physics...\n\\end{quote}\nThe objectives of EmQM13 (2013) were not very different:\n\\begin{quote}\nThe symposium invites the open exploration of an emergent quantum mechanics, a possible ``deeper level theory'' that interconnects three fields of knowledge: emergence, the quantum, and information. Could there appear a revised image of physical reality from recognizing new links between emergence, the quantum, and information? Could a novel synthesis pave the way towards a 21st century, ``super-classical'' physics? ...\n\\end{quote}\nHowever, the first objective  of EmQM15 (2015) is changed:\n\\begin{quote}\nThe symposium invites the open exploration of the quantum state as a reality. The resurgence of interest in ontological quantum theory, including both deterministic and indeterministic approaches, challenges long held assumptions...\n\\end{quote}\n\nI believe the quantum state is a reality. It is the reality. The only fundamental physical ontology is the quantum wave function.\n\nPlease allow me a sociology of science speculation for why this opinion is not in the consensus today. At the birth of quantum mechanics there was at least a decade of quantum mechanics without the wave function and the Scr\\\"odinger equation of its evolution. During this time Bohr with his charismatic behaviour convinced the physics community that the reality behind quantum mechanics cannot be grasped by human beings. A half century later,  physicists working on the foundations of quantum mechanics expressed growing dissatisfaction with Bohr's view, they  followed Bell \\cite{BellBeab} in search of ``local beables''. The ingenious model of Spekkens \\cite{Spek} gave a hope that our familiar everyday reality will  emerge from a ``deeper level theory''.\n\nHowever, the research triggered by these developments cooled down the expectations. It suggested that whatever the ``deeper theory'' is, it cannot be simpler than the standard quantum theory \\cite{Leifer} and it is strongly suggested that the quantum wave function is ``ontic'' \\cite{PBR,Hardy}, i.e., it is the ontology of quantum mechanics.\n\nMoreover, it was claimed that just a ``free will'' assumption is enough for a simple proof of an ontic nature of the quantum wave function \\cite{CR}. In my view, however, this argument is a metaphysical error. In science we {\\it assume} ``free will'' of an external agent who tests  theories we construct, so nothing can be derived from the ``free will'' assumption.\n\n It is considered ``politically incorrect'' to claim that physicists, at large, understand Nature. This was the view a little more than a century ago, the view which was proved to be very wrong. Two revolutions in physics took place in the last century: relativity theory and quantum theory. Classical physics provided very plausible explanations of many observable phenomena, but it was shown to be wrong in more precise experiments.\n\n Today there are no paradoxes similar to those encountered by classical physics. However, the ontology of the quantum wave function seems very far from what we observe with our senses, and this is what led the majority to believe that it cannot be the true description of Nature. I, however, think that the classical understanding of Nature at the beginning of the last century is roughly correct also for quantum theory. An atom is not made of point particles moving on definite trajectories, but  is a highly entangled quantum wave of electrons, protons, etc. This quantum wave is, however,  not relevant for the interaction with other systems, it exerts forces on other systems very much like a point-like body. The wave structure provides an explanation of the spectrum, which classical mechanics does not, but for explaining pressure or other macroscopic features, classical approximations work fine.\n\nIn the following I will present the view that all that exists is the wave function of the Universe which evolves according to the Schr\\\"odinger equation and without collapses during quantum measurements. This is the Many-worlds interpretation (MWI) of quantum mechanics \\cite{Everett}.\n\n\n\n\\section{Many worlds in a single Universe}\n\n\n\n\n In my view \\cite{SEP} the MWI is  the only interpretation which  can be viewed as  a possible final theory of physics. It is a deterministic theory  without action at a distance as all physics we know today. It  is  (still) not widely accepted since it requires a radical change in our view of Nature: there are many worlds which exist in parallel at the same space and time as our own.\n\nAs a physical theory, the MWI is a theory about the Wave Function of the physical Universe. The equations are those of the standard quantum mechanical formalism: the Schr\\\"odinger equation or its relativistic generalizations. The theory postulates that the only ontology is this Wave Function which evolves in a unitary way, i.e., there is no collapse of the Wave Function. Since the collapse process is the only random element in physics and also the only source for action at a distance, randomness and action at a distance are eliminated from physics in the MWI.\n\nClearly, the most general and complete description of  Nature is not the Wave Function of all particles in the Universe. Such a picture does not describe the creation and annihilation of particles and field theory is required. In such a framework a wave functional describing the amplitude for the local fields  at each point in space replaces the  Wave Function. However, I do not foresee that this entails significant conceptual changes or difficulties.  Surprisingly, little effort has been made to look for a rigorous definition of such a functional. Appreciation of the success of the MWI in resolving the paradoxes of  standard quantum mechanics may lead to further research in this direction.\n\nIn spite of the name, I view the MWI as a theory of a single Wave Function of a single physical Universe. This is the only ontology. Observables, which are frequently considered as basic building blocks of the standard formalism (and which are the\nbasis of algebraic approaches to quantum theory) are not part of the ontology. Thus, the uncertainty relations of quantum observables do not lead to randomness, the theory remains deterministic since the Wave Function evolves deterministically \\cite{qmdet}.\n\nThere are two parts to the MWI: physics and interpretation. Physics, the theory of the evolution of the Wave Function, is rigorously defined and tested up to the maximally possible precision. It is  a ``good'' theory: it has no paradoxes, it is complete, it  contradicts neither the spirit nor the letter of the special theory of relativity.  As a physical theory it is clearly   simpler than the spontaneous collapse theories \\cite{Pearle,GRW},   Bohmian mechanics \\cite{deBroglie,Bohm}, the consistent  histories approach \\cite{Grif}, or underlying probabilistic theories \\cite{Spek}.  Moreover, recent analyses of possibilities for alternative interpretations show that they cannot be more  parsimonious than the theory of the quantum Wave Function, see a recent experimental study \\cite{Rigbauer}.\n\nThe difficult part of the MWI is the interpretation, the explanation of how the Wave Function of the Universe describes the world we observe in 3-space while the Wave Function ``lives'' in the configuration space of $3N$ dimensions ($N$ is the number of particles in the world).\nThe role of interpretation is much larger in the MWI than in other interpretations of quantum mechanics. In most other interpretations the connection between ontology, e.g. Bohmian trajectories, or values of observables, is simple, transparent and immediate, so frequently it is not discussed at all, while in the MWI, it is the main part to be analyzed. Interpretation  belongs to the realm of human science with different standards and methodology. It has a much wider range of acceptable approaches.  FAPP (for all practical purposes) \\cite{Bell} definitions are good enough for discussing our experiences.\n\n\nThe main difficulty of connecting our experience with the ontology in the MWI is that it corresponds not just to our current experience, but also to a multitude of other experiences in parallel worlds. Clearly, there are very many parallel worlds now, although the number of worlds is not rigorously defined. However, even in the gedanken case in the framework of the MWI in which  the Wave Function now describes just one world in the Universe, the correspondence to our experience in this world is  not obvious. I will start with the analysis of such a single world.\n\n\\section{A World}\n\nThe concept of a world in the MWI does not belong to the physics part. It is not like a particle: the form of the Wave Function includes information about the number and the type of particles in the physical Universe. Since the Wave Function is all there is, it must also have information about worlds, but the definition of what a world means is not written in the Wave Function. It belongs to human science. A world is what a layman understands by the word ``world''. A quantum physicist might have a confusing concept of a world, he can think of a multiverse \\cite{Deut}, about particles in a superposition, etc., so the concept might not be clear. A world in the MWI is defined as a world of classical physics:\n\n\\begin{quote}\nA world is a collection of  objects in the Universe in definite states.\n\\end{quote}\n\n\n\nIn classical physics we can imagine that all particles have definite positions and velocities.\nHowever,  the positions of every molecule of Earth's atmosphere are not really relevant for describing a world in the MWI.  It is enough to specify definite states of all macroscopic objects. Moving to the quantum domain, we do not even have a description in terms of positions and velocities of  particles.  In the standard approach, the notion of observables plays an important   role. Our experience is described through a set of values of the observables. An observer is aware only of measured observables and their measurements collapse  the Wave Function to  definite eigenstates  which provide the correspondence.\n\n\nDavid Bohm, in support of his interpretation, noted that in every experiment we read the values of observables by observing   positions  in  space of the pointers of the measuring devices. Definite temperature corresponds to a particular position of a thermometer, etc. So, the world is specified by well localized positions of all macroscopic objects. The world is a concept in 3-space. It is a crucial point for explaining the connection between the quantum wave and our experience. In a theory with collapses at each quantum measurement there is a  single world in which the Wave Function is a product of wave functions  of all macroscopic objects in 3-space.  The classical picture of interaction according to which an object creates fields in 3-space which cause accelerations of other objects present in the location of the field is valid here. (Recently I found a local explanation of this form   for  an apparent counter example to this claim, the Aharonov-Bohm effect \\cite{VAB}.)\n\nNeuroscience does not have a clear explanation of human experiences. Considering experience as information flow allows to speculate that it might be defined in terms of spin wave function \\cite{Albert92}. This can lead to some modification of the picture, but apparently will not change it dramatically and currently there are no signs that our brain operates with spins.  The picture I present here is that our experiences supervene upon the space distribution of matter, so the Wave Function which allows us to build a three-dimensional picture of particles in 3-space specifies human experiences.\nThe density of particles is gauge independent and also properly transforms between different Lorentz observers. Thus, the explanation of our experience is unaffected by the ``narratability failure'' problem \\cite{Albert13}, the Wave Function description might be different for different Lorentz observers, but the local description remains the same.\n\n\nIn fact, in the standard approach with collapses at every quantum measurement ensuring a single world, the problem of correspondence between the ontology and our experience is not considered problematic. (The collapse process, however,  is a very serious problem for the physics part of the theory.) This connection between the collapsed Wave Function and our experience will be the basis of the connection with the experience in the framework of the MWI.\n\nThe exact Wave Function of a world is not rigorously defined. It is based on collapse, but there is no precise definition when exactly it happens. The collapse avoids superpositions of macroscopic objects in macroscopically different states, but the word ``macroscopic'' is a FAPP concept. Von Neumann proved that there is a very large flexibility where we put the cut between quantum and classical, i.e. when exactly the collapse takes place. In the collapsed wave we can see the classical world. The three-dimensional picture of  places where the wave density is large is the picture of the world we see around us. Instead of an {\\it ad hoc} von Neumann collapse, we can consider the spontaneously collapsed Wave Function of the GRW-Pearle type theory. The quantum wave resides essentially in 3-dimensions. It is connected to our experience in a transparent way. The macroscopic parts of  objects collapse to  pure unentangled states  due to  strong interaction with the environment.  States of electrons in atoms, and atom states in molecules, etc.  remain entangled, but micro states of particles are not relevant to the description of a world.\n\nThe state of an electron in an atom is not relevant for the description of a world, but when we describe a quantum interference experiment  with single particles, their state is important. One way to avoid description of micro objects is to note that the macroscopic description of macroscopic preparation and detection devices in the experiment includes the information about these particles. I think that it is more convenient to add a description of such micro particles to the picture of the world since it provides a clear characterization of the weak coupling of the particles with the environment. However, such micro particles require a special treatment, in particular, adding a backward evolving wave function specified by the final measurement \\cite{tisy,Vpast}.\n\n\nLet me summarize the main points. In a world, by definition, all macroscopic objects have definite macroscopic states.  In a world, there cannot be a Schr\\\"odinger's cat  in a closed box in a superposition of being alive and dead.\nI add to the mathematical formalism of the Universal Wave Function a postulate regarding the connection to our experience. The connection between experience and the world wave function which is an element of the Universal Wave Function is the same as in the collapse theories in which there is only one world wave function and single experience of every observer. The Wave Function of a world $|\\Psi_{\\rm world}\\rangle$ is a product state of well localized  (in 3-space) quantum states of all ``macroscopic'' objects multiplied by possibly entangled states of microscopic systems irrelevant for the macroscopic description of the world.\n\n\n\\section{The Wave Function of the Universe}\n\nThe correspondence between the Wave Function of a single world and our experience may not be as transparent as the correspondence between our experience and a classical world of particles moving on trajectories, or a world of Bohmian trajectories, but I still find it satisfactory. I do not believe that this is the picture of the Universe because there is no satisfactory physics which leads to such  Wave Function. The collapse process which is required here is foreign to all physics that I know. It has some kind of action at a distance and randomness and nobody has found elegant and simple equations for the collapse process. Physics forces me to reject the idea of collapse, and therefore, after every quantum measurement a superposition of wave functions of the single world wave function type is created:\n\\begin{equation}\\label{Univ}\n| \\Psi_{\\rm Universe}\\rangle=\\sum \\alpha_i | \\Psi_{{\\rm world}~i}\\rangle.\n\\end{equation}\n\nThe Wave Funciton of the Universe is a wave function in a configuration space $|\\Psi_{\\rm Universe} (x_1, x_2,...x_N)\\rangle$. Each one of the wave functions of the worlds  is essentially a product of well localized wave functions of macroscopic objects in 3-space $|\\Psi_{{\\rm world}~i}\\rangle= \\prod_j |\\Phi_i(X_j)\\rangle\\ |\\psi_i\\rangle$, where $|\\psi_i\\rangle$ signifies the wave function of other degrees of freedom: relative coordinates of electrons in atoms, weakly coupled particles like neutrinos, etc. Locality and strength of  interactions (with a buzz word ``decoherence'') ensure approximate uniqueness of the decomposition.\n\nDifferent worlds must have  different classical descriptions and therefore they correspond to orthogonal wave functions. Energies are   bounded, so even  well localized wave functions of macroscopic objects must have nonvanishing tails at macroscopic distances. This makes the above statement of orthogonality not so obvious. Most probably, the world wave functions of different worlds can be constructed to be orthogonal, but even if not, since the concept of a world is a human FAPP concept, a tiny overlap can and should be neglected.\n\nThe  Wave Function of the Universe $| \\Psi_{\\rm Universe}\\rangle$ can be decomposed into a superposition of the world wave functions $| \\Psi_{{\\rm world}~i}\\rangle$. The reason we do not experience superpositions is not because they do not exist, but because {\\it we} are not capable of experiencing  several different states simultaneously. The phrase ``different states'' means different places and the locality of physical interactions prevents conscious nonlocal creatures. A nonlocal creature which behaves differently depending on the relative phase $\\phi$  between parts at different locations, $\\frac{1}{\\sqrt 2} (|A\\rangle + e^{i\\phi} |B\\rangle)$, cannot be useful because local interactions wash out the relative phase almost immediately. This is the argument for the ``preferred basis'' of the decomposition of the Wave Function, it has to be the local basis.\n\nAnother argument for why the Universe cannot support worlds with nonlocal   creatures being in a superposition of two locations is that such a creature  will need much more resources. If such a nonlocal creature would like to eat, it will need two dinners instead of one. Both parts of the creature $ |A\\rangle$ and $|B\\rangle$ have to eat, but a superposition of a dinner in two locations $\\frac{1}{\\sqrt 2} (|A_{\\rm food}\\rangle + |B_{\\rm food}\\rangle)$ cannot provide a dinner for the creature through local interactions. Such interactions cannot transform a product state of the creature and the food into the required state $\\frac{1}{\\sqrt 2} (|A\\rangle|A_{\\rm food}\\rangle  + e^{i\\phi} |B\\rangle|B_{\\rm food}\\rangle)$.\n\nThe {\\it stability} of  world wave functions $| \\Psi_{{\\rm world}~i}\\rangle$  is the key issue.  Of course, at every quantum measurement the world wave function splits to several different (well localized) world wave functions. But this does not happen too fast and too often. If no observer has the time to be aware of a particular world, we have no reason to define such a world: remember, `world' is a human concept which is supposed to help explaining our experience.\nI do want to consider worlds at a particular time, but it is not enough that the picture drawn from a wave function looks like a world  (all macroscopic objects are well localized) at this moment. If, immediately after, it does not look like a world, such wave function should not be considered as corresponding to a world.\n\nWe can ask what was the past and what will be the future of a world.  Every quantum measurement splits the world, so a world in a particular time corresponds to a multitude of worlds in the future. Going backward in time we have a unique past. For many worlds at present there is the same past. So, if we consider worlds for periods of time we obtain ``overlapping'' worlds: they overlap in the past. (If we want to add some micro particles for describing the world, then splitting for them happens at the measurement {\\it after} the present time of the world we consider \\cite{tisy}.)\n\nAnother important concept which has to be clarified is ``I''. In which world am I? If I perform a quantum measurement what happens to me? If somebody else performs a quantum measurement will I be changed?\n\nBy definition, I have to be in a well defined state. All parts of my body, neurons in my brain, etc. have to be well localized. When I perform a quantum measurement with a few possible results and I observe an outcome, a few worlds are created with a different ``I'' in each world. All these ``I''s are descendants of the ``I'' before the measurement, so they have identical memories of what happened before the quantum experiment, but they have different knowledge about the result of the experiment.  ``I''s which have identical memories, but different locations of bodies, are also considered different. Such situations can be created when I am moved to different places, while asleep, according to the results  of a quantum experiment  \\cite{schizo}).\n\nIf somebody else makes a quantum measurement in a faraway location such that the information about the result does not arrive at my location, then I do not split. The world splits, but I remain part of all newly created worlds. There is no meaning for the question what is the result of this measurement in {\\it my} world. (This is one of the examples in which our language, developed during the time of a firm belief in existence of a single world, has difficulty in describing the situation.) So, it cannot be that different ``I''s are present in one world, but it is possible that the same ``I'' is present in several worlds.\n\nLet me summarize. I build the MWI on the basis of standard quantum mechanics. I ask physicists who accept Wave Function collapse to provide (a rough) description of the Wave Function of the physical Universe. This specifies what might be a world wave function. Next,  I decompose the Wave Function of the Universe as a superposition of such world wave functions. I postulate the correspondence with the experience as the one in   standard quantum mechanics. Thus, the presence of a term corresponding to a world $i$ in the decomposition, $\\alpha_i\\neq 0$, ensures the presence of such an experience. It is a question for  quantum physicists what should be the world wave function to ensure stability and ``classicality'' of this world.\n\n\n The wave function of a world $|\\Psi_{{\\rm world}~i}\\rangle$ specifies what we  ``feel'' in world $i$. What is the role of the coefficient $\\alpha_i$? The absolute value of the coefficient  specifies the illusion of probability as will be discussed in the next section. The phase is not relevant to us. We assume that worlds are different enough that we cannot make interference experiments between them. The worlds differ by macroscopic differences of states of macroscopic objects. The FAPP meaning of the word ``macroscopic'' can be defined exactly by this property: the states are macroscopically different if we cannot observe interference between them in a realistic experiment. The phase of $\\alpha_i$ can be relevant for a creature having unlimited technological power to perform interference experiments with macroscopic objects, i.e., interference between different worlds.\n\n\n \\section{Measure of existence of the world and the illusion of probability}\n\n The measure of existence of the world, $\\mu_i=|\\alpha_i|^2$ ,  provides the illusion of probability in our world. Before discussing the qualitative issue, why $ |\\alpha_i|^2$ and not something else, I will discuss the meaning of probability in the MWI.\n\n I write ``illusion'' of probability, because I consider a genuine concept of probability to require that there are several options and only one takes place. In case of quantum measurements all possible outcomes take place.\n\nThere is no randomness in the basic laws of physics. But determinism by itself does not prevent the concept of probability. We might be ignorant of some details which specify the outcome. In quantum mechanics there are no such details (which usually are named ``hidden variables''). We might have complete information about the system and the measuring device and it still will not help us to know what will be the outcome. More precisely, we know that all outcomes will take place and we say that each outcome will correspond to one of the newly created worlds.\n\nIf there is a collapse of the Wave Function to one of the world wave functions $|\\Psi_{{\\rm world}~i}\\rangle$, then we do have a legitimate concept of probability. An experience of an observer supervenes upon the world wave function. In the MWI, every term in the decomposition (\\ref{Univ}) corresponds to such experience. By construction, our experience in each particular world is the same as our experience in a physical universe with the collapse law in which  only one world exists. So, we have a complete illusion of probability: there is no difference between our experience and the experience of an agent with genuine probability.\n\n\nSince the MWI is a deterministic theory, there cannot be a genuine chance there, but we can arrange a special situation in which there will be  a genuine ignorance probability of an agent about the outcome of a quantum experiment. Before, I pointed out that in quantum mechanics we can know everything about the past without knowing what will be an outcome in the future.\n Now I will show that in the MWI we might know everything about the present, but still be ignorant about the outcome of already performed experiment.\n\n Such a situation is not easy to understand  because we are not used to considering a plurality of worlds. What might help is to imagine first a gedanken situation which is not related to quantum physics. Suppose that   creatures with super technology land on Earth. They can do what we would consider to be miracles. The creatures can create copies of Earth with everything on it and add it to the Solar system. They show their ability to people on Earth and they say that tomorrow morning there will be three identical Earth planets with all  inhabitants. In the morning I wake up and I do not know if I am the original, or I am one of the two copies. Three Earth worlds exist and there are three ``I''s. Each one does not know which ``I'' he is. The symmetry tells us that the probability to be each particular ``I'' is one third.  Each ``I'' will bet one third on the fact that he is the original and not a copy.\n\n In the case of a quantum measurement I can arrange a similar situation without super technology \\cite{schizo}. Before performing a quantum experiment which has three possible outcomes I take a sleeping pill and ask my friends to move me while I am asleep to one of three rooms according to the result of the experiment. My friends, instead of performing the experiment themselves, can  get instructions using the iPhone application ``Universe Splitter''  or the Tel Aviv World Splitter \\cite{TUWS}. When ``I'' in a particular room am  awake, but still have closed eyes, I  do not know in which room  am I. It is an unusual situation: I, more precisely, every copy of me, might know everything about the Universe, but still be ignorant about the outcome. I am ignorant about self-location in a particular world. This is my privileged property.   Any external observer does not have this probability concept. The question: ``In which world am I present?'' has no meaning for him. There are different ``I''s present in corresponding different worlds, but the external observer,   even if he is aware that the splitting has occurred,  belongs to all these worlds. Only when he contacts me, will he split his ``I'' according to the outcome of the quantum experiment.\n\n A widespread  approach to probability is the readiness to put  an intelligent bet on a particular outcome. Since the probability is relevant for all my descendants  as they    have a legitimate concept of ignorance probability, I suggest to associate it also with me at the time before the experiment. It is rational for me to place bets since all my descendants would like me to do so. They will get the rewards (and the losses) of the bet. Thus, considering the probability as an amount that an intelligent agent is ready to bet on a particular outcome \\cite{deFin}, we have a concept of probability in the framework of the MWI. The rule is that an experimentalist performing a quantum experiment should bet in proportion to the {\\it measure of existence} of the world with this outcome \\cite{SBAna}.\n\n In classical cases either a world exists or it does not. In case of several planet Earths, there is no way that (at a particular time) one planet exists more than the other. We must associate the same measure of existence with all planets: just 1.   The definition of the measure of existence of a quantum world, $\\mu_i=|\\alpha_i|^2$, allows different values. I postulate, that we should  bet according to this measure. This analog of the Born Rule  in the framework of the MWI is sometimes named the Born-Vaidman rule \\cite{Tap}.\n\n  Why accept this postulate? In my view, a good answer is that the records of the outcomes of quantum experiments in our world show that there is no general rule which   leads to   better results for an experimentalist betting on the outcomes of quantum experiments. One can add that the postulate is natural if we accept the idea of measure of existence. Indeed, arranging (artificially) that all worlds have equal measures of existence leads to a natural property of equal probability for all worlds as in the case of  multiple ``classical'' worlds (compare with my three Earths example). The measure of existence of a world  also quantifies the power of a world to interfere with other worlds if a quantum super technology will make interference experiments with macroscopic objects which are interference experiments between different worlds \\cite{schizo}.\n\n\n Another argument for accepting the Born-Vaidman Rule is that all the alternatives I try to imagine lead to contradiction with special relativity. If we have a wave function of a particle distributed in space, the sum of the probabilities to find it in some location adds to 1. Thus, if we can change the probability in one place  by some local operation, this will change the probability in another place. Changed probability is a signal, since a large ensemble of identical systems with identical actions allows transmission of a message. Local operations at a local part $O$ can change almost all local aspects of the wave function  except for changing its weight, $\\int_O\\Psi^*\\Psi dv$. So, it seems to be that the probability cannot depend on any other function of the wave function but its local weight.\n\n\n\n  \\section{Conclusions}\n\nThe MWI, first, tells us what the ontology is: it is the evolving Wave Function. Then it finds us in this ontology, as stable waves in human shapes, and explains  (through some natural postulates) our experience.\n\n The theory of the evolution of the Wave Function is an exact mathematical theory of a single physical Universe.\n  Avoiding discussion of values of variables makes it free of paradoxes, action at a distance, and randomness, the features which are foreign to all known areas of physics.\n\nThe second part of the MWI, the interpretation, is good only as a FAPP theory. It allows  various modifications, e.g., the original Everett relative state formulations provides a somewhat different decomposition to world wave functions.\n\nThe MWI is far from being the most clear  explanation of our experience and although it can probably  be improved, some other interpretations certainly explain it much better. However, these interpretations pay a very high price in spoiling the physical part of the theory which becomes so unnatural  that most physicists are not ready to accept them.\n\nMy impression is that the majority of physicists are not ready to accept the MWI because in this framework {\\it we}  are not in the center of the theory. The sun does not encircle us. Physicists are still hoping that there will be some new theory (interpretation) which will be better than the MWI. I doubt it, but I think that it should be at least accepted that meanwhile the MWI is the best alternative we have today.\n\nThis work has been supported in part by the  Israel Science Foundation  Grant No. 1311\/14.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Grothendieck-Riemann-Roch theorem is a fundamental statement in\nalgebraic geometry. It describes the behavior of the Chern character\nfrom algebraic $K$-theory to suitable cohomology theories (for\ninstance Chow groups), with respect to the push-forward operation by\nproper maps. It provides a vast generalization of the classical\nRiemann-Roch theorem on Riemann surfaces and the\nHirzebruch-Riemann-Roch theorem on compact complex manifolds. In their\ndevelopment of arithmetic intersection theory, Gillet and Soul\\'e were\nlead to extend the Grothendieck-Riemann-Roch theorem to the context of\narithmetic varieties. In this setting, vector bundles are equipped\nwith additional smooth hermitian metrics, for which an extension of\nalgebraic $K$-theory can be defined. There is a theory of\ncharacteristic classes for hermitian vector bundles, with values in\nthe so-called arithmetic Chow groups \\cite{GilletSoule:vbhm}. In\nanalogy to the classical algebraic geometric setting, it is natural to\nask about the behavior of the arithmetic Chern character with respect\nto proper push-forward. This is the question first addressed by\nGillet-Soul\\'e  \\cite{GilletSoule:aRRt} and later by\nGillet-R\\\"ossler-Soul\\'e\n\\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees}. These\nworks had to restrict to push-forward by morphisms which are smooth\nover the complex points of the arithmetic varieties. This assumption\nwas necessary in defining both push-forward on arithmetic $K$-theory\nand the arithmetic Chow groups. While on arithmetic Chow groups\npush-forward resides on elementary operations (direct images of cycles\nand fiber integrals of differential forms), they used holomorphic\nanalytic torsion on the arithmetic $K$-theory level\n\\cite{BismutKohler}.\n\nThe aim of this article is to extend the work of Gillet-Soul\\'e and\nGillet-R\\\"ossler-Soul\\'e to arbitrary projective morphisms of regular\narithmetic varieties. Hence we face to the difficulty of\nnon-smoothness of morphisms at the level of complex points. To\naccomplish our program, we have to introduce generalized arithmetic\nChow groups and arithmetic $K$-theory groups which afford proper\npush-forward functorialities for possibly non-smooth projective\nmorphisms. Loosely speaking, this is achieved by replacing smooth\ndifferential forms in the theory of Gillet and Soul\\'e by currents\nwith possibly non-empty wave front sets. To motivate the introduction\nof these currents, we remark that they naturally appear as\npush-forwards of smooth differential forms by morphisms whose critical\nset is non-empty. In this concrete example, the wave front set of the\ncurrents is controlled by the normal directions of the morphism. The\ndefinition of our generalized arithmetic Chow groups is a variant of\nthe constructions of Burgos-Kramer-K\\\"uhn\n\\cite{BurgosKramerKuehn:cacg}, specially their covariant arithmetic\nChow groups. As an advantage with respect to \\emph{loc. cit.}, the\npresentation we give simplifies the definition of proper push-forward,\nwhich is the main operation we have to deal with in the present\narticle. At the level of Chow groups, this operation relies on\npush-forward of currents and keeps track of the wave front sets. For\narithmetic $K$-groups, we replace the analytic torsion forms of\nBismut-K\\\"ohler by a choice of a generalized analytic torsion theory\nas developed in our previous work\n\\cite{BurgosFreixasLitcanu:GenAnTor}. While a generalized analytic\ntorsion theory is not unique, we proved it is uniquely determined by\nthe choice of a real genus. We establish an arithmetic\nGrothendieck-Riemann-Roch theorem for arbitrary projective morphisms,\nwhere this real genus replaces the $R$-genus of Gillet and Soul\\'e. We\ntherefore obtain the most general possible formulation of the\ntheorem. In particular, the natural choice of the 0 genus,\ncorresponding to what we called the homogenous theory of analytic\ntorsion, provides an exact Grothendieck-Riemann-Roch type formula,\nwhich is the formal translation of the classical algebraic geometric\ntheorem to the setting of Arakelov geometry. The present work is thus\nthe abutment of the articles \\cite{BurgosLitcanu:SingularBC} (by the\nfirst and third named authors) and\n\\cite{BurgosFreixasLitcanu:HerStruc}--\\cite{BurgosFreixasLitcanu:GenAnTor}.\n\nLet us briefly review the contents of this article. In section\n\\ref{section:GenArChow} we develop our new generalization of\narithmetic Chow groups, and consider as particular instances the\narithmetic Chow groups with currents of fixed wave front set. We study\nthe main operations, such as pull-back, push-forward and products. In\nsection \\ref{section:ArKTheory} we carry a similar program to\narithmetic $K$-theory. We also consider an arithmetic version of our\nhermitian derived categories \\cite{BurgosFreixasLitcanu:HerStruc},\nthat is specially useful to deal with complexes of coherent sheaves\nwith hermitian structures. The essentials on arithmetic characteristic\nclasses are treated in section \\ref{section:ArChar}. With the help of\nour theory of generalized analytic torsion, section\n\\ref{section:DirectImage} builds push-forward maps on the level of\narithmetic derived categories and arithmetic $K$-theory. The last\nsection, namely section \\ref{section:ARR}, is devoted to the statement\nand proof of the arithmetic Grothendieck-Riemann-Roch theorem for\narbitrary projective morphisms of regular arithmetic varieties. As an\napplication, we compute the main characteristic numbers of the\nhomogenous theory, a question that was left open in\n\\cite{BurgosFreixasLitcanu:GenAnTor}.\n\n\\section{Generalized arithmetic Chow groups}\\label{section:GenArChow}\n\nLet $(A,\\Sigma ,F_{\\infty})$ be an arithmetic ring\n\\cite{GilletSoule:ait}: that is, $A$ is an excellent regular\nNoetherian integral domain,\ntogether with a finite non-empty set of embeddings $\\Sigma$ of $A$\ninto ${\\mathbb C}$ and a linear \nconjugate involution $F_{\\infty}$ of the product ${\\mathbb C}^{\\Sigma}$ which\ncommutes with \nthe diagonal embedding of $A$. \nLet $F$ be the field of fractions of $A$.\nAn \\textit{arithmetic variety} $\\mathcal{X}$ is a\nflat and quasi-projective scheme over $A$ such that\n$\\mathcal{X}_{F}=\\mathcal{X}\\times \\Spec F$\nis smooth. Then $X_{{\\mathbb C}}:=\\coprod_{\\sigma \\in \\Sigma\n}\\mathcal{X}_{\\sigma }({\\mathbb C}) $ is a\ncomplex algebraic manifold, which is endowed with an\nanti-holomorphic automorphism $F_{\\infty}$. One also associates to\n$\\mathcal{X}$ the real variety\n$X=(X_{{\\mathbb C}},F_{\\infty})$. Whenever we have arithmetic varieties\n$\\mathcal{X},\\ \\mathcal{Y}, \\dots$ we will denote by $X_{{\\mathbb C}},\\\nY_{{\\mathbb C}}, \\dots$ the associated complex manifolds and by $X,\\\nY, \\dots$ the associated real manifolds.\n\nTo every regular arithmetic variety Gillet\nand Soul\\'e have associated arithmetic Chow groups, denoted $\\cha\n^{\\ast}(\\mathcal{X})$, and developed an\narithmetic intersection theory \\cite{GilletSoule:ait}. \n\nThe arithmetic Chow groups defined by Gillet and Soul\\'e are only\ncovariant for morphism that are smooth on the generic fiber. Moreover\nthey are not suitable to study the kind of singular metrics that\nappear naturally when dealing with non proper modular varieties.  \nIn order to have arithmetic Chow groups that are covariant with\nrespect to arbitrary proper morphism, or that are suitable to treat\ncertain kind of singular metrics,\nin \\cite{BurgosKramerKuehn:cacg}\ndifferent kinds\nof arithmetic Chow groups are constructed, depending on the choice of a\nGillet sheaf of \nalgebras $\\mathcal{G}$ and a $\\mathcal{G}$-complex $\\mathcal{C}$. We\ndenote by $\\cha^{\\ast}(\\mathcal{X},\\mathcal{C})$ the arithmetic Chow groups\ndefined in \\emph{op. cit.} Section 4.\n\nThe basic example of a Gillet algebra is the Deligne complex of\nsheaves of differential forms with logarithmic singularities  \n$\\mathcal{D}_{\\log}$, defined\nin \\cite[Definition 5.67]{BurgosKramerKuehn:cacg}; we refer to \\emph{op. cit.}\nfor the precise definition and properties. Therefore, to any\n$\\mathcal{D}_{\\log}$-complex we can associate arithmetic Chow\ngroups. In particular, considering $\\mathcal{D}_{\\log}$ itself as a\n$\\mathcal{D}_{\\log}$-complex, we obtain\n$\\cha^{\\ast}(\\mathcal{X},\\mathcal{D}_{\\log})$, \nthe arithmetic Chow groups defined in \\cite[Section\n6.1]{BurgosKramerKuehn:cacg}. When $\\mathcal{X}_{F}$ is projective,\nthese groups agree, up to a normalization factor, with the groups\ndefined by Gillet and Soul\\'e. \n\nThe groups $\\cha^{\\ast}(\\mathcal{X},\\mathcal{C})$ introduced in \\cite[Section\n6.1]{BurgosKramerKuehn:cacg} have several technical issues:\nthey depend on the sheaf structure of $\\mathcal{C}$ and not only on\nthe complex of global sections $\\mathcal{C}(X)$; moreover, they are\nnot completely satisfactory if the cohomology determined by\n$\\mathcal{C}$ does not satisfy a weak purity property; finally the\ndefinition of direct images is intricate. To overcome these\ndifficulties we introduce here a variant of the cohomological\narithmetic Chow groups that only depends on the complex of global\nsections of a $\\mathcal{D}_{\\log}$-complex. \n\n\\begin{definition}\\label{def:3} Let $\\mathcal{X}$ be an arithmetic variety,\n  $X_{{\\mathbb C}}=\\mathcal{X}_{\\Sigma }$ the associated complex manifold and\n  $X=(X_{{\\mathbb C}},F_{\\infty})$ the associated real manifold.\n  A \\emph{$\\mathcal{D}_{\\log}(X)$-complex} is a graded complex of\n  real vector spaces $C^{\\ast}(\\ast)$ provided with a morphism of\n  graded complexes\n  \\begin{displaymath}\n    \\cmap\\colon \\mathcal{D}_{\\log}^{\\ast}(X,\\ast)\\longrightarrow\n    C^{\\ast}(\\ast).  \n  \\end{displaymath}\n Given two $\\mathcal{D}_{\\log}(X)$-complexes $C$ and $C'$, we say that $C'$ is a $C$-complex if there is a commutative diagram of morphisms of graded complexes\n \\begin{displaymath}\n \t\\xymatrix{\n\t\t\\mathcal{D}_{\\log}^{\\ast}(X,\\ast)\\ar[r]^{c}\\ar[rd]^{c'}\t&C^{\\ast}(\\ast)\\ar[d]^{\\varphi}\\\\\n\t\t\t&C^{\\prime\\ast}(\\ast).\n\t}\n \\end{displaymath}\n In this situation, we say that $\\varphi$ is a morphism of $\\mathcal{D}_{\\log}(X)$-complexes.\n\\end{definition}\n\nWe stress the fact that a $\\mathcal{D}_{\\log}$-complex is a complex of\nsheaves while a $\\mathcal{D}_{\\log}(X)$-complex is a complex of vector\nspaces. \nIf $\\mathcal{C}$ is a $\\mathcal{D}_{\\log}$-complex\n  of real vector spaces, then the complex of global sections\n  $\\mathcal{C}^{\\ast}(X,\\ast)$ is  a $\\mathcal{D}_{\\log}(X)$-complex.\nWe are mainly interested in the\n  $\\mathcal{D}_{\\log}(X)$-complexes of Example \\ref{exm:1} made out of\n  differential forms and \n  currents. We will follow the conventions of \\cite[Section\n5.4]{BurgosKramerKuehn:cacg} regarding differential forms and\ncurrents. In particular, both the current associated to a differential form\nand the current associated to a cycle have implicit a power of the trivial\nperiod $2\\pi i$.\n\n\n\\begin{example} \\label{exm:1}\n\\begin{enumerate}\n\\item \\label{item:4} The Deligne complex $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)$ of\ndifferential forms on $X$ with arbitrary\nsingularities at infinity. Namely, if $E^{\\ast}(X_{{\\mathbb C}})$ is the Dolbeault\ncomplex \n(\\cite[Definition 5.7]{BurgosKramerKuehn:cacg}) of differential forms on\n$X_{{\\mathbb C}}$ then  \n\\begin{displaymath}\n  \\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X,\\ast)=\n\\mathcal{D}^{\\ast}(E^{\\ast}(X_{{\\mathbb C}}),\\ast)^{\\sigma},\n\\end{displaymath}\nwhere $\\mathcal{D}^{\\ast}(\\underline{\\ },\\ast)$ denotes the Deligne\ncomplex\n(\\cite[Definition 5.10]{BurgosKramerKuehn:cacg})\nassociated to a Dolbeault complex and $\\sigma $\nis the involution $\\sigma (\\eta)=\\overline\n{F_{\\infty}^{\\ast}\\eta}$ as in \\cite[Notation\n5.65]{BurgosKramerKuehn:cacg}.\n\nNote that $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)$ is the complex of\nglobal sections of the $\\mathcal{D}_{\\log}$-complex\n$\\mathcal{D}^{\\ast}_{{\\text{\\rm l, ll, a}}}$ that appears in \\cite[Section\n3.6]{BurgosKramerKuehn:accavb} with empty log-log singular locus. In\nparticular, by \\cite[Theorem \n3.9]{BurgosKramerKuehn:accavb} it satisfies the weak purity\ncondition. \n\\item \\label{item:5} The Deligne complex $\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\ast)$\n  of currents on $X$. Namely, if $D^{\\ast}(X_{{\\mathbb C}})$ is the Dolbeault\n  complex of currents on $X_{{\\mathbb C}}$ then\n\\begin{displaymath}\n  \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{\\ast}(X,\\ast)=\n\\mathcal{D}^{\\ast}(D^{\\ast}(X_{{\\mathbb C}}),\\ast)^{\\sigma}.\n\\end{displaymath}\nNote that here we are considering arbitrary\ncurrents on $X_{{\\mathbb C}}$ and not extendable currents as in \n\\cite[Definition 6.30]{BurgosKramerKuehn:cacg}. \n\\item \\label{item:6} Let $T^{\\ast}X_{{\\mathbb C}}$ be the cotangent bundle of\n  $X_{{\\mathbb C}}$. Denote by \n  $T_{0}^{\\ast}X_{{\\mathbb C}}=T^{\\ast}X_{{\\mathbb C}}\\setminus X_{{\\mathbb C}}$ \n  the cotangent bundle with the zero section deleted and let $S\\subset\n  T_{0}^{\\ast}X_{{\\mathbb C}}$ be a closed conical subset that is invariant under\n  $F_{\\infty}$. Let $D^{\\ast}(X_{{\\mathbb C}},S)$ be the\n  complex of currents whose wave front set is contained in $S$\n  \\cite[Section 4]{BurgosLitcanu:SingularBC}. The  Deligne complex\n  of currents on $X$ having the wave front set\n  included in the fixed set $S$ is given by\n\\begin{displaymath}\n  \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{\\ast}(X,S,\\ast)=\n\\mathcal{D}^{\\ast}(D^{\\ast}(X_{{\\mathbb C}},S),\\ast)^{\\sigma}.\n\\end{displaymath}\n\\end{enumerate}\nThe maps of complexes $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)\\to\n\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\ast)$ given by $\\eta\\mapsto\n[\\eta]$ is injective and makes of $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\ast)$ a $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)$-complex. We will use this map to identify\n$\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}$ with a subcomplex of\n$\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}$. Since $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)=\n\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\emptyset,\\ast)$ and\n$\\mathcal{D}^{\\ast}_{D,{\\text{\\rm a}}}(X,\\ast)= \n\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T^{\\ast}_{0}X,\\ast)$, examples\n\\ref{item:4} and \\ref{item:5} are particular cases of \\ref{item:6}.     \n\\end{example}\n\\begin{remark}\nWith these examples at hand, we can specialize the definition of\n$C$-complex (Definition \\ref{def:3}) to $C=\\mathcal{D}_{{\\text{\\rm a}}}(X)$. When\ndealing with hermitian structures on sheaves on non-necessarily proper\nvarieties, we will to work with\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes rather than\n$\\mathcal{D}_{\\log}(X)$-complexes. \n\\end{remark}\nWe will also follow the notation of \\cite[Section\n3]{BurgosKramerKuehn:cacg} regarding complexes. In particular we will\nwrite\n\\begin{displaymath}\n  \\widetilde C^{2p-1}(p)=C^{2p-1}(p)\/\\Img \\dd_{C}, \\quad\n  \\Zr C^{2p}(p)=\\Ker \\dd_{C}\\cap  C^{2p}(p).\n\\end{displaymath}\n\n\\begin{definition}\\label{def:1}\n  The \\emph{arithmetic Chow groups with $C$ coefficients} are defined\n  as \n  \\begin{equation}\\label{pfformula}\n    \\cha^p(\\mathcal{X},C)=\\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})\\times\n    \\widetilde C^{2p-1}(p)\/\\sim\n  \\end{equation}\n  where $\\sim$ is the equivalence relation generated by\n  \\begin{equation}\\label{equivdef}\n    (a(g),0)\\sim (0,c(g)).\n  \\end{equation}\n\\end{definition}\nIf $\\varphi:C\\rightarrow C'$ is a morphism of\n$\\mathcal{D}_{\\log}(X)$-complexes (so that $C'$ is a $C$-complex),\nthen there is a natural surjective morphism\n\\begin{equation}\n  \\cha^{p}(\\mathcal{X},C)\\times\\widetilde{C^{\\prime}}^{2p-1}(p)\n  \\longrightarrow\\cha^{p}(\\mathcal{X},C'). \\label{eq:10}  \n\\end{equation}\nIntroducing the equivalence relation generated by\n\\begin{displaymath}\n  ((0,c),0)\\equiv ((0,0),\\varphi(c)),\n\\end{displaymath}\nwe see that \\eqref{eq:10} induces an isomorphism\n\\begin{equation}\\label{eq:11}\n  \\cha^{p}(\\mathcal{X},C)\\times\\widetilde{C}^{\\prime 2p-1}(p)\/\n  \\equiv\\;\\overset{\\sim}{\\longrightarrow}\\cha^{p}(\\mathcal{X},C').\n\\end{equation}\n\nWe next unwrap Definition \\ref{def:1} in order to get simpler\ndescriptions of the arithmetic Chow groups associated to the complexes of\nExample \\ref{exm:1}. We start by recalling the construction of\n$\\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})$. The group of codimension $p$\narithmetic cycles is given by\n\\begin{displaymath}\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})=\n\\left\\{ (Z,\\widetilde g)\\in \\Zr^{p}(\\mathcal{X})\\times \n  \\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus \\mathcal{Z}^{p},p)\\left\\vert\n  \\begin{aligned}[c]\n    \\dd_{\\mathcal{D}}\\widetilde g&\\in \\mathcal{D}_{\\log}^{2p}(X,p)\\\\\n    \\cl(Z)&=[(\\dd_{\\mathcal{D}}\\widetilde g,\\widetilde g)]\n  \\end{aligned}\n  \\right.\\right\\},\n\\end{displaymath}\nwhere $\\Zr^{p}(\\mathcal{X})$ is the group of codimension $p$\nalgebraic cycles of $\\mathcal{X}$, $\\mathcal{Z}^{p}$ is the ordered\nsystem of codimension at least $p$ closed subsets of $X$,\n\\begin{displaymath}\n\\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus\n\\mathcal{Z}^{p},p) =\\lim_{\\substack{\\longrightarrow\\\\W\\in \\mathcal{Z}^{p}}}\n\\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus W,p),  \n\\end{displaymath}\nand $\\cl(Z)$ and\n$[(\\dd_{\\mathcal{D}}\\widetilde g,\\widetilde g)]$ denote the class in\nthe real Deligne-Beilinson cohomology group\n$H^{2p}_{\\mathcal{D},\\mathcal{Z}^{p}}(X,{\\mathbb R}(p))$ with supports on\n$\\mathcal{Z}^{p}$ of the cycle $Z$ and the pair\n$(\\dd_{\\mathcal{D}}\\widetilde g,\\widetilde g)$ respectively.\n\nFor each codimension $p-1$ irreducible variety $W$ and each rational\nfunction $f\\in K(W)$, there is a class $[f]\\in\nH^{2p-1}_{\\mathcal{D}}(X\\setminus |\\dv f|,{\\mathbb R}(p))$. Hence a\nclass $\\bmap([f])\\in\n\\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus\n\\mathcal{Z}^{p},p)$ that is denoted $\\mathfrak{g}(f)$. Then\n$\\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ is the group generated by the elements of\nthe form $\\diva(f)=(\\dv(f),\\mathfrak{g}(f))$.\n\nThen\n\\begin{displaymath}\n  \\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})=\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\left\/\n\\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\right. .\n\\end{displaymath}\n\nWe will use the following well stablished notation.\nIf $B$ is any subring of ${\\mathbb R}$ we will denote by\n$\\Zr^{p}_{B}(\\mathcal{X})=\\Zr^{p}(\\mathcal{X})\\otimes B$, by\n$\\za^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})$ the group with the same\ndefinition as $\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ with\n$\\Zr^{p}_{B}(\\mathcal{X})$ instead of $\\Zr^{p}(\\mathcal{X})$, and we\nwrite\n$\\rata^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})=\n\\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\otimes B$. Finaly we write\n\\begin{equation*}\n  \\cha^p_{B}(\\mathcal{X},\\mathcal{D}_{\\log})=\n  \\za^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})\\left\/\n\\rata^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})\\right.\n\\end{equation*}\nNote that\n$\\cha^p_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\log})=\\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})\\otimes\n{\\mathbb Q}$ but, in general, $\\cha^p_{{\\mathbb R}}(\\mathcal{X},\\mathcal{D}_{\\log})\\not\n= \\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})\\otimes {\\mathbb R}$. We will use the\nsame notation for all variants of the arithmetic Chow groups.\n\nNow, in the definition of $\\cha^{p}(\\mathcal{X},C)$ we can first\nchange coefficients and then take rational equivalence. We define\n\\begin{displaymath}\n  \\za^p(\\mathcal{X},C)=\\za^p(\\mathcal{X},\\mathcal{D}_{\\log})\\times\n  \\widetilde C^{2p-1}(p)\/\\sim\n\\end{displaymath}\nwhere again $\\sim$ is the equivalence relation generated by\n$(a(g),0)\\sim (0,c(g))$.\n\nThere are maps\n\\begin{alignat*}{2}\n\\zeta_{C}&\\colon\n\\za^{p}(\\mathcal{X},C)\\longrightarrow \\Zr^{p}(\\mathcal{X}),\n& \\zeta_{C}((Z,\\widetilde g),\\widetilde c)&=Z,\n\\\\\n\\amap_{C}&\\colon \\widetilde{C}^{2p-1}(p)\\longrightarrow\\za^{p}\n(\\mathcal{X},C),\n& \\amap_{C}(\\widetilde c)&=((0,0),-\\widetilde c),\\\\\n\\omega_{C}&\\colon \\za^{p}(\\mathcal{X},C)\\longrightarrow \\Zr C^{2p}(p), &\n\\qquad \\omega_{C}((Z,\\widetilde g),\\widetilde c)&=\n\\cmap(\\dd_{\\mathcal{D}}\\widetilde g)+\\dd_{C}\\widetilde c.\n\\end{alignat*}\nWe also consider the map \n$$\\bmap_{C}\\colon H^{2p-1}(C^{\\ast}(p))\\to \\za^{p} (\\mathcal{X},C)$$\nobtained by composing $\\amap_{C}$ \nwith the inclusion $H^{2p-1}(C^{\\ast}(p))\\to \\widetilde C^{2p-1}(p)$,\nand the map $$\\rho _{C}\\colon \\CH^{p,p-1}(\\mathcal{X})\\to\nH^{2p-1}(C^{\\ast}(p))$$ obtained by composing the regulator map $\\rho\n\\colon \\CH^{p,p-1}(\\mathcal{X})\\to H^{2p-1}_{\\mathcal{D}}(X,{\\mathbb R}(p))$\nin \\cite[Notation 4.12]{BurgosKramerKuehn:cacg}\nwith the map $\\cmap\\colon H^{2p-1}_{\\mathcal{D}}(X,{\\mathbb R}(p))\\to\nH^{2p-1}(C^{\\ast}(p))$. We will also denote by $\\rho _{C}$ the\nanalogous map with target $\\widetilde C^{2p-1}(p)$.\n\nThere are induced maps\n\\begin{align*}\n\\zeta_{C}&\\colon\n\\cha^{p}(\\mathcal{X},C)\\longrightarrow\\CH^{p}(\\mathcal{X}),\n\\\\\n\\amap_{C}&\\colon \\widetilde{C}^{2p-1}(p)\\longrightarrow\\cha^{p}\n(\\mathcal{X},C),\\\\\n\\omega_{C}&\\colon \\cha^{p}(\\mathcal{X},C)\\longrightarrow\\Zr C^{2p}(p).\n\\end{align*}\n\n\n\\begin{lemma}\\label{lemm:1}\n  \\begin{enumerate}\n  \\item \\label{item:1} Let $\\rata ^{p}(\\mathcal{X},C)$ denote the\n    image of $\\rata \n    ^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ in the group \n    $\\za^p(\\mathcal{X},C)$. Then\n    \\begin{displaymath}\n      \\cha^p(\\mathcal{X},C)=\n      \\za^{p}(\\mathcal{X},C)\\left\/\n        \\rata^{p}(\\mathcal{X},C)\\right. .      \n    \\end{displaymath}\n  \\item \\label{item:2} There is an exact sequence\n    \\begin{equation}\\label{eq:3}\n      0\\to \\widetilde C^{2p-1}(p)\\overset{\\amap_{C}}{\\longrightarrow\n      }\\za^{p}(\\mathcal{X},C)\n      \\overset{\\zeta_{C}}{\\longrightarrow }\n      \\Zr^{p}(\\mathcal{X}) \\to 0.\n    \\end{equation}\n  \\item \\label{item:3} There are exact sequences\n\\begin{equation}\\label{eq:1}\n\\CH^{p,p-1}(\\mathcal{X})\\overset{\\rho_{C}}{\\longrightarrow }\n\\widetilde{C}^{2p-1}(p)\\overset{\\amap_{C}}{\\longrightarrow }\n\\cha^p(\\mathcal{X},C) \n\\overset{\\zeta_{C}}{\\longrightarrow }  \\CH^p(\\mathcal{X})\\to 0,\n\\end{equation}\nand\n\\begin{multline}\\label{eq:2}\n  \\CH^{p,p-1}(\\mathcal{X})\\overset{\\rho_{C}}{\\longrightarrow }\n  H^{2p-1}(C^{\\ast}(p))\\overset{\\bmap_{C}}{\\longrightarrow }\\cha^p(\\mathcal{X},C)\n  \\overset{\\zeta_{C}\\oplus \\omega _{C}}{\\longrightarrow }\n \\\\  \\CH^p(\\mathcal{X})\\oplus\\Zr C^{2p}(p)\n\\longrightarrow  H^{2p}(C^{\\ast}(p)) \\to 0.\n\\end{multline}\n \\end{enumerate}\n  \\end{lemma}\n  \\begin{proof}\n    \\ref{item:1} Follows easily from the definition.\n\n    \\ref{item:2} By \\cite[Proposition 5.5]{Burgos:CDB} there is an\n    exact sequence\n    \\begin{displaymath}\n      0\\to \\widetilde\n      {\\mathcal{D}}_{\\log}^{2p-1}(X,p)\\overset{\\amap}{\\longrightarrow \n      }\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\n      \\overset{\\zeta}{\\longrightarrow }\n      \\Zr^{p}(\\mathcal{X}) \\to 0.      \n    \\end{displaymath}\n    From it and the definition of\n    $\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ we derive the exactness of\n    \\eqref{eq:3}.  \n\n    \\ref{item:3} Follows from the exact sequences of \\cite[Theorem\n    7.3]{Burgos:CDB} and the definition of $\\cha^p(\\mathcal{X},C)$.\n  \\end{proof}\n\nThe contravariant functoriality of\n$\\cha^{\\ast}(\\mathcal{X},\\mathcal{D}_{\\log})$ is easily translated to\nother coefficients.\nLet $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a morphism of regular arithmetic\nvarieties. \nLet $C$ be a $\\mathcal{D}_{\\log}(X)$-complex and $C'$ a\n$\\mathcal{D}_{\\log}(Y)$-complex, such that\nthere exists a map of complexes $f^{\\ast}:C'^{\\ast}(\\ast)\\to\nC^{\\ast}(\\ast)$ that makes the \nfollowing diagram commutative:\n\\begin{displaymath}\n  \\xymatrix{\n  \\mathcal{D}_{\\log}^{\\ast}(Y)\\ar[r]^{f^{\\ast}}\\ar[d] & \\mathcal{D}_{\\log}^{\\ast}(X) \\ar[d] \\\\\n  C'^{\\ast}(Y)\\ar[r]_{f^{\\ast}} & C^{\\ast}(X).\n}\n\\end{displaymath}\nThen we define\n$$f^{\\ast}((\\mathcal{Z},g),c)=(f^{\\ast}(\\mathcal{Z},g),f^{\\ast}(c)).$$\nIt is easy to see that this map is well defined, because the pull-back map\n$f^{\\ast}:\\cha^{\\ast}(\\mathcal{Y})\\to \\cha^{\\ast}(\\mathcal{X})$ (for\nthe Chow groups corresponding to \n$\\mathcal{D}_{\\log}$) is compatible to the map $\\amap$. \n\nBefore stating concrete examples of this contravariant functoriality\nwe need some notation (\\cite[Theorem 8.2.4]{Hormander:MR1065993},\n see also \\cite[Section 4]{BurgosLitcanu:SingularBC}). Let\n $f_{{\\mathbb C}}\\colon X_{{\\mathbb C}}\\to Y_{{\\mathbb C}}$ denote \nthe induced map of complex manifolds. Let $N_{f}$ be the set of normal\ndirections of $f_{{\\mathbb C}}$,  that is\n \\begin{displaymath}\n   N_{f}=\\{(f(x),\\xi)\\in T_{0}^{\\ast}Y_{{\\mathbb C}}\\mid \\dd\n   f_{{\\mathbb C}}^{t}\\xi=0\\}. \n \\end{displaymath}\n Let $S\\subset\n  T^{\\ast}_{0}Y_{{\\mathbb C}}$ be a closed conical subset invariant under\n  $F_{\\infty}$. When\n  $N_{f}\\cap S=\\emptyset$, the function  $f$ is said to be transverse\n  to $S$. In this case we write\n  \\begin{displaymath}\n    f^{\\ast}S=\\{(x,\\dd f_{{\\mathbb C}}^{t}\\xi)\\mid (f(x),\\xi)\\in S\\}.\n  \\end{displaymath}\n  It is a closed conical subset of $T^{\\ast}_{0}X_{{\\mathbb C}}$ invariant under\n  $F_{\\infty}$.\n\n\\begin{proposition} \\label{prop:3}\n  Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a morphism of regular arithmetic\n  varieties.\n  \\begin{enumerate}\n  \\item There is a pull-back morphism $f^{\\ast}\\colon\n    \\cha^{p}(\\mathcal{Y},\\mathcal{D}_{{\\text{\\rm a}}}(Y)) \\to\n    \\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n  \\item Let $N_{f}$ be the set of normal directions of $f_{{\\mathbb C}}$\n    and $S\\subset\n    T^{\\ast}_{0}Y_{{\\mathbb C}}$ a closed conical subset invariant under\n    $F_{\\infty}$. If $N_{f}\\cap\n    S=\\emptyset$, then there is a pull-back morphism\n    \\begin{displaymath} f^{\\ast}\\colon\n      \\cha^{p}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S)) \\to\n      \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}S)).\n    \\end{displaymath}\n  \\item If $f_{F}$ is smooth (hence $N_{f}=\\emptyset$) then there is a\n    pull-back morphism \n    $$f^{\\ast}\\colon\n    \\cha^{p}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y)) \\to\n    \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)).$$\n  \\end{enumerate}\n\\end{proposition}\n\n\nSimilarly the multiplicative properties of\n$\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\log})$ can be transferred to\nother coefficients.  \nLet $C$,  $C'$ and $C''$ be\n$\\mathcal{D}_{\\log}(X)$-complexes such that there is a commutative\ndiagram of morphisms of complexes\n\\begin{displaymath}\n  \\xymatrix{\n  \\mathcal{D}_{\\log}(X)\\otimes \\mathcal{D}_{\\log}(X) \\ar[r]^-{\\bullet} \\ar[d] & \\mathcal{D}_{\\log} (X)\\ar[d] \\\\\n  C\\otimes C' \\ar[r]_-{\\bullet} & C''.\n}\n\\end{displaymath}\nThen we define a product\n$$\\cha^p(\\mathcal{X},C)\\times \\cha^q(\\mathcal{X},C')\n\\to \\cha^{p+q}_{{\\mathbb Q}}(\\mathcal{X},C'')$$\nby\n\\begin{equation}\\label{product}\n((\\mathcal{Z},g),c)\\cdot((\\mathcal{Z}',g'),c')=\n((\\mathcal{Z},g)\\cdot(\\mathcal{Z}',g'),c\\bullet\\cmap(\\omega(g'))+\n\\cmap(\\omega(g))\\bullet c'+\\dd_{C}c\\bullet c').\n\\end{equation}\nAs a consequence we obtain the following result.\n\\begin{proposition} \\label{prop:2}\n  Let $\\mathcal{X}$ be a regular arithmetic variety.\n  \\begin{enumerate}\n  \\item $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$ is an\n    associative commutative graded ring.\n  \\item $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$ is a\n    module over\n    $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n  \\item Let $S,S'$ be closed conic subsets of $T^{\\ast}_{0}X_{{\\mathbb C}}$ that are\n    invariant under $F_{\\infty}$. Then\n    $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))$ is a\n    module over $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n    Moreover, if $S\\cap (-S')=\\emptyset$, there is a graded bilinear map\n    \\begin{multline*}\n      \\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\\times\n      \\cha^q(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S')) \\longrightarrow  \\\\\n      \\cha^{p+q}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S\\cup S'\\cup (S+S'))).\n    \\end{multline*}\n  \\item The product is compatible with the pull-back of Proposition\n    \\ref{prop:3}. \n  \\end{enumerate}\n\\end{proposition}\n\n\n\nWe now turn our attention towards direct images. The definition of\ndirect images for general $\\mathcal{D}_{\\log}$-complexes is quite\nintricate, involving the notion of covariant $f$-pseudo-morphisms (see\n\\cite[Definition 3.71]{BurgosKramerKuehn:cacg}). By contrast, we will\ngive a another description of the groups\n$\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$ for which the\ndefinition of push-forward is much simpler.\n\nBy \\cite[3.8.2]{Burgos:Gftp} we know that any\n$\\mathcal{D}_{\\log}$-Green form is locally integrable. Therefore there\nis a well defined map\n\\begin{displaymath}\n  \\varphi\\colon \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\longrightarrow \n  \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n\\end{displaymath}\ngiven by $(Z,\\widetilde g)\\to (Z,\\widetilde{[g]})$ for any\nrepresentative $g$ of $\\widetilde g$.  The previous map can be extended\nto a map\n\\begin{displaymath}\n  \\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}\\colon\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))\\longrightarrow  \n  \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde  {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n\\end{displaymath}\ngiven by $\\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}((Z,\\widetilde g),\\widetilde\nh)=(Z,\\widetilde{[g]}+\\widetilde h)$.\n\nThe following result is clear.\n\\begin{lemma}\\label{lemm:3} The\n  map $\\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}$ is an isomorphism. \n\\end{lemma}\n  This lemma gives us a more concrete description of the\n  group $\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$. In fact,\n  we will identify it with the group $\\Zr^{p}(\\mathcal{X})\\oplus\n  \\widetilde {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)$ when necessary. Some\n  care has to be taken when\n  doing this identification. For instance\n  \\begin{equation}\n    \\label{eq:5}\n    \\omega\n    _{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}(\\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}^{-1}(Z,\\widetilde\n    g))= \\dd_{\\mathcal{D}}g+\\delta _{Z}.\n  \\end{equation}\n \nLet now $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\nregular arithmetic varieties of relative dimension $e$. Using the\nabove identification, we define\n\\begin{equation}\\label{eq:4}\n  f_{\\ast}\\colon\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))\\longrightarrow \n  \\za^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y))\n\\end{equation}\nby $f_{\\ast}(Z,\\widetilde g)=(f_{\\ast}Z, \\widetilde{f_{\\ast} g})$,\nwhere $g$ is any representative of $\\widetilde g$, and  $f_{\\ast}(g)$ is\nthe usual direct image of currents given by\n$f_{\\ast}(g)(\\eta)=g(f^{\\ast}\\eta)$.  \n\n\\begin{proposition} \\label{prop:4}\n  The map $f_{\\ast}$ in \\eqref{eq:4} sends the group $\n  \\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$ to the group\n  $\\rata^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y))$. Therefore\n  it induces a map\n  \\begin{displaymath}\n    \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))\\longrightarrow \n  \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y)).\n  \\end{displaymath}\n\\end{proposition}\n\nIn order to transfer this push-forward to other coefficients we\nintroduce two extra properties for a $\\mathcal{D}_{\\log}(X)$-complex $C$.\n\\begin{description}\n\\item{(H1)} There is a commutative diagram of injective morphisms of\n  complexes \n\\begin{displaymath}\n  \\xymatrix{\n    \\mathcal{D}_{\\log} ^{\\ast}(X,\\ast)\\ar[r]^{\\cmap}\\ar[dr]&\n    C^{\\ast}(\\ast)\\ar[d]^{\\cmap'}   \\\\\n  & \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}} ^{\\ast}(X,\\ast).\n}\n\\end{displaymath}\nSince $\\cmap'$ is injective we will usually identify $C$ with its image by\n$\\cmap'$.\n\\item{(H2)} The map $\\cmap'$ induces isomorphisms\n  \\begin{align*}\n    H^{n}(C^{\\ast}(p))\\cong H^{n}(\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{\\ast}(X,p))\n  \\end{align*}\nfor all $p\\ge 0$ and $n=2p-1,2p$.\n\\end{description}\n\nThe conditions (H1) and (H2) have two consequences. First\nif $\\eta \\in D_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)$ is a current such that\n$\\dd_{\\mathcal{D}}\\eta\\in C^{2p}(p)$, there exist\n$a\\in\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-2}(X,p)$ such that \n$$\\dd_{\\mathcal{D}}a+\\eta\\in C^{2p-1}(p).$$\nSecond, the induced map $\\widetilde C^{2p-1}(p)\\to \\widetilde\n{\\mathcal{D}}^{2p-1}(X,p)$ is injective.  \n\nLet $C$ be a  $\\mathcal{D}_{\\log}(X)$-complex satisfying (H1).\nConsider the diagram\n\\begin{displaymath}\n  \\xymatrix{\n  \\cha^p(\\mathcal{X},C) \\ar[ddr]_{i}\\ar[dr]_{j}\\ar^{\\omega _{C}}[drr] &&\\\\\n  & A \\ar[r] \\ar[d] & \\Zr C^{2p}(p)\\ar[d] \\\\\n  & \\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)) \\ar[r]_-{\\omega} & \\Zr\n  \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p}(p) \n}\n\\end{displaymath}\nwhere $A$ is defined by the cartesian square, $i$ is induced by (H1)\nand $j$ is induced by $i$ and $\\omega _{C}$.\n\\begin{lemma} \\label{lemm:2}\n  If $C$ also satisfies (H2) then $j$ is an isomorphism.\n\\end{lemma}\n\\begin{proof}\nBy the injectivity of $\\widetilde C^{2p-1}(p)\\to \\widetilde\n{\\mathcal{D}}^{2p-1}(X,p)$ and Lemma \\ref{lemm:1}~\\ref{item:3}, the map $i$ is\ninjective. Hence  the map $j$ is injective and we only need to prove\nthat $j$ is surjective.\n\nLet $x=((Z,\\widetilde g),\\eta)\\in A$. This means that $(Z,\\widetilde\ng)\\in\\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$, $\\eta\\in \\Zr\nC^{2p}(p)$ and $\\omega _{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}(Z,\\widetilde\ng)=\\dd_{\\mathcal{D}}g+\\delta _{Z}=\\eta$. Let $g'\\in \n\\mathcal{D}^{2p-1}_{\\log}(X\\setminus |Z|,p)$ be a Green form for\n$Z$. Then $g-[g']\\in \\mathcal{D}^{2p-1}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,p)$ satisfies \n$\\dd_{\\mathcal{D}}(g-[g'])\\in C^{2p}(p)$. By (H2) there is $a\\in\n\\mathcal{D}^{2p-2}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,p)$ such that\n$g'':=g-[g']+\\dd_{\\mathcal{D}}a\\in C^{2p-1}(p)$.\nConsider the element $x'=((Z,\\widetilde g'),\\widetilde g'')\\in\n\\cha^{p}(\\mathcal{X},C)$. To see that $j(x')=x$ we have to check that \n$\\omega _{C}(x')=\\eta$ and $i(x')=(Z,\\widetilde g)$. We compute\n\\begin{gather*}\n  \\omega _{C}(x')=\\cmap(\\dd_{\\mathcal{D}}g')+\\dd_{C}g''=\n  \\dd_{\\mathcal{D}}[g']+\\delta _{Z}+\\dd_{\\mathcal{D}} g-\\dd_{\\mathcal{D}}[g']+\n  \\dd_{\\mathcal{D}}\\dd_{\\mathcal{D}}a=\\eta,\\\\\n  i(x')=(Z,([g']+g-[g']+\\dd_{\\mathcal{D}}a)^{\\sim})=(Z,\\widetilde g),\n\\end{gather*}\nconcluding the proof of the lemma.\n\\end{proof}\nWe can rephrase the lemma as follows.\n\\begin{theorem}\\label{pf=pb} \nLet $C$ be a $\\mathcal{D}_{\\log}(X)$-complex that satisfies (H1) and\n(H2). Then the map $\\varphi$ can be extended to an injective map \n\\begin{displaymath}\n  \\varphi_{C}\\colon \\za^{p}(\\mathcal{X},C)\\longrightarrow \n  \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde D_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{displaymath}\nMoreover\n\\begin{equation}\n  \\label{eq:6}\n  \\Img(\\varphi_{C})=\\left\\{(Z,\\widetilde g)\\in\n    \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde\n    {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p) \\mid \n    \\dd_{\\mathcal{D}}g+\\delta _{Z}\\in\n    C^{2p}(p)\\right\\}.   \n\\end{equation}\n\\end{theorem}\nIn view of this theorem, if $C$ satisfies (H1) and (H2), we will\nidentify $\\za^{p}(\\mathcal{X},C)$ with the right hand side of equation\n\\eqref{eq:6}. \n\n\n\\begin{proposition}\\label{prop:1} The\n  $\\mathcal{D}_{\\log}(X)$-complexes   $\\mathcal{D}_{{\\text{\\rm a}}}(X)$,\n  $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ and $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)$ satisfy\n  (H1) and (H2). Therefore we can identify\n\\begin{align*}\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))&\\cong\n  \\left\\{(Z,\\widetilde g) \\in \\Zr^{p}\\oplus \\widetilde\n    {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n    \\mid\n    \\dd_{\\mathcal{D}}g+\\delta _{Z}\\in\n    \\mathcal{D}^{2p}_{{\\text{\\rm a}}}(X,p)\\right\\}\\\\\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))&\\cong\n  \\left\\{(Z,\\widetilde g) \\in \\Zr^{p}\\oplus \\widetilde\n    {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n    \\mid \n    \\dd_{\\mathcal{D}}g+\\delta _{Z}\\in\n    \\mathcal{D}^{2p}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S,p)\\right\\}\\\\\n  \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))&\\cong\n  \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde\n    {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n  The result for $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ follows from the Poincar\\'e\n  $\\overline \\partial$-Lemma for currents\n  \\cite[Pag. 382]{GriffithsHarris:pag}. The result for\n  $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ follows from the Poincar\\'e\n  $\\overline \\partial$-Lemma for currents with fixed wave front set\n  \\cite[Theorem 4.5]{BurgosLitcanu:SingularBC}. The statement for\n  $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)$ is Lemma \\ref{lemm:3}.\n\\end{proof}\n\n\\begin{remark}\n  The previous proposition gives us a more concrete description of the\n  groups $\\za^{p}(\\mathcal{X},C)$ for $C=\\mathcal{D}_{{\\text{\\rm a}}}(X)$,\n  $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ and $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)$ and\n  also a more concrete description of the groups\n  $\\cha^{p}(\\mathcal{X},C)$. In particular it easily implies that the\n  groups $\\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$, $p\\ge 0$, agree\n  (up to a \n  normalization factor) with the arithmetic Chow groups of Gillet-Soul\\'e\n  $\\cha^{p}(\\mathcal{X})$ and that the groups\n  $\\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$, $p\\ge 0$, agree\n  (again up to a \n  normalization factor) with the\n  arithmetic Chow groups of Kawaguchi-Moriwaki $\\cha^{p}_{D}(\\mathcal{X})$, introduced in\n  \\cite{KawaguchiMoriwaki:isfav}.   \n\\end{remark}\n\nWe want to transfer the push-forward of Proposition \\ref{prop:4} to\ncomplexes satisfying (H1) and (H2). \nLet again $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\nregular arithmetic varieties of relative dimension $e$. Let \n$C$ be a $\\mathcal{D}_{\\log}(X)$-complex and $C'$ a\n$\\mathcal{D}_{\\log}(Y)$-complex, both satisfying (H1) and (H2). We assume  \nthere exists a map of complexes $f_{\\ast}\\colon C^{\\ast}(\\ast)\\to\nC'^{\\ast-2e}(\\ast-e)$ that makes the \nfollowing diagram commutative: \n\\begin{equation}\\label{eq:18}\n  \\xymatrix{\n  C^{\\ast}(\\ast)\\ar[r]^-{f_{\\ast}} \\ar[d] & C'^{\\ast-2e}(\\ast-e) \\ar[d] \\\\\n  \\mathcal{D}_{\\text{{\\rm cur}}}^{\\ast}(X,\\ast)\\ar[r]_-{f_{\\ast}} & \\mathcal{D}_{\\text{{\\rm cur}}}^{\\ast-2e}(Y,\\ast-e).\n} \n\\end{equation}\nThen, for $p\\ge 0$, we define maps\n\\begin{displaymath}\n  f_{\\ast}\\colon \\za^{p}(\\mathcal{X},C)\\to \\za^{p-e}(\\mathcal{Y},C') \n\\end{displaymath}\nthat, using the identification of Theorem \\ref{pf=pb} are given by\n\\begin{equation}\\label{dir_im_gen}\nf_{\\ast}(Z,\\widetilde g)=(f_{\\ast}Z,\\widetilde {f_{\\ast}(g)}).\n\\end{equation}\nBy Proposition \\ref{prop:4}, these maps send $\n\\rata^{p}(\\mathcal{X},C)$ to \n$\\rata^{p-e}(\\mathcal{Y},C')$. Therefore they\ninduce morphisms\n\\begin{displaymath}\n  f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},C)\\longrightarrow \n  \\cha^{p-e}(\\mathcal{Y},C').\n\\end{displaymath}\n\nBefore stating concrete examples of this covariant functoriality\nwe need some notation.  Let\n $f_{{\\mathbb C}}\\colon X_{{\\mathbb C}}\\to Y_{{\\mathbb C}}$ denote \nthe induced proper map of complex manifolds.\n If $S\\subset\n  T^{\\ast}_{0}X_{{\\mathbb C}}$ is a closed conical subset invariant under\n  $F_{\\infty}$, then we write\n  \\begin{displaymath}\n    f_{\\ast}S=\\{(f(x),\\xi)\\in T^{\\ast}_{0}Y_{{\\mathbb C}}\\mid (x,\\dd\n    f^{t}\\xi)\\in S\\}\\cup N_{f}. \n  \\end{displaymath}\n  It is a closed conical subset of $T^{\\ast}_{0}Y_{{\\mathbb C}}$ invariant under\n  $F_{\\infty}$.\n\n  \\begin{proposition}\\label{prop:8}\n    Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$, $g\\colon \\mathcal{Y}\\to\\mathcal{Z}$ be proper morphism of\nregular arithmetic varieties of relative dimension $e$ and $e'$, and $S\\subset\n  T^{\\ast}_{0}X_{{\\mathbb C}}$ a closed conical subset invariant under\n  $F_{\\infty}$. Then there are maps\n  \\begin{displaymath}\n    f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\n    \\to \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}S)),\n  \\end{displaymath}\n  and similarly\n  \\begin{displaymath}\n  \tg_{\\ast}\\colon \\cha^{p'}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}S))\n    \\to \\cha^{p'-e'}(\\mathcal{Z},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Z,g_{\\ast}f_{\\ast}S)).\n  \\end{displaymath}\n  Furthermore, the relation $(g\\circ f)_{\\ast}=g_{\\ast}\\circ f_{\\ast}$ is satisfied.\n  \\end{proposition}\n\nAs a particular case of the above proposition we obtain the following\ncases.\n\\begin{corollary}\\label{cor:1}\n     Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\nregular arithmetic varieties of relative dimension $e$.\n\\begin{enumerate}\n\\item  There are maps\n  \\begin{displaymath}\n    f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n    \\to \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,N_{f})).\n  \\end{displaymath}\n\\item If $N_{f}=\\emptyset$, then there are maps \n  \\begin{displaymath}\n    f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n    \\to \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{{\\text{\\rm a}}}(Y)).\n  \\end{displaymath}\n\\end{enumerate}\n\\end{corollary}\n\nThe functoriality and the multiplicative structures satisfy the\nfollowing compatibility properties.\n\n\\begin{theorem}\\label{thm:1}\n  Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a morphism of regular arithmetic\n  varieties. Let $N_{f}$ be the set of normal directions of $f_{{\\mathbb C}}$ and\n  $S,\\,S'\\subset\n  T^{\\ast}_{0}Y_{{\\mathbb C}}$  closed conical subsets invariant under\n  $F_{\\infty}$. Then\n  \\begin{displaymath}\n    f^{\\ast}(S\\cup\n        S'\\cup \n        (S+S'))=f^{\\ast}(S)\\cup f^{\\ast}(\n        S')\\cup \n        (f^{\\ast}(S)+f^{\\ast}(S'))).\n  \\end{displaymath}\n  If $N_{f}\\cap(S\\cup S'\\cup S+S')=\\emptyset$ and $S\\cap\n  (-S')=\\emptyset$ then $f^{\\ast}(S)\\cap\n  (-f^{\\ast}(S'))=\\emptyset$. In\n  this case, if $\\alpha \\in\n  \\cha^p(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))$ and $\\beta \\in\n  \\cha^q(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S'))$ then\n  \\begin{displaymath}\n    f^{\\ast}(\\alpha \\cdot \\beta )=f^{\\ast}(\\alpha )\\cdot\n    f^{\\ast}(\\beta )\\in\n    \\cha^{p+q}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S\\cup\n    S'\\cup \n    (S+S')))). \n  \\end{displaymath}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:2}\n    Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\n    regular arithmetic \n  varieties of relative dimension $e$. Let $N_{f}$ be the set of\n  normal directions of $f_{{\\mathbb C}}$,\n  $S\\subset\n  T^{\\ast}_{0}X_{{\\mathbb C}}$ and   $S'\\subset\n  T^{\\ast}_{0}Y_{{\\mathbb C}}$  closed conical subsets invariant under\n  $F_{\\infty}$. Then\n  \\begin{displaymath}\n    f_{\\ast}(S\\cup f^{\\ast}(\n        S')\\cup \n        (S+f^{\\ast}(S')))\\subset f_{\\ast}(S)\\cup\n        S'\\cup \n        (f_{\\ast}(S)+S')).\n  \\end{displaymath}\n  If $f_{\\ast}(S)\\cap (-S')=\\emptyset$ then $N_{f}\\cap\n  S'=\\emptyset$ and $S\\cap (-f^{\\ast}(S'))=\\emptyset$. In this case, if\n  $\\alpha \\in \n  \\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))$ and $\\beta \\in\n      \\cha^q(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S'))$ then\n      \\begin{displaymath}\n        f_{\\ast}(\\alpha \\cdot f^{\\ast}(\\beta) )=f_{\\ast}(\\alpha )\\cdot\n        \\beta \\in\n        \\cha^{p+q-e}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(S)\\cup\n        S'\\cup \n        (f_{\\ast}(S)+S'))). \n      \\end{displaymath} \n\\end{theorem}\n\n\\section{Arithmetic $K$-theory and derived categories}\\label{section:ArKTheory}\n\\subsection{Arithmetic $K$-theory}\nAs for arithmetic Chow groups, the arithmetic $K$-groups of\nGillet-Soul\\'e can be generalized to include more general\ncoefficients at the archimedean places. Because the\ndefinition of arithmetic $K$-groups involves hermitian vector bundles\nwhose metrics have arbitrary singularities at infinity, we are\nactually forced to consider $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex\ncoefficients. The reader is referred to\n\\cite[Sec. 4.2]{BurgosKramerKuehn:accavb} for the construction of the\ngroups $\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})$, which provide\nthe base of our extension (see also Definition \\ref{def:2} below). An\narithmetic Chern character allows to compare these generalized\narithmetic $K$-groups and the generalized arithmetic Chow groups. The\narithmetic Chern character is automatically compatible with pull-back\nand products whenever defined.\n\n\\begin{definition}\\label{def:2}\nLet $\\mathcal{X}$ be an arithmetic variety and $C^{\\ast}(\\ast)$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex with structure morphism $\\cmap:\\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X,\\ast)\\rightarrow C^{\\ast}(\\ast)$. The arithmetic $K$ group of $\\mathcal{X}$ with $C$ coefficients is the abelian group $\\widehat{K}_{0}(\\mathcal{X},C)$ generated by pairs $(\\overline{\\mathcal{E}},\\eta)$, where $\\overline{\\mathcal{E}}$ is a smooth hermitian vector bundle on $\\mathcal{X}$ and $\\eta\\in\\bigoplus_{p\\geq 0}\\widetilde{C}^{2p-1}(p)$, modulo the relations\n\\begin{displaymath}\n\t(\\overline{\\mathcal{E}}_{1},\\eta_{1})+(\\overline{\\mathcal{E}}_{2},\\eta_{2})=(\\overline{\\mathcal{E}},\\cmap(\\widetilde{\\ch}(\\overline{\\varepsilon}))+\\eta_{1}+\\eta_{2}),\n\\end{displaymath}\nfor every exact sequence\n\\begin{displaymath}\n\t\\overline{\\varepsilon}\\colon\\quad 0\\longrightarrow\\overline{\\mathcal{E}}_{1}\\longrightarrow\\overline{\\mathcal{E}}\\longrightarrow\\overline{\\mathcal{E}}_{2}\\longrightarrow 0\n\\end{displaymath}\nwith Bott-Chern secondary class $\\widetilde{\\ch}(\\overline{\\varepsilon})$.\n\\end{definition}\nAn equivalent construction can be given in the same lines as for the generalized arithmetic Chow groups. For this, observe there is a natural morphism\n\\begin{equation}\\label{eq:7}\n\t\\begin{split}\n\t\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})&\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C)\\\\\n\t\t[\\overline{\\mathcal{E}},\\eta]&\\longmapsto [\\overline{\\mathcal{E}},\\cmap(\\eta)]\n\t\\end{split}\n\\end{equation}\ninduced by $\\cmap:\\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X,\\ast)\\rightarrow C^{\\ast}(\\ast)$, and also\n\\begin{equation}\\label{eq:8}\n\t\\begin{split}\n\t\t\\bigoplus_{p\\geq 0}\\widetilde{C}^{2p-1}(p)&\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C)\\\\\n\t\t\\eta&\\longmapsto [0,\\eta].\n\t\\end{split}\n\\end{equation}\nOne easily sees the maps \\eqref{eq:7}--\\eqref{eq:8} induce a natural isomorphism of groups\n\\begin{equation}\\label{eq:9}\n\t\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})\\times\\bigoplus_{p\\geq 0}\\widetilde{C}^{2p-1}(p)\/\\equiv\\;\n\t\\overset{\\cong}{\\longrightarrow}\\widehat{K}_{0}(\\mathcal{X},C),\n\\end{equation}\nwhere $\\equiv$ is the equivalence relation generated by\n\\begin{displaymath}\n\t((\\overline{\\mathcal{E}},\\eta),0)\\equiv ((\\overline{\\mathcal{E}},0),\\cmap(\\eta)).\n\\end{displaymath}\nGeneralized arithmetic $K$-groups for suitable complexes have\npull-backs and products. Let $f:\\mathcal{X}\\to\\mathcal{Y}$ be a morphism\nof arithmetic varieties, and suppose given $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and\n$\\mathcal{D}_{{\\text{\\rm a}}}(Y)$ complexes $C$ and $C'$ respectively, for which\nthere is a commutative diagram \n\\begin{equation}\\label{eq:12}\n  \\xymatrix{\n    \\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(Y)\\ar[d]\\ar[r]^{f^{\\ast}}\t\t\n    &\\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X)\\ar[d]\\\\\n    C^{\\prime\\ast}\\ar[r]_{f^{\\ast}}\n    &C^{\\ast}.\n  }\n\\end{equation}\nBy description \\eqref{eq:9} and the contravariant functoriality of\n$\\widehat{K}_{0}(\\underline{\\ },\\mathcal{D}_{{\\text{\\rm a}}})$, we see there is\nan induced morphism of groups \n\\begin{displaymath}\n\tf^{\\ast}:\\widehat{K}_{0}(\\mathcal{Y},C^{\\prime})\n        \\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C).\n\\end{displaymath}\nFor this kind of functoriality, an analog statement to Proposition\n\\ref{prop:3} holds, and we leave to the reader the task of stating\nit. As for products, let $C$, $C'$ and $C''$ be\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes with a product $C\\otimes\nC'\\overset{\\bullet}{\\rightarrow} C''$ and a commutative diagram\n\\begin{equation}\\label{eq:19}\n\t\\xymatrix{\n\t\t\\mathcal{D}_{{\\text{\\rm a}}}(X)\\otimes\\mathcal{D}_{{\\text{\\rm a}}}(X)\\ar[r]^-{\\bullet}\\ar[d]\t&\\mathcal{D}_{{\\text{\\rm a}}}(X)\\ar[d]\\\\\n\t\tC\\otimes C'\\ar[r]_-{\\bullet}\t&C''.\n\t}\n\\end{equation}\nThen there is an induced product at the level of $\\widehat{K}_{0}$\n\\begin{displaymath}\n\t\\widehat{K}_{0}(\\mathcal{X},C)\\times\\widehat{K}_{0}(\\mathcal{X},C')\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C'')\n\\end{displaymath}\ndescribed by the rule\n\\begin{displaymath}\n\t[\\overline{\\mathcal{E}},\\eta]\\cdot [\\overline{\\mathcal{E}}',\\eta']=\n\t\t[\\overline{\\mathcal{E}}\\otimes\\overline{\\mathcal{E}}',c(\\ch(\\overline{\\mathcal{E}}))\\bullet\\eta'+c'(\\ch(\\overline{\\mathcal{E}}'))\\bullet\\eta\n\t\t+\\dd_{C}\\eta\\bullet\\eta'].\n\\end{displaymath}\nWith respect to this laws, the groups $\\widehat{K}_{0}$ enjoy of the\nanalogue properties to Proposition \\ref{prop:2}. \n\nFinally, we discuss on the arithmetic Chern character. If\n$\\mathcal{X}$ is a regular arithmetic variety, we recall there is an\nisomorphism of rings \\cite[Thm. 4.5]{BurgosKramerKuehn:accavb}, namely \n\\begin{displaymath}\n  \\chh:\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})_{{\\mathbb Q}}\\longrightarrow\n  \\bigoplus_{p\\geq 0}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}).\n\\end{displaymath}\nIf $C$ is a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, by the presentations\n\\eqref{eq:11} of $\\cha^{\\ast}(\\mathcal{X},C)$ and \\eqref{eq:9} of\n$\\widehat{K}_{0}$ it is clear that $\\chh$ extends to an  isomorphism\nof groups \n\\begin{displaymath}\n\t\\chh:\\widehat{K}_{0}(\\mathcal{X},C)_{{\\mathbb Q}}\\longrightarrow\\bigoplus_{p\\geq 0}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C).\n\\end{displaymath}\nSuppose now that $C,C',C''$ are $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes with\na product $C\\otimes C'\\rightarrow C''$ as above. Then, it is easily\nseen that there is a commutative diagram of morphisms of groups\n\\begin{displaymath}\n  \\xymatrix{\n    \\widehat{K}_{0}(\\mathcal{X},C)_{{\\mathbb Q}}\\times\\widehat{K}_{0}(\\mathcal{X},C')_{{\\mathbb Q}}\n    \\ar[r]^{\\hspace{1cm}\\cdot}\\ar[d]_{(\\chh,\\chh)}\n    &\\widehat{K}_{0}(\\mathcal{X},C'')_{{\\mathbb Q}}\\ar[d]^{\\chh}\\\\ \n    \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C)\\times\\bigoplus_{p}\n    \\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C')\\ar[r]^{\\hspace{1.3cm\\cdot}}\n    &\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C''). \n  }\n\\end{displaymath}\nThis is in particular true for complexes of currents with controlled wave front set, a result that we next record.\n \\begin{proposition}\nLet $S, S'$ be closed conical subsets of $T_{0}^{\\ast}X_{{\\mathbb C}}$\ninvariant under the action of complex conjugation, with $S\\cap\n(-S')=\\emptyset$. Define $T=S\\cup S'\\cup (S+S')$. If $\\alpha \\in\n\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))_{{\\mathbb Q}}$ and\n$\\beta \\in\n\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S'))_{{\\mathbb Q}}$, then\n\\begin{displaymath}\n  \\chh(\\alpha )\\cdot \\chh(\\beta )=\\chh(\\alpha \\cdot \\beta )\\in\n  \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T)). \n\\end{displaymath}\n\\end{proposition}\nAnother feature of the Chern character is its compatibility with\npull-back functoriality. Let $f:\\mathcal{X}\\rightarrow\\mathcal{Y}$ be\na morphism of regular arithmetic varieties. Let $C,C'$ be\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$ complexes,\nrespectively, together with a morphism of complexes\n$f^{\\ast}:C'\\rightarrow C$ satifying the commutativity\n\\eqref{eq:12}. Then there is a commutative diagram\n\\begin{displaymath}\n  \\xymatrix{\n    \\widehat{K}_{0}(\\mathcal{Y},C')_{{\\mathbb Q}}\\ar[r]^{f^{\\ast}}\\ar[d]_{\\chh}\n    &\\widehat{K}_{0}(\\mathcal{X},C)_{{\\mathbb Q}}\\ar[d]^{\\chh}\\\\ \n    \\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{Y},C')\\ar[r]^{f^{\\ast}}\n    &\\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{X},C).\n  }\n\\end{displaymath}\nAgain, the proof is a simple consequence for the known compatibility\nin the case of the $\\mathcal{D}_{{\\text{\\rm a}}}$ arithmetic Chow groups. In\nparticular we have the following proposition.\n\\begin{proposition}\\label{prop:6}\n  Let $f\\colon \\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism of regular\n  arithmetic varieties, and $S\\subset T_{0}^{\\ast}Y_{{\\mathbb C}}$ a closed\n  conical subset, invariant under complex conjugation and disjoint\n  with the normal directions $N_{f}$ of $f_{{\\mathbb C}}$. Then there is a\n  commutative diagram\n\\begin{displaymath}\n  \\xymatrix{\n    \\widehat{K}_{0}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))_{{\\mathbb Q}}\\ar[r]^{f^{\\ast}}\\ar[d]_{\\chh}\t\n    &\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S))_{{\\mathbb Q}}\\ar[d]^{\\chh}\\\\\n    \\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))\\ar[r]^{f^{\\ast}}\n    &\\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S)). \n  }\n\\end{displaymath}\n\\end{proposition}\n\\subsection{Arithmetic derived categories}\nFor the problem of defining direct images on arithmetic $K$-theory it\nis useful to deal with arbitrary complexes of coherent sheaves instead\nof locally free sheaves. We therefore introduce an arithmetic\ncounterpart of our theory of hermitian structures on derived\ncategories of coherent sheaves, developed in\n\\cite{BurgosFreixasLitcanu:HerStruc}, and the $\\hDb$ categories in\n\\cite{BurgosFreixasLitcanu:GenAnTor}. We then compare this\nconstruction to the arithmetic $K$ groups. \n\n\\begin{definition}\nLet $S$ be a scheme. We denote by $\\Db(S)$ the derived category of cohomological complexes of quasi-coherent sheaves with bounded coherent cohomology.\n\\end{definition}\nIf $\\mathcal{X}$ is a regular arithmetic variety, every object of\n$\\Db(\\mathcal{X})$ is quasi-isomorphic to a bounded cohomological\ncomplex of locally free sheaves. One checks there is a well defined\nmap \n\\begin{displaymath}\n\t\\Ob\\Db(\\mathcal{X})\\longrightarrow K_{0}(\\mathcal{X})\n\\end{displaymath}\nthat sends an object $\\mathcal{F}^{\\ast}$ to the class\n$\\sum_{i}(-1)^{i}[\\mathcal{E}^{i}]$, where $\\mathcal{E}^{\\ast}$ is\nquasi-isomorphic to $\\mathcal{F}^{\\ast}$, and that is compatible with\nderived tensor products on $\\Db(\\mathcal{X})$ and the ring structure\non $K_{0}(\\mathcal{X})$. Our aim is thus to extend this picture and\nincorporate hermitian structures. \n\nLet us consider the complex quasi-projective manifold $X_{{\\mathbb C}}$\nassociated to an arithmetic variety $\\mathcal{X}$. It comes equipped\nwith the conjugate-linear involution $F_{\\infty}$. Recall the data\n$X=(X_{{\\mathbb C}},F_{\\infty})$ uniquely determines a smooth quasi-projective\nscheme $X_{{\\mathbb R}}$ over ${\\mathbb R}$, whose base change to ${\\mathbb C}$ is isomorphic\nto $X_{{\\mathbb C}}$, and such that its natural automorphism given by complex\nconjugation gets identified to $F_{\\infty}$. The abelian category of\nquasi-coherent (resp. coherent) sheaves over $X_{{\\mathbb R}}$ is equivalent\nto the category of quasi-coherent (resp. coherent) sheaves over\n$X_{{\\mathbb C}}$, equivariant with respect to the action of\n$F_{\\infty}$. Namely, giving a quasi-coherent (resp. coherent) sheaf\non $X_{{\\mathbb R}}$ is equivalent to giving a quasi-coherent (resp. coherent)\nsheaf $\\mathcal{F}$ on $X_{{\\mathbb C}}$, together with a morphism of\nsheaves\n\\begin{displaymath}\n\t\\mathcal{F}\\longrightarrow F_{\\infty\\ast}\\mathcal{F}\n\\end{displaymath}\ncompatible with the conjugate-linear morphism\n\\begin{displaymath}\n\tF_{\\infty}^{\\sharp}:{\\mathcal O}_{X_{{\\mathbb C}}}\\longrightarrow F_{\\infty\\ast}{\\mathcal O}_{X_{{\\mathbb C}}}\n\\end{displaymath}\ninduced by the morphism of ${\\mathbb R}$-schemes $F_{\\infty} \\colon\nX_{{\\mathbb C}}\\rightarrow X_{{\\mathbb C}}$. A similar condition characterizes\nmorphisms of quasi-coherent (resp. coherent) sheaves. Therefore, we\ndenote the bounded derived category of coherent sheaves on $X_{{\\mathbb R}}$\njust by $\\Db(X)$.\n\nThe theory of hermitian structures on the bounded derived category of\ncoherent sheaves on a complex algebraic manifold developed in\n\\cite{BurgosFreixasLitcanu:HerStruc} can be adapted to the real\nsituation of $\\Db(X)$, by considering hermitian structures invariant\nunder the action of complex conjugation. All the results in\n\\emph{loc. cit.} carry over to the real case. We denote by $\\oDb(X)$\nthe category whose objects are objects of $\\Db(X)$ endowed with a\nhermitian structure (\\emph{loc. cit.}, Def. 3.10) invariant under\ncomplex conjugation, and whose morphisms are just morphisms in\n$\\Db(X)$. Thus, every object $\\overline{\\mathcal{F}}^{\\ast}$ in $\\oDb(X)$ is\nrepresented by a quasi-isomorphism\n$\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, where\n$\\overline{\\mathcal{E}}^{\\ast}$ is a bounded complex of hermitian locally\nfree sheaves on $X_{{\\mathbb C}}$, equivariant under $F_{\\infty}$. There is an\nobvious forgetful functor $\\mathfrak{F}:\\oDb(X)\\rightarrow\\Db(X)$,\nthat makes of $\\oDb(X)$ a principal fibered category over $\\Db(X)$,\nwith structural group $\\overline{\\KA}(X)$, the group of hermitian structures\nover the 0 object \\cite[Def. 2.34,\nThm. 3.13]{BurgosFreixasLitcanu:HerStruc}.\n\nBase change to ${\\mathbb R}$ induces a covariant functor\n$\\Db(\\mathcal{X})\\rightarrow\\Db(X)$.\n\n\\begin{definition}\nWe define the category $\\oDb(\\mathcal{X})$ as the fiber product category\n\\begin{displaymath}\n  \\xymatrix{\n    \\Db(\\mathcal{X})\\times_{\\Db(X)}\\oDb(X)\\ar[r]\\ar[d]\t&\\oDb(X)\\ar[d]^{\\mathfrak{F}}\\\\\n    \\Db(\\mathcal{X})\\ar[r]\t&\\Db(X).\t\n  }\n\\end{displaymath}\nWe still denote by $\\mathfrak{F}$ the forgetful functor $\\oDb(\\mathcal{X})\\rightarrow\\Db(\\mathcal{X})$.\n\\end{definition}\nBy construction, $\\mathfrak{F}$ makes of $\\oDb(\\mathcal{X})$ a principal fibered category over $\\Db(\\mathcal{X})$, with structure group $\\overline{\\KA}(X)$. In particular, $\\overline{\\KA}(X)$ acts on $\\oDb(\\mathcal{X})$.\n\nThe Bott-Chern secondary character $\\cht$ can be defined at the level of $\\overline{\\KA}$ groups \\cite[Sec. 4, Def. 4.6]{BurgosFreixasLitcanu:HerStruc}. In our situation, we actually have a morphism of groups\n\\begin{equation}\\label{eq:13}\n\t\\cht:\\overline{\\KA}(X)\\longrightarrow\\bigoplus_{p}\\widetilde{\\mathcal{D}}_{{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{equation}\nMore generally, if $C$ is a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, we may consider a secondary Chern character with values in $\\bigoplus_{p}\\widetilde{C}^{2p-1}(p)$, that we denote $\\cht_{C}$. In particular, $\\overline{\\KA}(X)$ acts on $\\bigoplus_{p}\\widetilde{C}^{2p-1}(p)$ through $\\cht_{C}$.\n\\begin{definition}\nThe arithmetic derived category $\\hDb(\\mathcal{X},C)$ is defined as the cartesian product\n\\begin{displaymath}\n\t\\oDb(\\mathcal{X})\\times_{\\overline{\\KA}(X),\\cht_{C}}\\bigoplus_{p}\\widetilde{C}^{2p-1}(p).\n\\end{displaymath}\n\\end{definition}\n\\begin{remark}\\label{rem:1}\nIf $\\mathcal{X}$ is regular, then every object of $\\hDb(\\mathcal{X},C)$ can be represented by \n\\begin{displaymath}\n\t(\\mathcal{F}^{\\ast},\\overline{\\mathcal{E}}_{{\\mathbb C}}\\dashrightarrow\\mathcal{F}^{\\ast}_{{\\mathbb C}},\\widetilde{\\eta})\n\\end{displaymath}\nwhere $\\mathcal{E}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$ is any quasi-isomorphism from a bounded complex of locally free sheaves over $\\mathcal{X}$. Indeed, $\\mathcal{X}$ is assumed to be regular, so that $\\mathcal{F}^{\\ast}$ is quasi-isomorphic to a bounded complex of locally free sheaves $\\mathcal{E}^{\\ast}$. One then endows the $\\mathcal{E}^{i}$ with smooth hermitian metrics invariant under complex conjugation, and takes into account that $\\overline{\\KA}(X)$ acts transitively on the hermitian structures on $\\mathcal{F}^{\\ast}_{{\\mathbb C}}$. We introduce the simplified notation $(\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F},\\widetilde{\\eta})$ for such representatives.\n\\end{remark}\nThe categories $\\hDb(\\mathcal{X},C)$ are the arithmetic analogues to those we introduced in \\cite[Sec. 4]{BurgosFreixasLitcanu:GenAnTor}, and have similar properties. We are only going to review some of them.\n\nFor the next proposition, we recall that any group can be considered\nas a category, with morphisms given by the group law. This in\nparticular the case of $\\widehat{K}_{0}(\\mathcal{X},C)$.\n\\begin{theorem}\nIf $\\mathcal{X}$ is regular, there is a natural functor\n\\begin{displaymath}\n  \\hDb(\\mathcal{X},C)\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C),\n\\end{displaymath}\nthat, at the level of objects is given by\n\\begin{displaymath}\n  (\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F},\\widetilde{\\eta})\\longmapsto\n  \\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},\\widetilde{\\eta}\\right],\n\\end{displaymath}\nand at the level of morphism sends any $f\\in \\Hom_{\\hDb(\\mathcal{X},C)}(A,B)$ to $[B]-[A]$.\n\\end{theorem}\n\\begin{proof}\nIt is enough to see that the defining assignment does not depend on\nthe representatives. For this, take two objects\n$(\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta})$\nand\n$(\\overline{\\mathcal{E}}^{\\prime\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta}^{\\prime})$\ngiving raise to the same class. We have an equivalence of objects\n\\begin{displaymath}\n  \\begin{split}\n    (\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta})\\sim&\n    (\\overline{\\mathcal{E}}^{\\prime\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta}^{\\prime})\\\\\n    &\\sim\n    (\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta}^{\\prime}+c(\\widetilde{\\ch}(\\Id\\colon\n    \\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast}))).\n  \\end{split}\n\\end{displaymath}\nConsequently\n\\begin{displaymath}\n\t\\widetilde{\\eta}=\\widetilde{\\eta}^{\\prime}+c(\\widetilde{\\ch}(\\Id\\colon\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast})).\n\\end{displaymath}\nHence, the image of $(\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta})$ is equivalently written\n\\begin{displaymath}\n\t\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},\\widetilde{\\eta}^{\\prime}+c(\\widetilde{\\ch}(\\Id\\colon\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast}))\\right].\n\\end{displaymath}\nWe thus have to show the equality in $\\widehat{K}_{0}(\\mathcal{X},C)$\n\\begin{displaymath}\n\t[0,c(\\widetilde{\\ch}(\\Id\\colon\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast}))]\\overset{?}{=}\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{\\prime\n            i},0\\right]-\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},0\\right].\n\\end{displaymath}\nBy \\cite[Lemma 3.5]{BurgosFreixasLitcanu:HerStruc}, the quasi-isomorphism $\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{E}}^{\\prime\\ast}$ inducing the identity on $\\mathcal{F}^{\\ast}$ can be lifted to a diagram\n\\begin{displaymath}\n  \\xymatrix{\n    &\\overline{\\mathcal{E}}^{''\\ast}\\ar[ld]_{a}\\ar[rd]^{b}\t&\\\\\n    \\overline{\\mathcal{E}}^{\\ast}\t&\t&\\overline{\\mathcal{E}}^{\\prime\\ast},\n  }\n\\end{displaymath}\nwhere $a$ and $b$ are quasi-isomorphisms and $\\ocone(a)$ is meager \\cite[Def. 2.9]{BurgosFreixasLitcanu:HerStruc}. On the one hand, by the characterization \\cite[Thm. 2.13]{BurgosFreixasLitcanu:HerStruc} of meager complexes, one can show\n\\begin{displaymath}\n\t\\left[\\sum_{i}(-1)^{i}\\ocone(a)^{i},0\\right]=0.\n\\end{displaymath}\nOn the other hand, we have an exact sequence of complexes\n\\begin{displaymath}\n\t0\\rightarrow\\overline{\\mathcal{E}}^{''\\ast}\\rightarrow\\ocone(a)\\rightarrow\\overline{\\mathcal{E}}^{\\ast}[1]\\rightarrow 0,\n\\end{displaymath}\nwhose constituent rows are orthogonally split. This shows\n\\begin{equation}\\label{eq:14}\n\t\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{'' i},0\\right]-\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},0\\right]=\\left[\\sum_{i}(-1)^{i}\\ocone(a)^{i},0\\right]=0.\n\\end{equation}\nSimilarly we have\n\\begin{equation}\\label{eq:15}\n\t\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{'i},0\\right]-\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{'' i},0\\right]=\\left[\\sum_{i}(-1)^{i}\\ocone(b)^{i},0\\right].\n\\end{equation}\nBut the complex underlying $\\ocone(b)$ is acyclic, so that\n\\begin{equation}\\label{eq:16}\n\t\\left[\\sum_{i}(-1)^{i}\\ocone(b)^{i},0\\right]=[0,c(\\widetilde{\\ch}(\\ocone(b))].\n\\end{equation}\nFinally, by \\cite[Def. 3.14, Thm. 4.11]{BurgosFreixasLitcanu:HerStruc} we have\n\\begin{equation}\\label{eq:17}\n\t\\widetilde{\\ch}(\\Id:\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{'\\ast})=\\widetilde{\\ch}(\\ocone(b)).\n\\end{equation}\nPutting \\eqref{eq:14}--\\eqref{eq:17} together allows to conclude.\n\\end{proof}\n\\begin{notation}\nLet $\\mathcal{X}$ be a regular arithmetic variety. Then we still denote the image of an object $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]\\in\\Ob\\hDb(\\mathcal{X},C)$ by the morphism of the proposition by $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$. We call this image the class of $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$ in arithmetic $K$-theory.\n\\end{notation}\n\\begin{remark}\nTwo tightly isomorphic objects in $\\hDb(\\mathcal{X},C)$ have the same class in arithmetic $K$-theory.\n\\end{remark}\nLet $f:\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism of regular arithmetic varieties and let $C$, $C'$ be $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$ complexes respectively, with a commutative diagram as in \\eqref{eq:12}. Then there is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{Y},C')\\ar[r]^{f^{\\ast}}\\ar[d]\t&\\hDb(\\mathcal{X},C)\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{Y},C')\\ar[r]_{f^{\\ast}}\t&\\widehat{K}_{0}(\\mathcal{X},C).\n\t}\n\\end{displaymath}\nThe following statement is a particular case.\n\\begin{proposition}\nLet $f:\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism of regular arithmetic varieties, and $S\\subset T_{0}^{\\ast}Y_{{\\mathbb C}}$ a closed conical subset  invariant under complex conjugation, disjoint with the normal directions $N_{f}$ of $f_{{\\mathbb C}}$. Then there is a commutative diagram\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))\\ar[r]^-{f^{\\ast}}\\ar[d]\t&\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X, f^{\\ast}(S)))\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))\\ar[r]_-{f^{\\ast}}\t&\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S))).\n\t}\n\\end{displaymath}\n\\end{proposition}\nIf $C,C',C''$ are $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes with a product $C\\otimes C'\\rightarrow C''$ compatible with the product of $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and $\\mathcal{X}$ is regular, then there is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{X},C)\\times\\hDb(\\mathcal{X},C')\\ar[r]^-{\\otimes} \\ar[d]\t&\\hDb(\\mathcal{X},C'')\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{X},C)\\times\\widehat{K}_{0}(\\mathcal{X},C')\\ar[r]\t&\\widehat{K}_{0}(\\mathcal{X},C''),\n\t}\n\\end{displaymath}\nwhere the derived tensor product $\\otimes$ is defined by\n\\begin{displaymath}\n\t[\\overline{\\mathcal{F}},\\eta]\\otimes [\\overline{\\mathcal{G}},\\nu]=\n    [\\overline{\\mathcal{F}}\\otimes\\overline{\\mathcal{G}},\n    c(\\ch(\\overline{\\mathcal{F}}))\\bullet\\nu+\n    \\eta\\bullet c'(\\ch(\\overline{\\mathcal{G}}))+\\dd_{C}\\eta\\bullet\\nu]\n\\end{displaymath}\n\\begin{proposition}\nLet $S, S'$ be closed conical subsets of $T_{0}^{\\ast}X_{{\\mathbb C}}$ invariant under the action of complex conjugation, with $S\\cap (-S')=\\emptyset$. Define $T=S\\cup S'\\cup (S+S')$. If $\\mathcal{X}$ is regular, then there is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\\times\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S'))\\ar[r]^-{\\otimes} \\ar[d]\t\n\t\t\t&\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T))\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\\times\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S'))\\ar[r]\t\n\t\t&\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T)).\n\t}\n\\end{displaymath}\nFurthermore, it is compatible with pull-back $f^{\\ast}$ whenever defined.\n\\end{proposition}\nFinally, the class functor and the arithmetic Chern character on arithmetic $K$ groups, allow to extend it to arithmetic derived categories.\n\\begin{notation}\\label{not:1}\nLet $\\mathcal{X}$ be a regular arithmetic variety and $C$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ complex. We denote by\n\\begin{displaymath}\n\t\\chh:\\hDb(\\mathcal{X},C)\\longrightarrow\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C)\n\\end{displaymath}\nthe arithmetic Chern character on $\\hDb(\\mathcal{X},C)$, obtained as\nthe composition of the class functor and $\\chh$ on\n$\\widehat{K}_{0}(\\mathcal{X},C)$.\n\\end{notation}\n\\section{Arithmetic characteristic classes}\\label{section:ArChar}\nLet $\\mathcal{X}$ be a regular arithmetic variety. From the previous\nsections, there exists a natural functor \n\\begin{displaymath}\n  \\oDb(\\mathcal{X})\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n\\end{displaymath}\nthat factors through $\\hDb(\\mathcal{X},C)$, and there is a ring isomorphism\n\\begin{displaymath}\n  \\chh:\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))_{{\\mathbb Q}}\\longrightarrow\n  \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)).\n\\end{displaymath}\nWe therefore obtain a functor\n\\begin{displaymath}\n  \\chh:\\oDb(\\mathcal{X})\\longrightarrow\n  \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n\\end{displaymath}\nautomatically satisfying several compatibilities with the operations\nin $\\oDb(\\mathcal{X})$ and distinguished triangles. More generally, in\nthis section we construct arithmetic characteristic classes attached\nto real additive or multiplicative genera. The case of the arithmetic\nTodd class will be specially relevant, since it is involved in the\narithmetic Riemann-Roch theorem. Our construction relies on the one\ngiven by Gillet-Soul\\'e \\cite{GilletSoule:vbhm}.\n\nLet $B$ be a subring of ${\\mathbb R}$ and $\\varphi\\in B[[x]]$ a real power series, defining an\nadditive genus. For each hermitian vector bundle $\\overline {\\mathcal{E}}$, in\n\\cite{GilletSoule:vbhm}, there is attached a class\n$\\widehat{\\varphi}(\\overline{\\mathcal{E}})\\in \\cha_{B}\n^{\\ast}(\\mathcal{X})$. By the isomorphism \\cite[Theorem\n3.33]{BurgosKramerKuehn:accavb} we obtain a class \n$\\widehat{\\varphi}(\\overline{\\mathcal{E}})\\in \\cha_{B}\n^{\\ast}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.  \n\n\nFor every finite complex of smooth hermitian vector\nbundles $\\overline{\\mathcal{E}}^{\\ast}$, we put\n\\begin{displaymath}\n  \\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})=\n  \\sum_{i}(-1)^{i}\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{i})\\in\n  \\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)).  \n\\end{displaymath}\nLet us now consider an object $\\overline{\\mathcal{F}}^{\\ast}$ in\n$\\oDb(\\mathcal{X})$. We choose an auxiliary quasi-isomor\\-phism\n$\\psi:\\mathcal{E}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, where\n$\\mathcal{E}^{\\ast}$ is a bounded complex of locally free sheaves on\n$\\mathcal{X}$. This is possible since $\\mathcal{X}$ is regular by\nassumption. We also fix auxiliary smooth hermitian metrics on the\nindividual terms $\\mathcal{E}^{i}$. We thus obtain an isomorphism\n$\\overline{\\psi}\\colon\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{F}}^{\\ast}$\nthat in general is not tight. The lack of tightness is measured by a\nclass $[\\overline{\\psi}_{{\\mathbb C}}]\\in\\overline{\\KA}(X)$, that we simply denote\n$[\\overline{\\psi}]$ \\cite[Sec. 3]{BurgosFreixasLitcanu:HerStruc}. Recall\nthat Bott-Chern secondary classes can be defined at the level of\n$\\overline{\\KA}(X)$ (see \\emph{loc. cit.} Sec. 4, and especially the\ncharacterization given in Prop. 4.6). In particular we have a class\n\\begin{displaymath}\n  \\widetilde{\\varphi}(\\overline{\\psi}):=\\widetilde{\\varphi}([\\overline{\\psi}])\n  \\in\\bigoplus_{p}\\widetilde{\\mathcal{D}}_{{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{displaymath}\n\\begin{lemma}\nThe class\n\\begin{equation}\\label{eq:22}\n  \\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})+\n  \\amap(\\widetilde{\\varphi}(\\overline{\\psi}))\\in\n  \\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n\\end{equation}\ndepends only on $\\overline{\\mathcal{F}}^{\\ast}$.\n\\end{lemma}\n\\begin{proof}\nLet\n$\\psi^{\\prime}\\colon\\mathcal{E}^{\\prime\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$\nbe another finite locally free resolution, and choose arbitrary\nmetrics on the $\\mathcal{E}^{\\prime i}$. We can construct a\ncommutative diagram of complexes in $\\Db(\\mathcal{X})$\n\\begin{equation}\\label{eq:27}\n  \\xymatrix{\n    &\\mathcal{E}^{\\prime\\prime\\ast}\\ar[ld]_{\\alpha}\\ar[rd]^{\\beta}\n    &\\\\\n    \\mathcal{E}^{\\ast}\\ar@{-->}[d]_{\\psi}\t&\n    &\\mathcal{E}^{\\prime\\ast}\\ar@{-->}[d]^{\\psi^{\\prime}}\\\\ \n    \\mathcal{F}^{\\ast}\\ar[rr]^{\\Id}\t&\t&\\mathcal{F}^{\\ast}, \n  }\n\\end{equation}\nwhere $\\mathcal{E}^{\\prime\\prime}$ is also a finite complex of locally\nfree sheaves, that we endow with smooth hermitian metrics, and\n$\\alpha$, $\\beta$ are quasi-isomorphisms. Because the exact sequences\n\\begin{align*}\n\t&0\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\ast}\\rightarrow\\ocone(\\alpha)\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}[1]\\rightarrow 0,\\\\\n\t&0\\rightarrow\\overline{\\mathcal{E}}^{\\ast}\\rightarrow\\ocone(\\beta)\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}[1]\\rightarrow 0\n\\end{align*}\nhave orhtogonally split constituent rows, we find\n\\begin{align}\n\t&\\widehat{\\varphi}(\\ocone(\\alpha))=\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}),\\\\ \\label{eq:23}\n\t& \\widehat{\\varphi}(\\ocone(\\beta))=\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}).\n\\end{align}\nAlso, the complexes $\\cone(\\alpha)$ and $\\cone(\\beta)$ are acyclic, so that\n\\begin{align}\n\t&\\widehat{\\varphi}(\\ocone(\\alpha))=\\amap(\\widetilde{\\varphi}(\\ocone(\\alpha)))=\\amap(\\widetilde{\\varphi}(\\overline{\\alpha})),\\\\\n\t&\\widehat{\\varphi}(\\ocone(\\beta))=\\amap(\\widetilde{\\varphi}(\\ocone(\\beta)))=\\amap(\\widetilde{\\varphi}(\\overline{\\beta})), \\label{eq:26}\n\\end{align}\nwhere we took into account the very definition of the class of an isomorphism in $\\oDb(X)$. From the relations \\eqref{eq:23}--\\eqref{eq:26} we derive\n\\begin{equation}\\label{eq:28}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t=\\amap(\\widetilde{\\varphi}(\\overline{\\alpha})-\\widetilde{\\varphi}(\\overline{\\beta}))=\\amap(\\widetilde{\\varphi}(\\overline{\\alpha}\\circ\\overline{\\beta}^{-1})),\n\\end{equation}\nwhere we plugged $\\widetilde{\\varphi}(\\overline{\\alpha}\\circ\\overline{\\beta}^{-1})=\\widetilde{\\varphi}(\\overline{\\alpha})-\\widetilde{\\varphi}(\\overline{\\beta})$ \\cite[Prop. 4.13]{BurgosFreixasLitcanu:HerStruc}. But by diagram \\eqref{eq:27} we have $\\overline{\\alpha}\\circ\\overline{\\beta}^{-1}=\\overline{\\psi}^{-1}\\circ\\overline{\\psi}^{\\prime}$.  This fact combined with \\eqref{eq:28} implies\n\\begin{displaymath}\n\t\\begin{split}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t=&\\amap(\\widetilde{\\varphi}(\\overline{\\alpha})-\\widetilde{\\varphi}(\\overline{\\beta}))\\\\\n\t&=\\amap(\\widetilde{\\varphi}(\\overline{\\psi}^{-1}\\circ\\overline{\\psi}^{\\prime}))\\\\\n\t&\\hspace{0.4cm}=\\amap(\\widetilde{\\varphi}(\\overline{\\psi}^{\\prime}))-\\amap(\\widetilde{\\varphi}(\\overline{\\psi})).\n\t\\end{split}\n\\end{displaymath}\nThis completes the proof of the lemma.\n\\end{proof}\n\\begin{definition}\nThe notations being as above, we define\n\\begin{displaymath}\n  \\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast}):=\n  \\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})+\\amap(\\widetilde{\\varphi}(\\overline{\\psi}))\n  \\in\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)),\n\\end{displaymath}\n\\end{definition}\nThe additive arithmetic characteristic classes are indeed additive with respect to direct sum, and are compatible with pull-back by morphisms of arithmetic varieties. However, the most important property of additive arithmetic characteristic classes is the behavior with respect to distinguished triangles. The reader may review \\cite[Def. 3.29, Thm. 3.33, Def. 4.17, Thm. 4.18]{BurgosFreixasLitcanu:HerStruc} for definitions and main properties, especially for the class in $\\overline{\\KA}(X)$ and secondary class of a distinguished triangle.\n\\begin{theorem}\\label{thm:3}\nLet us consider a distinguished triangle in $\\oDb(\\mathcal{X})$:\n\\begin{displaymath}\n\t\\overline{\\tau}\\colon\\quad \\overline{\\mathcal{F}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{G}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{H}}^{\\ast}\n\t\\dashrightarrow\\overline{\\mathcal{F}}^{\\ast}_{0}[1].\n\\end{displaymath}\nThen we have\n\\begin{displaymath}\n\t\\widehat{\\varphi}(\\mathcal{F}^{\\ast})-\\widehat{\\varphi}(\\mathcal{G}^{\\ast})+\\widehat{\\varphi}(\\mathcal{H}^{\\ast})\n\t=\\amap(\\widetilde{\\varphi}(\\overline{\\tau}))\n\\end{displaymath}\nin $\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$. In particular, if $\\overline{\\tau}$ is tightly distinguished, we have\n\\begin{displaymath}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{G}}^{\\ast})=\\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast})+\\widehat{\\varphi}(\\overline{\\mathcal{H}}^{\\ast}).\n\\end{displaymath}\n\\end{theorem}\n\\begin{proof}\nIt is possible to find a diagram\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\overline{\\eta}\\colon  &\\mathcal{E}^{\\ast}\\ar@{-->}[d]^{f}\\ar[r]^{\\alpha}\t\t&\\mathcal{E}^{\\prime\\ast}\\ar@{-->}[d]^{g}\\ar[r]\t&\\cone(\\alpha)\\ar@{-->}[d]^{h}\\ar[r]\t&\\mathcal{E}^{\\ast}[1]\\ar@{-->}[d]^{f[1]}\\\\\n\t\t\\overline{\\tau}\\colon\t&\\mathcal{F}^{\\ast}\\ar@{-->}[r]\t&\\mathcal{G}^{\\ast}\\ar@{-->}[r]\t&\\mathcal{H}^{\\ast}\\ar@{-->}[r]\t&\\mathcal{F}^{\\ast}[1],\n\t}\n\\end{displaymath}\nwhere $\\mathcal{E}^{\\ast}$, $\\mathcal{E}^{\\prime\\ast}$ are bounded complexes of locally free sheaves and the vertical arrows are isomorphisms in $\\Db(\\mathcal{X})$. We choose arbitrary smooth hermitian metrics on the $\\mathcal{E}^{i}$, $\\mathcal{E}^{\\prime j}$, and put the orthogonal sum metric on $\\cone(\\alpha)$. Then, by construction of the arithmetic characteristic classes, we have\n\\begin{displaymath}\n\t\\begin{split}\n\t\t\\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{G}}^{\\ast})\n\t\t+\\widehat{\\varphi}(\\overline{\\mathcal{H}}^{\\ast})=&\n\t\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t\t+\\widehat{\\varphi}(\\ocone(\\alpha))\\\\\n\t\t&+a(\\widetilde{\\varphi}(\\overline{f})-\\widetilde{\\varphi}(\\overline{g})+\\widetilde{\\varphi}(\\overline{h})).\n\t\\end{split}\n\\end{displaymath}\nBecause the exact sequence\n\\begin{displaymath}\n\t0\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\ast}\\rightarrow\\ocone(\\alpha)\\rightarrow\\overline{\\mathcal{E}}^{\\ast}[1]\\rightarrow 0\n\\end{displaymath}\nhas orthogonally split constituent rows, we observe\n\\begin{displaymath}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t\t+\\widehat{\\varphi}(\\ocone(\\alpha))=0.\n\\end{displaymath}\nMoreover, by \\cite[Thm 3.33 (vii)]{BurgosFreixasLitcanu:HerStruc} the equality\n\\begin{displaymath}\n\t\\widetilde{\\varphi}(\\overline{f})-\\widetilde{\\varphi}(\\overline{g})+\\widetilde{\\varphi}(\\overline{h})\n\t=\\widetilde{\\varphi}(\\overline{\\tau})-\\widetilde{\\varphi}(\\overline{\\eta}),\n\\end{displaymath}\nholds, and $\\widetilde{\\varphi}(\\overline{\\eta})=0$ since $\\overline{\\eta}$ is tightly distinguished. The theorem now follows.\n\\end{proof}\nWe may also say a few words on multiplicative arithmetic\ncharacteristic classes. We follow the discussion in\n\\cite[Sec. 5]{BurgosFreixasLitcanu:HerStruc}. Assume from now on that ${\\mathbb Q}\\subset B$. Let $\\psi\\in B[[x]]$ be a formal power series with \n$\\psi^{0}=1$. We denote also by $\\psi$ the associated multiplicative\ngenus. \nThen\n\\begin{displaymath}\n\t\\varphi=\\log(\\psi)\\in B[[x]]\n\\end{displaymath}\ndefines an additive genus, to which we can associate an addtive\narithmetic characteristic genus $\\widehat{\\varphi}$. Then, given an\nobject $\\overline{\\mathcal{F}}^{\\ast}$ in $\\oDb(\\mathcal{X})$, we have a\nwell defined class \n\\begin{displaymath}\n  \\widehat{\\psi}_{m}(\\overline{\\mathcal{F}}^{\\ast}):=\\exp(\\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast}))\n  \\in\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)).\n\\end{displaymath}\nObserve this construction uses the ring structure of\n$\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$, hence\nthe regularity of $\\mathcal{X}$ and the product structure of\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$. In case $\\psi$ has rational coefficients,\n$\\widehat{\\psi}(\\overline{\\mathcal{F}}^{\\ast})$ takes values in  \n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$. We\nalso recall that there is a multiplicative secondary class associated\nto $\\psi$, denoted $\\widetilde{\\psi}_{m}$, that can be expressed in\nterms of $\\widetilde{\\varphi}$:\n\\begin{displaymath}\n  \\widetilde{\\psi}_{m}(\\theta)=\\frac{\\exp(\\varphi(\\theta))-1}{\\varphi(\\theta)}\\widetilde{\\varphi}(\\theta),\n\\end{displaymath}\nwhere $\\theta$ is any class in $\\overline{\\KA}(X)$. If $\\overline{\\tau}$ is a distinguished triangle in $\\oDb(\\mathcal{X})$, we will simply write $\\widetilde{\\psi}_{m}(\\overline{\\tau})$ instead of $\\widetilde{\\psi}_{m}([\\overline{\\tau}])$.\n\n\\begin{theorem}\\label{thm:4}\nLet $\\psi$ be a multiplicative genus, with degree 0 component\n$\\psi^{0}=1$. Then, for every distinguished triangle in\n$\\oDb(\\mathcal{X})$ \n\\begin{displaymath}\n\t\\overline{\\tau}\\colon\t\\overline{\\mathcal{F}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{G}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{H}}^{\\ast}\n\t\\dashrightarrow\\overline{\\mathcal{F}}[1]\n\\end{displaymath}\nwe have the relation\n\\begin{displaymath}\n\t\\widehat{\\psi}(\\overline{\\mathcal{F}}^{\\ast})^{-1}\\widehat{\\psi}(\\overline{\\mathcal{G}})\\widehat{\\psi}(\\overline{\\mathcal{H}})^{-1}-1=\n\t\\amap(\\widetilde{\\psi}_{m}(\\overline{\\tau})).\n\\end{displaymath}\nIn particular, if $\\overline{\\tau}$ is tightly distinguished, the equality\n\\begin{displaymath}\n\t\\widehat{\\psi}(\\overline{\\mathcal{G}}^{\\ast})=\\widehat{\\psi}(\\overline{\\mathcal{F}}^{\\ast})\\widehat{\\psi}(\\overline{\\mathcal{H}}^{\\ast})\n\\end{displaymath}\nholds.\n\\end{theorem}\n\\begin{proof}\nIt is enough to exponentiate the relation provided by Theorem \\ref{thm:3} and observe that, due to the mutliplicative law in $\\bigoplus_{p}\\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$, one has\n\\begin{displaymath}\n\t\\exp(\\amap(\\widetilde{\\varphi}(\\overline{\\tau})))=1+\\amap\\left(\\frac{\\exp(\\varphi(\\overline{\\tau}))-1}{\\exp(\\varphi(\\overline{\\tau}))}\\widetilde{\\varphi}(\\overline{\\tau})\\right).\n\\end{displaymath}\n\\end{proof}\n\\begin{example}\n\\begin{enumerate}\n\\item The arithmetic Chern class, is the additive class\n  attached to the additive genus $\\ch(x)=e^{x}$. It coincides\n  with the character $\\chh$ of the previous section. \n\\item The arithmetic Todd class, is the multiplicative class attached to the genus\n  \\begin{displaymath}\n    \\Td(x)=\\frac{x}{1-e^{-x}}.\n  \\end{displaymath}\n  Observe that the formal series of $\\Td(x)$ has constant coefficient 1.\n\\end{enumerate}\n\\end{example}\n\\begin{remark}\nIf $C$ is a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, we can define arithmetic\ncharacteristic classes with values in\n$\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},C)$, just taking their image by the\nnatural morphism\n$\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\\rightarrow\n\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},C)$. We will use the same\nnotations to refer to these classes.\n\\end{remark}\n\n\n\\section{Direct images and generalized analytic torsion}\\label{section:DirectImage}\nIn the preceding section we introduced the arithmetic $K$-groups and\nthe arithmetic derived categories. They satisfy elementary\nfunctoriality properties and are related by a class functor. A missing\nfunctoriality is the push-forward by \\emph{arbitrary} projectives\nmorphisms of arithmetic varieties. Similarly to the work of\nGillet-R\\\"ossler-Soul\\'e\n\\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees},\nwe will define direct images after choosing a generalized analytic\ntorsion theory in the sense of\n\\cite{BurgosFreixasLitcanu:GenAnTor}. Our theory is more general in\nthat we don't require our morphisms to be smooth over the generic\nfiber. At the archimedean places, we are thus forced to work with\ncomplexes of currents with controlled wave front sets. In this level\nof generality, the theory of arithmetic Chow groups, arithmetic\n$K$-theory and arithmetic derived categories has already been\ndiscussed. Let us recall that a generalized analytic torsion theory is\nnot unique, but according to \\cite[Thm. 7.7 and\nThm. 7.14]{BurgosFreixasLitcanu:GenAnTor} it is classified by a real\nadditive genus.\n\nOur theory of generalized analytic torsion classes involves the notion of relative metrized complex \\cite[Def. 2.5]{BurgosFreixasLitcanu:GenAnTor}. In the sequel we will need a variant on real smooth quasi-projective schemes $X=(X_{{\\mathbb C}},F_{\\infty})$. With respect to \\emph{loc. cit.}, this amounts to imposing an additional invariance under the action of $F_{\\infty}$.\n\n\\begin{definition}\nA real relative metrized complex is a triple $\\overline{\\xi}=(\\overline{f}, \\overline{\\mathcal{F}}^{\\ast},\\overline{f_{\\ast}\\mathcal{F}^{\\ast}})$, where\n\\begin{itemize}\n\t\\item $\\overline{f}:X\\rightarrow Y$ is a projective morphism of real smooth quasi-projective varieties, together with a hermitian structure on the tangent complex $T_{f}$, invariant under the action of complex conjugation;\n\t\\item $\\overline{\\mathcal{F}}^{\\ast}$ is an object in $\\oDb(X)$;\n\t\\item $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ is an object in $\\oDb(Y)$ lying over $f_{\\ast}\\mathcal{F}^{\\ast}$.\n\\end{itemize}\n\\end{definition}\nThe following lemma is checked by a careful proof reading of the construction of generalized analytic torsion classes in \\cite{BurgosFreixasLitcanu:GenAnTor}.\n\\begin{lemma}\nLet $T$ be a theory of generalized analytic torsion classes. Then, for every real relative metrized complex $\\overline{\\xi}=(\\overline{f}\\colon X\\to Y,\\overline{\\mathcal{F}}^{\\ast},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast})$, $T(\\overline{\\xi})$ is a class of real currents,\n\\begin{displaymath}\n\tT(\\overline{\\xi})\\in\\bigoplus_{p}\\widetilde{\\mathcal{D}}^{2p-1}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,N_{f},p),\n\\end{displaymath}\nwhere $N_{f}$ is the cone of normal directions to $f$.\n\\end{lemma}\nIt will be useful to have an adaptation of the category\n$\\overline{\\Sm}_{\\ast\/{\\mathbb C}}$ \\cite[Sec. 5]{BurgosFreixasLitcanu:HerStruc} to\nreal quasi-projective schemes, that we denote\n$\\overline{\\Sm}_{\\ast\/{\\mathbb R}}$. In this category, the objects are real smooth\nquasi-projective schemes, and the morphisms are projective morphisms\nwith a hermitian structure invariant under complex conjugation. The\ncomposition law is then described in \\emph{loc. cit.}, Def. 5.7, with\nthe help of the hermitian cone construction. We will follow the\nnotation introduced in\n\\cite[Def. 2.12]{BurgosFreixasLitcanu:GenAnTor}. \n\\begin{notation}\n  Let $\\overline{f}\\colon X\\to Y$ be in $\\overline{\\Sm}_{\\ast\/{\\mathbb R}}$, of pure\n  relative dimension $e$, $C$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, $C'$ a\n  $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$-complex and $f_{\\ast}:C\\rightarrow C'$ a\n  morphism of fitting into the commutative diagram\n  \\eqref{eq:18}. Assume furthermore that $C$ is a\n  $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-module with a product law $\\bullet$ as in\n  \\eqref{eq:19}. Then we put\n\\begin{displaymath}\n  \\begin{split}\n    \\overline{f}_{\\flat}:C^{\\ast}(\\ast)&\\longrightarrow C^{\\prime\\ast-2e}(p-e)\\\\\n    \\eta&\\longmapsto f_{\\ast}(\\eta\\bullet \\Td(T_{\\overline{f}})).\n  \\end{split}\n\\end{displaymath}\nThis morphism induces a corresponding morphism on\n$\\widetilde{C}^{\\ast}(\\ast)$, for which we use the same notation. \n\\end{notation}\nFor an arithmetic ring $A$, we introduce $\\overline{\\Reg}_{\\ast\/A}$ the\ncategory of  quasi-proj\\-ective regular arithmetic varieties\nover $A$, with projective morphisms endowed (at the archimedean\nplaces) with a hermitian structure invariant under complex\nconjugation. By construction, there is a natural base change functor \n\\begin{displaymath}\n\t\\overline{\\Reg}_{\\ast\/A}\\longrightarrow\\overline{\\Sm}_{\\ast\/{\\mathbb R}}.\n\\end{displaymath}\nGiven $\\overline{f}\\colon\\mathcal{X}\\to\\mathcal{Y}$ a morphism in $\\overline{\\Reg}_{\\ast\/A}$ and objects $\\overline{\\mathcal{F}}^{\\ast}$, $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ in $\\oDb(\\mathcal{X})$ and $\\oDb(\\mathcal{Y})$, respectively, we may consider the corresponding real relative metrized complex, that we will abusively write $\\overline{\\xi}=(\\overline{f},\\overline{\\mathcal{F}},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast})$, and its analytic torsion class $T(\\overline{\\xi})$. We may also write $\\overline{f}_{\\flat}$ instead of $\\overline{f}_{{\\mathbb C}\\,\\flat}$, etc. \n\nWe are now in position to construct the arithmetic counterpart of \\cite[Eq. (10.6)]{BurgosFreixasLitcanu:GenAnTor}, namely the direct image functor on arithmetic derived categories, as well as a similar push-forward on arithmetic $K$-theory.\n\\begin{definition}\nLet $\\overline{f}\\colon\\mathcal{X}\\to\\mathcal{Y}$ be a morphism in\n$\\overline{\\Reg}_{\\ast\/A}$, $C$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, $C'$ a\n  $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$-complex, both satisfying the\nhypothesis (H1) and (H2), and $f_{\\ast}:C\\rightarrow C'$ a\n  morphism fitting into a commutative diagram like\n  \\eqref{eq:18}.\n  \\begin{enumerate}\n  \\item  We define the functor\n\\begin{displaymath}\n\t\\overline{f}_{\\ast}\\colon\\hDb(\\mathcal{X},C)\\longrightarrow\\hDb(\\mathcal{Y},C'),\n\\end{displaymath}\nacting on objects by the assignment\n\\begin{displaymath}\n\t[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]\\longmapsto [\\overline{f_{\\ast}\\mathcal{F}}^{\\ast},\\overline{f}_{\\flat}(\\widetilde{\\eta})-c'(T(\\overline{f},\\overline{\\mathcal{F}}^{\\ast},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}))].\n\\end{displaymath}\nHere $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ carries an arbitrary choice of hermitian structure. The action on morphisms of $\\overline{f}_{\\ast}$ is just the usual action $f_{\\ast}$ on morphisms of $\\Db(\\mathcal{X})$.\n\n\\item  We define a morphism of groups\n\\begin{displaymath}\n\t\\begin{split}\n\t\tf_{\\ast}:\\widehat{K}_{0}(\\mathcal{X},C)&\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C')\\\\\n\t\t[\\overline{\\mathcal{E}},\\widetilde{\\eta}]&\\longmapsto [\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{\\prime i},\\overline{f}_{\\flat}(\\widetilde{\\eta})-T(\\overline{f},\\overline{\\mathcal{E}},\\overline{f_{\\ast}\\mathcal{E}}^{\\ast})],\n\t\\end{split}\n\\end{displaymath}\nwhere we choose an arbitrary quasi-isomorphism\n$\\mathcal{E}^{\\prime\\ast}\\dashrightarrow f_{\\ast}\\mathcal{E}$ and\narbitrary smooth hermitian metrics on the $\\mathcal{E}^{\\prime\n  i}$. Here $f_{\\ast}\\mathcal{E}$ denotes the derived direct image of\nthe single locally free sheaf $\\mathcal{E}$.\n\n  \\end{enumerate}\n\\end{definition}\nNotice the previous definition makes sense by the anomaly formulas satisfied by analytic torsion theories \\cite[Prop. 7.4]{BurgosFreixasLitcanu:GenAnTor}. \nBoth push-forwards are compatible through the class map from arithmetic derived categories to arithmetic $K$-theory.\n\\begin{theorem}\\label{thm:6}\nThere is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{X},C)\\ar[r]^-{\\overline{f}_{\\ast}}\\ar[d]\t&\\hDb(\\mathcal{Y},C')\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{X},C)\\ar[r]\t_-{\\overline{f}_{\\ast}}\t&\\widehat{K}_{0}(\\mathcal{Y},C').\n\t}\n\\end{displaymath}\nThis is in particular true for $C=\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ and $C'=\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(S))$, where $S\\subset T_{0}^{\\ast}X$ is a closed conical subset invariant under the action of complex conjugation.\n\\end{theorem}\n\\begin{proof}\nLet $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$ be an object in\n$\\hDb(\\mathcal{X},C)$. By Remark \\ref{rem:1}, we can suppose the\nhermitian structure on $\\overline{\\mathcal{F}}^{\\ast}$ is given by a\nquasi-isomorphism\n$\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, where\n$\\overline{\\mathcal{E}}^{\\ast}$ is a finite complex of locally free sheaves,\neach one endowed with a smooth hermitian metric. For every $i$, we\nhave to endow $f_{\\ast}\\mathcal{E}^{i}$ with a hermitian structure\ngiven by a finite locally free resolution\n$\\overline{\\mathcal{E}}^{i\\ast}\\dashrightarrow f_{\\ast}\\mathcal{E}^{i}$ and\na choice of arbitrary smooth hermitian metric on every piece\n$\\mathcal{E}^{ij}$. From the data\n$\\overline{\\mathcal{E}}^{i\\ast}\\dashrightarrow f_{\\ast}\\mathcal{E}^{i}$,\nevery $i$, the procedure of\n\\cite[Def. 3.39]{BurgosFreixasLitcanu:HerStruc} produces a hermitian\nstructure on $f_{\\ast}\\mathcal{E}^{\\ast}$, via the hermitian cone\nconstruction. Observe the construction of \\emph{loc. cit.} can be done\nin $\\oDb(\\mathcal{Y})$. Combined with the fixed quasi-isomorphism\n$\\mathcal{E}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, we thus obtain\na hermitian structure on $f_{\\ast}\\mathcal{F}^{\\ast}$, that we denote\nby $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$. \n\nThe class of $[\\overline{\\mathcal{F}},\\widetilde{\\eta}]$ in $\\hDb(\\mathcal{X},C)$ is thus\n\\begin{displaymath}\n\t\\sum_{i}(-1)^{i}[\\overline{\\mathcal{E}}^{i},0] + [0,\\widetilde{\\eta}].\n\\end{displaymath}\nIts image under $\\overline{f}_{\\ast}$ is\n\\begin{equation}\\label{eq:20}\n\t\\sum_{i}(-1)^{i}\\left((\\sum_{j}(-1)^{j}[\\overline{\\mathcal{E}}^{ij},0])-[0,c'(T(\\overline{f},\\overline{\\mathcal{E}}^{i}, \\overline{f_{\\ast}\\mathcal{E}}^{i}))]\\right) + [0,\\overline{f}_{\\flat}(\\widetilde{\\eta})].\n\\end{equation}\nThe class of $\\overline{f}_{\\ast}[\\overline{\\mathcal{F}},\\widetilde{\\eta}]$ in $\\hDb(\\mathcal{Y},C')$ is\n\\begin{equation}\\label{eq:21}\n\t[\\overline{f_{\\ast}\\mathcal{F}}^{\\ast},\\overline{f}_{\\flat}(\\widetilde{\\eta})-c'(T(\\overline{f},\\overline{\\mathcal{F}},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}))].\n\\end{equation}\nBy the choice of the hermitian structures on $\\overline{\\mathcal{F}}^{\\ast}$ and $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ and by Theorem \\cite[Prop. 7.6]{BurgosFreixasLitcanu:GenAnTor}, we have\n\\begin{displaymath}\n\tT(\\overline{f},\\overline{\\mathcal{F}},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast})=\\sum_{i}(-1)^{i} T(\\overline{f},\\overline{\\mathcal{E}}^{i},\\overline{f_{\\ast}\\mathcal{E}}^{i}).\n\\end{displaymath}\nTherefore the class in arithmetic $K$-theory of \\eqref{eq:21} equals \\eqref{eq:20}.\n\\end{proof}\nThere are several compatibilities between direct images, inverse images and derived tensor product. We now state them without proof, referring the reader to \\cite[Thm. 10.7]{BurgosFreixasLitcanu:GenAnTor} for the details.\n\\begin{proposition}\\label{prop:5}\n Let $\\overline{f}\\colon\\mathcal{X}\\to\\mathcal{Y}$ and $\\overline{g}\\colon\\mathcal{Y}\\to \\mathcal{Z}$ be morphisms in $\\overline{\\Reg}_{\\ast\/A}$. Let $S\\subset T^{\\ast}X_{0}$ and $T\\subset T^{\\ast}Y_{0}$ be closed conical subsets.\n\\begin{enumerate}\n\t\\item (Functoriality of push-forward) We have the relation\n  \t\t\\begin{displaymath}\n    \t\t\t(\\overline{g}\\circ \\overline{f})_{\\ast}=\\overline{g}_{\\ast}\\circ\\overline{f}_{\\ast},\n   \t\t\\end{displaymath}\n\t\tas functors $\\hDb(\\mathcal{X},S)\\rightarrow\\hDb(\\mathcal{Z},g_{\\ast}f_{\\ast}S)$.\n  \\item (Projection formula) Assume that $T\\cap N_{f}=\\emptyset$ and that $T+f_{\\ast}S$ does not cross the zero section of $T^{\\ast}Y$. Let $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$ be in $\\hDb(\\mathcal{X},S)$ and $[\\overline{\\mathcal{F}}^{\\prime\\ast},\\widetilde{\\eta}^{\\prime}]$ in $\\hDb(\\mathcal{Y},T)$. Then\n\t\\begin{displaymath}\n     \t\t\\overline{f}_{\\ast}([\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}] \\otimes\n      \t\tf^{\\ast}[\\overline{\\mathcal{F}}^{\\prime\\ast},\\widetilde{\\eta}^{\\prime}])\n     \t\t= \\overline{f}_{\\ast}[\\overline{\\mathcal{F}},\\widetilde{\\eta}]\\otimes [\\overline{\\mathcal{F}}^{\\prime},\\widetilde{\\eta}^{\\prime}]\n    \\end{displaymath}\n    in $\\hDb(\\mathcal{Y},W)$, where\n    \\begin{displaymath}\n    \tW=f_{\\ast}(S + f^{\\ast}T)\\cup f_{\\ast}S\\cup f_{\\ast}f^{\\ast}T.\n    \\end{displaymath}\n\t\\item There are analogous relations on the level of arithmetic $K$-theory, compatible with the class functor from arithmetic derived categories.\n\\end{enumerate}\n\\end{proposition}\n\\begin{remark}\nStrictly speaking, the equalities provided by the previous statement should be canonical isomorphisms, but as usual we abuse the notations and pretend they are equalities.\n\\end{remark}\n\\section{Arithmetic Grothendieck-Riemann-Roch}\\label{section:ARR}\n\\subsection{Statement and reductions}\n\\paragraph{Hermitian tangent complexes.} Let $\\overline{f}\\colon\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism in $\\overline{\\Reg}_{\\ast\/A}$. We explain how to construct the associated hermitian tangent complex $T_{\\overline{f}}$. This is an object in $\\oDb(\\mathcal{X})$, well defined up to tight isomorphism. \n\nBecause $\\mathcal{X}$, $\\mathcal{Y}$ are regular schemes and $f$ is projective, it is automatically a l.c.i. morphism. The tangent complex of $f$ is an object in $\\Db(\\mathcal{X})$, well defined up to isomorphism. Consider a factorization\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\mathcal{X}\\ar@{^{(}->}[r]^-{i}\\ar[rd]\t&\\mathcal{Z}\\ar[d]^{\\pi}\\\\\n\t\t&\\mathcal{Y},\n\t}\n\\end{displaymath}\nwith $i$ being a closed regular immersion and $\\pi$ a smooth morphism. For instance, one may choose $\\mathcal{Z}={\\mathbb P}^{n}_{\\mathcal{Y}}$, for some $n$. Let us denote by $\\mathcal{I}$ the ideal defining the closed immersion $i$. Then $\\mathcal{I}\/\\mathcal{I}^{2}$ is a locally free sheaf on $\\mathcal{X}$ and, as customary, we define the normal bundle $N_{\\mathcal{X}\/\\mathcal{Z}}=(\\mathcal{I}\/\\mathcal{I}^{2})^{\\vee}$. There is a morphism of coherent sheaves\n\\begin{displaymath}\n\t\\varphi\\colon i^{\\ast}T_{\\mathcal{Z}\/\\mathcal{Y}}\\longrightarrow N_{\\mathcal{X}\/\\mathcal{Z}},\n\\end{displaymath}\nnamely the dual of the differential map $d:\\mathcal{I}\/\\mathcal{I}^{2}\\to i^{\\ast}\\Omega_{\\mathcal{Z}\/\\mathcal{Y}}$. We consider $T_{\\mathcal{Z}\/\\mathcal{Y}}$ as a complex concentrated in degree 0, and $N_{\\mathcal{X}\/\\mathcal{Z}}$ as a complex concentrated in degree one. We then put\n\\begin{displaymath}\n\tT_{f}:=\\cone(\\varphi)[-1].\n\\end{displaymath}\nThe isomorphism class of $T_{f}$ in $\\Db(\\mathcal{X})$ is independent of the factorization. \n\nThe base change to ${\\mathbb C}$ of $T_{f}$ is naturally isomorphic to the tangent complex $T_{f_{{\\mathbb C}}}:TX_{{\\mathbb C}}\\rightarrow f_{{\\mathbb C}}^{\\ast}Y_{{\\mathbb C}}$, which is equipped with a hermitian structure by assumption. Therefore, the data provided by the constructed complex $T_{f}$ and the hermitian structure on $\\overline{f}$ determine an object $T_{\\overline{f}}$ in $\\oDb(\\mathcal{X})$, which is well defined up to tight isomorphism. By Theorem \\ref{thm:4}, the arithmetic Todd class of $T_{\\overline{f}}$ is unambiguously defined.\n\\begin{definition}\nThe arithmetic Todd class of $\\overline{f}$ is\n\\begin{displaymath}\n  \\widehat{\\Td}(\\overline{f}):=\\widehat{\\Td}(T_{\\overline{f}})\\in\n  \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)). \n\\end{displaymath}\n\\end{definition}\n\\begin{theorem}\nLet $\\overline{f}\\colon\\mathcal{X}\\rightarrow\\mathcal{Y}$,\n$\\overline{g}\\colon\\mathcal{Y}\\rightarrow\\mathcal{Z}$ be morphisms in\n$\\overline{\\Reg}_{\\ast\/A}$. Then we have an equality \n\\begin{displaymath}\n  \\widehat{\\Td}(\\overline{g}\\circ\\overline{f})=f^{\\ast}\\widehat{\\Td}(\\overline{g})\\cdot\\widehat{\\Td}(\\overline{f})\n\\end{displaymath}\nin $\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n\\end{theorem}\n\\begin{proof}\nBy construction of $T_{\\overline{f}}$ and definition of the composition rule of morphisms in $\\overline{\\Reg}_{\\ast\/A}$, there a is tightly distinguished triangle in $\\oDb(\\mathcal{X})$\n \\begin{displaymath}\n\tT_{\\overline{f}}\\dashrightarrow T_{\\overline{g}\\circ \\overline{f}}\\dashrightarrow f^{\\ast}T_{\\overline{g}}\\dashrightarrow T_{\\overline{f}}[1].\n\\end{displaymath}\nWe conclude by an application of Theorem \\ref{thm:4}.\n\\end{proof}\n\\paragraph{Statement.} The arithmetic Grothendieck-Riemann-Roch theorem describes the behavior of the arithmetic Chern character with respect to the push-forward functor. Recall that the definition of the push-forward functor depends on the choice of a theory of generalized analytic torsion classes. In its turn, such a theory corresponds to a real additive genus. \n\\begin{theorem}\\label{thm:5}\nLet $\\overline{f}\\colon\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism in $\\overline{\\Reg}_{\\ast\/A}$. Fix a closed conical subset $W$ of $T_{0}^{\\ast}X$ and a theory of generalized analytic torsion classes $T$, whose associated real additive genus is $S$. Then, the derived direct image functor $$\\overline{f}_{\\ast}\\colon\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,W))\\longrightarrow\\hDb(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(W)))$$ attached to $T$ satisfies the equation\n\\begin{equation}\\label{eq:29}\n\t\\chh(\\overline{f}_{\\ast}\\alpha)=f_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{f}))-\\amap(f_{\\ast}(\\ch(\\mathcal{F}^{\\ast}_{{\\mathbb C}})\\Td(T_{f_{{\\mathbb C}}})S(T_{f_{{\\mathbb C}}})),\n\\end{equation}\nfor every object\n$\\alpha=[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$. The equality\ntakes place in the arithmetic Chow group\n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(W)))$.\n\\end{theorem}\nRecall that the genus $S$=0 corresponds to the homogeneous theory\n$T^{h}$, which is characterized by satisfying an additional\nhomogeneity property \\cite[Sec. 9]{BurgosLitcanu:SingularBC}. Roughly\nspeaking, this condition is exactly the one guaranteeing an\narithmetic Grothendieck-Riemann-Roch theorem without\ncorrection term.\n\\begin{corollary}\nThe direct image functor $\\overline{f}^{h}_{\\ast}$ attached to the\nhomogeneous generalized analytic torsion theory $T^{h}$ satisfies an\nexact Grothendieck-Riemann-Roch type formula:\n\\begin{displaymath}\n  \\chh(\\overline{f}^{h}_{\\ast}\\alpha)=f_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{f})).\n\\end{displaymath}\n\\end{corollary}\nA Grothendieck-Riemann-Roch type theorem in Arakelov geometry was\nfirst proven by Gillet-Soul\\'e \\cite{GilletSoule:aRRt}, for the degree\n1 part of the Chern character (namely the determinan of the\ncohomology) and under the restriction on the morphism $f$ to be smooth\nover ${\\mathbb C}$. Also they can only deal with hermitian vector\nbundles. They used the holomorphic analytic torsion\n\\cite{BismutGilletSoule:at}, \\cite{BismutGilletSoule:atII},\n\\cite{BismutGilletSoule:atIII} and deep results of Bismut-Lebeau\n\\cite{BismutLebeau:CiQm} on the compatibility of analytic torsion with\nclosed immersions. The holomorphic analytic torsion was later\ngeneralized by Bismut-K\\\"ohler \\cite{BismutKohler}, to the holomorphic\nanalytic torsion forms, that transgress the whole\nGrothendieck-Riemann-Roch theorem for K\\\"ahler submersions, at the\nlevel of differential forms. The extension of the arithmetic\nGrothendieck-Riemann-Roch theorem to the full Chern character and\ngenerically smooth morphisms was finally proven by\nGillet-R\\\"ossler-Soul\\'e\n\\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees}. They\napplied the analogue to the Bismut-Lebeau immersion theorem, for\nanalytic torsion forms, established in the monograph by Bismut\n\\cite{Bismut:Asterisque}.\n\nTheorem \\ref{thm:5} provides an extension of the previous results in\nseveral directions. First, we allow the morphism to be an arbitrary\nprojective morphism, non necessarily generically smooth. Second, we\ncan deal with metrized objects in the bounded derived category of\ncoherent sheaves. Third, we provide all the possible forms of such a\ntheorem, by introducing our theory of generalized analytic torsion\nclasses, thus explaining  the topological correction term. \n\n\\paragraph{Reductions.} The proof of our version of the arithmetic\nGrothendieck-Riemann-Roch theorem follows the pattern of the classical\napproach in algebraic geometry. Namely, it proceeds by factorization\nof the morphism $\\overline{f}$ into a regular closed immersion and a trivial\nprojective bundle projection. The advantage of working with the\nformalism of hermitian structures on the derived category of coherent\nsheaves makes the whole procedure more transparent and direct, in\nparticular avoiding the appearance of several secondary classes. Also\nthe cocyle type relation expressing the behaviour of generalized\nanalytic torsion with respect to composition of morphisms in\n$\\overline{\\Sm}_{\\ast\/{\\mathbb R}}$ (see the axioms\n\\cite[Sec. 7]{BurgosFreixasLitcanu:GenAnTor}) is well suited to this\nfactorization argument. \n\nBy the classification of generalized analytic torsion theories, it is\nenough to prove Theorem \\ref{thm:5} for the homogenous theory\n$T^{h}$. From now on we fix this choice, and hence all derived direct\nimage functors will be with respect to this theory. \n\nLet $\\overline{f}$ be a morphism in $\\overline{\\Reg}_{\\ast\/A}$, and consider a factorization\n\\begin{displaymath}\n  \\xymatrix{\n    \\mathcal{X}\\ar@{^{(}->}[r]^{i}\\ar[rd]^{f}\n    &\\mathcal{{\\mathbb P}}^{n}_{\\mathcal{Y}}\\ar[d]^{\\pi}\\\\\n    &\\mathcal{Y}.\n  }\n\\end{displaymath}\nThe tangent complex of $\\pi$ is canonically isomorphic to $p^{\\prime\\ast}T_{{\\mathbb P}^{n}_{A}\/A}$, where \n\\begin{displaymath}\n\t\\xymatrix{\n\t\t{\\mathbb P}^{n}_{\\mathcal{Y}}\\ar[r]^-{p^{\\prime}}\\ar[d]_{\\pi}\t&{\\mathbb P}^{n}_{A}\\ar[d]^{\\pi^{\\prime}}\\\\\n\t\t\\mathcal{Y}\\ar[r]_-{p}\t&\\Spec A.\n\t}\n\\end{displaymath}\nWe may thus endow $\\pi$ with the pull-back by $p^{\\prime}$ of the Fubini-Study metric on $T_{{\\mathbb P}^{n}_{A}\/A}$ \\cite[Sec. 5]{BurgosFreixasLitcanu:GenAnTor}. We write $\\overline{\\pi}$ for the resulting morphism in $\\overline{\\Reg}_{\\ast\/A}$. By \\cite[Lemma 5.3]{BurgosFreixasLitcanu:HerStruc}, there exists a unique hermitian structure on $i$ such that $\\overline{f}=\\overline{\\pi}\\circ\\overline{i}$. Then we recall that we have the equality of functors\n\\begin{displaymath}\n\tf_{\\ast}=\\overline{\\pi}_{\\ast}\\circ\\overline{i}_{\\ast}\n\\end{displaymath}\nand the equality of arithmetic Todd genera\n\\begin{displaymath}\n\t\\widehat{\\Td}(\\overline{f})=i^{\\ast}\\widehat{\\Td}(\\overline{\\pi})\\widehat{\\Td}(\\overline{i}).\n\\end{displaymath}\n\\begin{lemma}\nIt is enough to prove Theorem \\ref{thm:5} for $\\overline{i}$ and $\\overline{\\pi}$ individually.\n\\end{lemma}\n\\begin{proof}\nLet us assume the theorem known for $\\overline{i}$ and for $\\overline{\\pi}$. Because the theory is known for $\\overline{\\pi}$, we may apply it to the object $\\overline{i}_{\\ast}\\alpha$ in $\\oDb({\\mathbb P}^{n}_{\\mathcal{Y}})$, to obtain\n\\begin{equation}\\label{eq:30}\n\t\\chh(\\overline{f}_{\\ast}\\alpha)=\\chh(\\overline{\\pi}_{\\ast}(\\overline{i}_{\\ast}\\alpha))=\n\t\t\\pi_{\\ast}(\\chh(\\overline{i}_{\\ast}\\alpha)\\widehat{\\Td}(\\overline{\\pi})).\n\\end{equation}\nBecause the theorem is known for $\\overline{i}$, we also have\n\\begin{equation}\\label{eq:31}\n\t\\begin{split}\n\t\t\\chh(\\overline{i}_{\\ast}\\alpha)\\widehat{\\Td}(\\overline{\\pi})=&i_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{i}))\\widehat{\\Td}(\\overline{\\pi})\\\\\n\t\t&=i_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{i})i^{\\ast}\\widehat{\\Td}(\\overline{\\pi}))\\\\\n\t\t&\\hspace{0.3cm}=i_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{f})).\n\t\\end{split}\n\\end{equation}\nObserve we used the projection formula in arithmetic Chow groups\n(Theorem \\ref{thm:2}), and that this equality holds in\n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(W)))$,\nbecause the definition of direct image of wave front sets and the fact\nthat $\\pi$ is smooth yield $f_{\\ast}(W)=\\pi_{\\ast}(i_{\\ast}(W))$. We\nconclude by putting \\eqref{eq:30}--\\eqref{eq:31} together and using\nthe fact $f_{\\ast}=\\pi_{\\ast} i_{\\ast}$ on arithmetic Chow groups\n(Proposition \\ref{prop:8}).\n\\end{proof}\n\\begin{lemma} \\label{lemm:4}\nTheorem \\ref{thm:5} holds for $\\overline{i}$.\n\\end{lemma}\n\\begin{proof}\nIn \\cite[Thm 10.28]{BurgosLitcanu:SingularBC}, the authors prove the\ntheorem for direct images by closed immersions, defined on arithmetic\n$K$ groups. They suppose as well that the hermitian structure on\n$T_{i}$ is given by a smooth hermitian metric on\n$N_{\\mathcal{X}\/{\\mathbb P}^{n}_{\\mathcal{Y}}}$. Finally, the result in\n\\emph{loc,. cit.} holds in\n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}}, {\\text{\\rm a}}}(Y))$. \n\nBy the anomaly formula provided by Theorem \\ref{thm:4} and the anomaly\nformula of generalized analytic torsion theories\n\\cite[Prop. 7.4]{BurgosFreixasLitcanu:GenAnTor} for change of\nhermitian structure on the tangent complex, one can extend \\cite[Thm\n10.28]{BurgosLitcanu:SingularBC} to arbitrary hermitian structures on\n$i$, in particular the one we fixed. Also, a careful proof reading of\nthe proof in \\emph{loc. cit.} shows that the theorem can be adapted to\nallow\n$\\alpha\\in\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,W))$,\ntaking values in $\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}({\\mathbb P}^{n}_{Y},\\mathcal{D}_{\\text{{\\rm cur}},\n  {\\text{\\rm a}}}({\\mathbb P}^{n}_{Y},i_{\\ast}W))$. Finally, to get the result for\n$\\overline{i}_{\\ast}$ on the arithmetic derived category, we use the class\nfunctor to arithmetic $K$-theory, the commutativity Theorem\n\\ref{thm:6} and that the Chern character on arithmetic derived\ncategory factors, by construction, through arithmetic $K$ groups (see\nNotation \\ref{not:1}).\n\\end{proof}\n\n\\begin{lemma}\nTo prove Theorem \\ref{thm:5} for $\\overline{\\pi}$, it is enough to prove it\nwhen $\\mathcal{Y}=\\Spec A$ and $\\alpha=\\overline{{\\mathcal O}(k)}$, $-n\\leq k\\leq 0$,\nwhere the chosen hermitian structure on ${\\mathcal O}(k)$ is the Fubini-Study\nmetric. \n\\end{lemma}\n\\begin{proof}\nThe derived category $\\Db({\\mathbb P}^{n}_{\\mathcal{Y}})$ is generated by\ncoherent sheaves of the form $\\pi^{\\ast}\\mathcal{G}\\otimes\np^{\\prime\\ast}{\\mathcal O}(k)$\n\\cite[Cor. 4.11]{BurgosFreixasLitcanu:GenAnTor}. By the anomaly\nformulas \\cite[Prop. 7.4, Prop. 7.6]{BurgosFreixasLitcanu:GenAnTor},\nwe thus reduce to prove the theorem for $\\alpha$ of the form\n$\\pi^{\\ast}\\overline{\\mathcal{G}}\\otimes p^{\\prime\\ast}\\overline{{\\mathcal O}(k)}$, for any\nmetric on $\\mathcal{G}$. On the one hand, by the formulas in Prop\n\\ref{prop:5}, the multiplicativity and pull-back functoriality\n(Prop. \\ref{prop:6}) of the Chern character, we have \n\\begin{displaymath}\n  \\begin{split}\n    \\chh(\\overline{\\pi}_{\\ast}\\alpha)=&\\chh(\\overline{\\mathcal{G}}\\otimes\\overline{\\pi}_{\\ast}p^{\\prime\\ast}\\overline{{\\mathcal O}(k)})\\\\\n    =&\\chh(\\overline{\\mathcal{G}})\\chh(\\overline{\\pi}_{\\ast}p^{\\prime\\ast}\\overline{{\\mathcal O}(k)})\\\\\n    &\\hspace{0.3cm}=\\chh(\\overline{\\mathcal{G}})\\chh(p^{\\ast}\\overline{\\pi}^{\\prime}_{\\ast}\\overline{{\\mathcal O}(k)})\\\\\n    &\\hspace{0.6cm}=\\chh(\\overline{\\mathcal{G}})p^{\\ast}\\chh(\\overline{\\pi}^{\\prime}_{\\ast}\\overline{{\\mathcal O}(k)}).\n  \\end{split}\n\\end{displaymath}\nHere we recall $\\pi^{\\prime}:{\\mathbb P}^{n}_{A}\\rightarrow\\Spec A$ is the\nstructure morphism, that we endow with the Fubini-Study metric. On the\nother hand, we similarly prove\n\\begin{displaymath}\n  \\begin{split}\n    \\pi_{\\ast}(\\chh(\\pi^{\\ast}\\overline{\\mathcal{G}}\\otimes p^{\\prime\\ast}\\overline{{\\mathcal O}(k)})\\widehat{\\Td}(\\overline{\\pi}))\n    =&\\chh(\\overline{\\mathcal{G}})\\pi_{\\ast}(p^{\\prime\\ast}\\chh(\\overline{{\\mathcal O}(k)})p^{\\prime\\ast}\\widehat{\\Td}(\\overline{\\pi}^{\\prime}))\\\\\n    &=\\chh(\\overline{\\mathcal{G}})p^{\\ast}\\pi^{\\prime}_{\\ast}(\\chh(\\overline{{\\mathcal O}(k)})\\widehat{\\Td}(\\overline{\\pi^{\\prime}})).\n  \\end{split}\n\\end{displaymath}\nWe thus reduce to prove the theorem for $\\overline{\\pi}^{\\prime}$ and\n$\\alpha=\\overline{{\\mathcal O}(k)}$, as was to be shown.\n\\end{proof}\n\\paragraph{The case of projective spaces.} To prove Theorem\n\\ref{thm:5} in full generality, it remains to treat the case of the\nprojection $\\overline{\\pi}\\colon{\\mathbb P}^{n}_{A}\\rightarrow\\Spec A$, where we\nendow $\\pi$ with the Fubini-Study metric. Since this projection is the\npull-back of the projection\n$\\overline{\\pi}\\colon{\\mathbb P}^{n}_{{\\mathbb Z}}\\rightarrow\\Spec {\\mathbb Z}$ it is enough to\ntreat the case when $A={\\mathbb Z}$. \n\nFurthermore, we showed that\nit is enough to consider $\\alpha$ of the form $\\overline{{\\mathcal O}(k)}$, $-n\\leq\nk\\leq 0$, with the Fubini-Study metric as well and any metric of the\ndirect image $\\pi_{\\ast}{\\mathcal O}(k)$.\nIn \\cite[Def. 5.7]{BurgosFreixasLitcanu:GenAnTor}, we introduced the\nmain characteristic numbers of $T^{h}$,\n\\begin{displaymath}\n  t_{n,k}^{h}=T^{h}(\\overline{\\pi},\\overline{{\\mathcal O}(k)},\\overline{\\pi_{\\ast}{\\mathcal O}(k)}),\\\n  -n\\le k \\le 0,\n\\end{displaymath}\nwhere $\\pi_{\\ast}{\\mathcal O}(k)$ was endowed with its $L^{2}$ metric. Since we\nwill only consider the homogeneous analytic torsion we will shorthand\n$t^{h}_{n,k}=t_{n,k}$. \n\n We will\ndenote by $\\overline 0$ the trivial vector bundle with trivial hermitian\nstructure and, for any $X$, we denote by $\\overline {\\mathcal O}_{X}$ the structural\nsheaf with the metric $\\|1\\|=1$.\nFor\n$-n\\le k < 0$, the complex $\\overline{\\pi_{\\ast}{\\mathcal O}(k)}$ can be represented\nby $\\overline 0$, while \nthe complex $\\overline{\\pi_{\\ast}{\\mathcal O}(0)}$ can be represented by the\nstructural sheaf ${\\mathcal O}_{{\\mathbb Z}}$ with the metric $\\|1\\|=1\/n!$. Since this\nmetric depends on $n$ it will be simpler to consider the complexes\n$\\overline{\\pi_{\\ast}{\\mathcal O}(k)}'=\\overline{\\pi_{\\ast}{\\mathcal O}(k)}$ for $-n\\le k < 0$ and \n$\\overline{\\pi_{\\ast}{\\mathcal O}(k)}'=\\overline {\\mathcal O}_{{\\mathbb Z}}$.\nThen we write \n\\begin{displaymath}\n  t'_{n,k}=T^{h}(\\overline{\\pi},\\overline{{\\mathcal O}(k)},\\overline{\\pi_{\\ast}{\\mathcal O}(k)}'),\\\n  -n\\le k \\le 0.\n\\end{displaymath}\nThese characteristic numbers satify\n\\begin{displaymath}\n  t'_{n,k}=\n  \\begin{cases}\n    t_{n,k},&\\text{ if }-n\\le k<0,\\\\\n    t_{n,0}-(1\/2)\\log(n!),& \\text{ if }k=0.\n  \\end{cases}\n\\end{displaymath} \nClearly it is equivalent to work with the characteristic numbers\n$t_{n,k}$ or with $t'_{n,k}$.\n\nIn order to finish the the proof of Theorem \\ref{thm:5} it only\nremains to show the following particular cases.\n\n\\begin{theorem}\\label{thm:7}\nFor every $0\\leq k\\leq n$, we have the equality in $\\cha^{1}(\\Spec\n{\\mathbb Z})={\\mathbb R}$ \n\\begin{equation}\\label{eq:32}\n  \\amap(t'_{n,-k})=\\chh(\\overline{\\pi_{\\ast}{\\mathcal O}(-k)}')-\\\n  \\pi_{\\ast}(\\chh(\\overline{{\\mathcal O}(-k)})\\widehat{\\Td}(\\overline{\\pi})).\n\\end{equation}\n\\end{theorem}\nThe proof will proceed by induction. The next proposition treats the first case.\n\\begin{proposition}\\label{prop:7}\nEquation \\eqref{eq:32} holds for $k=0$.\n\\end{proposition}\n\\begin{proof}\nThe proof exploits the behavior of generalized analytic torsion with\nrespect to composition of morphisms. Let us consider the diagram\n\\begin{displaymath}\n  \\xymatrix{\n    {\\mathbb P}^n_{{\\mathbb Z}} \\ar[rd]_{\\Id} \\ar[r]^{\\Delta\\ \\ \\ }   & {\\mathbb P}^n_{{\\mathbb Z}}\n    \\underset{{\\mathbb Z}}{\\times}{\\mathbb P}^n_{{\\mathbb Z}} \\ar[r]^{p_1}\\ar[d]^{p_2} &\n    {\\mathbb P}^n_{{\\mathbb Z}} \\ar[d]^{\\pi}\\\\ \n    & {\\mathbb P}^n_{{\\mathbb Z}} \\ar[r]_{\\pi_{1}} & \\Spec {\\mathbb Z} .}\n\\end{displaymath}\nThe Fubini-Studi metric on the tangent space $T_{{\\mathbb P}^{n}_{{\\mathbb C}}}$\nis invariant under complex conjugation. It induces a \nmetric on the tangent space of the product of projective spaces. On\neach morphism on the above diagram we consider the relative hermitian\nstructure deduced by the hermitian metrics on each tangent space. With\nthis choice \n\\begin{equation}\n  \\label{eq:24}\n  \\overline \\Id=\\overline p_{2}\\circ \\overline {\\Delta },\n\\end{equation}\nwhere the composition of relative hermitian structures is defined in\n\\cite[Definition 5.7]{BurgosFreixasLitcanu:HerStruc}. On the relative\ntangent bundle  $T_{p_{2}}$\nwe consider the metric induced by the metric on $T_{{\\mathbb P}^{n}\\times\n  {\\mathbb P}^{n}}$.\nSince the\nshort exact sequence\n\\begin{displaymath}\n  0\\longrightarrow\n  \\overline T_{p_{2}}\\longrightarrow\n  \\overline T_{{\\mathbb P}^{n}\\times {\\mathbb P}^{n}}\\longrightarrow\n  p_{2}^{\\ast}\\overline T_{{\\mathbb P}^{2}} \\longrightarrow 0\n\\end{displaymath}\nis orthogonaly split,\nwe deduce that the metric we consider on $\\overline p_{2}$ agrees with the\nmetric on the vector bundle\n$\\overline T_{p_{2}}$ and this, in turn agrees with the metric on\n$p_{1}^{\\ast }T_{{\\mathbb P}^{n}}$. \n\nLet $\\overline Q$ be the tautological quotient bundle on ${\\mathbb P}^{n}_{{\\mathbb Z}}$ with\nany hermitian metric invariant under complex conjugation. We denote by\n$K$ the Koszul resolution of the diagonal and $\\overline K$ the same\nresolution with the induced metrics. Namely $\\overline K$ is the complex\n\\begin{displaymath}\n  0\\to p_{2}^{\\ast}\\Lambda ^{n}\\overline Q^{\\vee}\\otimes p_{1}^{\\ast}\\overline\n  {{\\mathcal O}(-n)}\n\\to \\dots\\to p_{2}^{\\ast}\\overline Q^{\\vee}\\otimes p_{1}^{\\ast}\\overline\n  {{\\mathcal O}(-1)}\\to \\overline {\\mathcal O}_{{\\mathbb P}^{n}\\times{\\mathbb P}^{n}}\\to 0,\n\\end{displaymath}\nand there is an isomorphism in the derived category $K\\to \\Delta\n_{\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}$. Since the theories of homogeneous torsion classes\nfor closed immersions and for projective spaces are compatible\n\\cite[Definition 6.2]{BurgosFreixasLitcanu:GenAnTor} the equation\n\\begin{equation}\n  \\label{eq:25}\n  T(\\overline p_{2},\\overline K, \\overline {\\mathcal O}_{{\\mathbb P}^{n}})+ \\overline p_{2\\flat}(T(\\overline \\Delta\n  ,\\overline {\\mathcal O}_{{\\mathbb P}^{n}},\\overline K))=0\n\\end{equation}\nholds. Then\n\\begin{align*}\n  T(\\overline p_{2},\\overline K, \\overline {\\mathcal O}_{{\\mathbb P}^{n}})&=\n  T(\\overline p_{2},\\overline {\\mathcal O}_{{\\mathbb P}^{n}\\times {\\mathbb P}^{n}},\\overline {\\mathcal O}_{{\\mathbb P}^{n}})+\n  \\sum_{i=1}^{n}(-1)^{i}T(\\overline p_{2}, p_{2}^{\\ast}\\Lambda ^{i}\\overline\n  Q^{\\vee}\\otimes p_{1}^{\\ast}\\overline\n  {{\\mathcal O}(-i)},\\overline 0)\\\\\n  &=\\pi_{1}^{\\ast}T(\\overline \\pi ,\\overline {\\mathcal O}_{ {\\mathbb P}^{n}},\\overline\n  {\\mathcal O}_{{\\mathbb Z}})\\bullet \\ch(\\overline {\\mathcal O}_{{\\mathbb P}^{n}})+\\\\\n  &\\qquad \\qquad \\sum_{i=1}^{n}(-1)^{i}\\pi_{1}^{\\ast}T(\\overline \\pi ,\\overline {{\\mathcal O}\n    (-i)},\\overline 0)\\bullet \\ch(\\Lambda ^{i}\\overline Q^{\\vee}) \\\\\n  &=\\sum_{i= 0}^{n}(-1)^{i}t_{n,-i}\\bullet \\ch(\\Lambda ^{i}\\overline\n  Q^{\\vee}). \n\\end{align*}\n\nUsing that\n\\begin{displaymath}\n  \\pi _{1\\ast}(\\ch(\\Lambda ^{i}\\overline\n  Q^{\\vee})\\Td(\\overline \\pi _{1}))=\n  \\begin{cases}\n    1, &\\text{ if }i=0,\\\\\n    0, &\\text{ otherwise,} \n  \\end{cases}\n\\end{displaymath}\nwe deduce\n\\begin{equation}\n  \\label{eq:33}\n  \\overline \\pi _{1\\flat}(T(\\overline p_{2},\\overline K, \\overline {\\mathcal O}_{{\\mathbb P}^{n}}))=t_{n,0}.\n\\end{equation}\n\nBy the arithmetic Riemann-Roch theorem for closed immersions\n\\begin{displaymath}\n  \\amap(T(\\overline \\Delta\n  ,\\overline {\\mathcal O}_{{\\mathbb P}^{n}},\\overline\n  K))=\n  \\sum_{i=0}^{n}(-1)^{i}p_{2}^{\\ast}\\chh(\\Lambda ^{i}\\overline\n  Q^{\\vee})\\cdot\n  p_{1}^{\\ast}\\chh(\\overline{{\\mathcal O}(-i)})-\\Delta _{\\ast}(\\chh(\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\Delta )).\n\\end{displaymath}\nFor $i>0$\n\\begin{multline*}\n  \\pi _{1\\ast}\\left(p_{2\\ast}\\left(p_{2}^{\\ast}\\chh(\\Lambda ^{i}\\overline\n  Q^{\\vee})p_{1}^{\\ast}\\chh(\\overline {{\\mathcal O}(-i)})\\widehat {\\Td}(\\overline{p_{2}})\\right)\n  \\widehat {\\Td}(\\overline{\\pi} _{1})\\right)\\\\\n  =\\pi _{1\\ast}(\\chh(\\Lambda ^{i}\\overline\n  Q^{\\vee})\\widehat {\\Td}(\\overline{\\pi} _{1}))\\cdot \\pi _{\\ast}(\\chh(\\overline\n  {{\\mathcal O}(-i)})\\widehat{\\Td}(\\overline \\pi )).\n\\end{multline*}\nSince\n\\begin{displaymath}\n  \\zeta (\\pi _{1\\ast}(\\chh(\\Lambda ^{i}\\overline\n  Q^{\\vee})\\widehat {\\Td}(\\overline \\pi _{1})))=\n \\zeta(\\pi _{\\ast}(\\chh(\\overline\n  {{\\mathcal O}(-i)})\\widehat{\\Td}(\\overline \\pi )))=0\n\\end{displaymath}\nand $\\Ker \\zeta $ is a square zero ideal of the arithmetic Chow ring,\nwe deduce that, for $i>0$\n\\begin{equation}\\label{eq:36}\n  \\pi _{1\\ast}\\left(p_{2\\ast}\\left(p_{2}^{\\ast}\\chh(\\Lambda ^{i}\\overline\n  Q^{\\vee})p_{1}^{\\ast}\\chh(\\overline {{\\mathcal O}(-i)})\\widehat {\\Td}(\\overline{p_{2}})\\right)\n  \\widehat {\\Td}(\\overline \\pi _{1})\\right)=0.\n\\end{equation}\nFor $i=0$ we compute\n\\begin{displaymath}\n  \\pi _{1\\ast}\\left(p_{2\\ast}\\left(p_{2}^{\\ast}\\chh(\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}})p_{1}^{\\ast}\\chh(\\overline {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat {\\Td}(\\overline{p_{2}})\\right)\n  \\widehat {\\Td}(\\overline \\pi _{1})\\right)=\\pi _{\\ast}(\\chh(\\overline\n{\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\pi))^{2}.\n\\end{displaymath}\nUsing that $\\chh(\\overline {\\mathcal O}_{{\\mathbb P}^{n}})=1$ and that $\\pi\n_{\\ast}(\\widehat{\\Td}(\\overline \\pi))-1\\in \\Ker \\zeta $ we obtain that\n\\begin{equation}\\label{eq:35}\n  \\pi _{\\ast}(\\chh(\\overline\n{\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\pi))^{2}=-1+2\\pi\n_{\\ast}(\\widehat{\\Td}(\\overline \\pi )).\n\\end{equation}\nFurthermore, by the choice of metrics on the relative tangent complexes\n\\begin{equation}\\label{eq:34}\n  \\pi _{1\\ast}p_{2\\ast}\\Delta _{\\ast}(\\chh(\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\Delta )\\widehat{\\Td}(\\overline\n  p_{2})\\widehat{\\Td}(\\overline\\pi _{1}))=\n\\pi _{\\ast}(\\chh(\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline\\pi )).\n\\end{equation}\nUsing equations \\eqref{eq:36}, \\eqref{eq:35} and \\eqref{eq:34} we\ndeduce that\n\\begin{equation}\n  \\label{eq:37}\n  \\amap(\\overline\\pi _{1\\flat}(\\overline p_{2\\flat}(T(\\overline \\Delta ,\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}},\\overline K))))=\\pi _{\\ast}(\\chh(\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline\\pi ))-\\chh(\\overline {\\mathcal O}_{{\\mathbb Z}}).\n\\end{equation}\nBy equations \\eqref{eq:25}, \\eqref{eq:33} and \\eqref{eq:37} we conclude\n\\begin{displaymath}\n  \\amap(t'_{n,0})=\\chh(\\overline {\\mathcal O}_{{\\mathbb Z}})-\\pi _{\\ast}(\\chh(\\overline\n  {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline\\pi ))\n\\end{displaymath}\nproving the proposition.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm:7}] We now proceed with the\n  induction step. We assume that equation \\eqref{eq:32} holds for some\n  $k\\ge 0$ and all $n\\ge k$. Fix now $n\\ge k+1$. Consider the diagram \n\\begin{displaymath}\n  \\xymatrix{\n    {\\mathbb P}^{n-1}_{{\\mathbb Z}} \\ar[rd]_{\\pi _{n-1}} \\ar[r]^{s}   & {\\mathbb P}^n_{{\\mathbb Z}}\n    \\ar[d]^{\\pi _{n}}\\\\ \n    & \\Spec {\\mathbb Z},}\n\\end{displaymath}\n  where $s$ is the closed immersion induced by a section $s$ of\n  ${\\mathcal O}_{{\\mathbb P}^{n}}(1)$. As in the proof of Proposition \\ref{prop:7}, we\n  consider the relative hermitian \n  structures defined by the Fubini-Study metric on the tangent\n  bundles. In this way $\\overline \\pi _{n-1}=\\overline \\pi _{n}\\circ \\overline s$.\n\n  We consider the Koszul complex\n  \\begin{displaymath}\n    K_{n,k}\\colon {\\mathcal O}_{{\\mathbb P}^{n}}(-k-1)\\overset{s}{\\longrightarrow }\n    {\\mathcal O}_{{\\mathbb P}^{n}}(-k)\n  \\end{displaymath}\n  and we denote by $\\overline K_{n,k}$ the same complex provided with the\n  Fubini-Study metrics.\n\n  By the transitivity \\cite[Definition\n  7.1]{BurgosFreixasLitcanu:GenAnTor} of the homogeneous theory\n  of analytic torsion, the equation\n  \\begin{multline*}\n    T^{h}(\\overline \\pi _{n-1},\\overline {{\\mathcal O}(-k)},\\overline {\\pi _{n-1\\ast}{\\mathcal O}(-k)}')=\\\\\n    T^{h}(\\overline \\pi _{n},\\overline K_{n,k},\\overline {\\pi _{n-1\\ast}{\\mathcal O}(-k)}')+\n    \\overline\\pi _{n\\flat}(T^{h}(\\overline s,\\overline {{\\mathcal O}(-k)},\\overline K_{n,k})).\n  \\end{multline*}\n  By definition $T^{h}(\\overline \\pi _{n-1},\\overline {{\\mathcal O}(-k)},\\overline {\\pi\n    _{n-1\\ast}{\\mathcal O}(-k)}')=t'_{n-1,-k}$. By the choice of metrics, the\n  exact sequence\n  \\begin{equation} \\label{eq:38}\n    0\\to \\overline {\\pi _{n\\ast}{\\mathcal O}(-k-1)}'\n    \\to \\overline {\\pi _{n\\ast}{\\mathcal O}(-k)}' \\to \\overline {\\pi\n      _{n-1\\ast}{\\mathcal O}(-k)}'\\to 0\n  \\end{equation}\n  is orthogonal split (all the terms appearing in this short exact\n  sequence are either $\\overline 0$ or $\\overline\n  {\\mathcal O}_{{\\mathbb Z}}$). \n  This implies that\n  \\begin{multline*}\n    T^{h}(\\overline \\pi _{n},\\overline K_{n,k},\\overline {\\pi _{n-1\\ast}{\\mathcal O}(-k)}')=\\\\\n    T^{h}(\\overline \\pi _{n},\\overline{{\\mathcal O}(-k)},\\overline {\\pi _{n\\ast}{\\mathcal O}(-k)}')\n    -T^{h}(\\overline \\pi _{n},\\overline{{\\mathcal O}(-k-1)},\\overline {\\pi\n      _{n\\ast}{\\mathcal O}(-k-1)}')=\\\\\n    t'_{n,-k}-t'_{n,-k-1}.\n  \\end{multline*}\n  By Lemma \\ref{lemm:4} (the case of closed immersions)\n  \\begin{displaymath}\n    T^{h}(\\overline s,\\overline {{\\mathcal O}(-k)},\\overline K_{n,k})=\\chh(\\overline {{\\mathcal O}(-k)})-\n    \\chh(\\overline{{\\mathcal O}}(-k-1))-\\overline{s}_{\\flat}(\\chh(\\overline{{\\mathcal O}(-k)})).\n  \\end{displaymath}\nThis implies that\n\\begin{multline*}\n  \\overline{\\pi} _{n\\flat}(T^{h}(\\overline s,\\overline {{\\mathcal O}(-k)},\\overline K_{n,k}))=\\\\ \\overline{\\pi}\n  _{n\\flat}\\chh(\\overline {{\\mathcal O}(-k)})- \n    \\overline{\\pi} _{n\\flat} \\chh(\\overline{{\\mathcal O}}(-k-1))\n    -\\overline{\\pi} _{n-1\\flat}\\chh(\\overline{{\\mathcal O}(-k)}).\n\\end{multline*}\nSumming up, we deduce\n\\begin{multline}\\label{eq:39}\n  \\amap(t'_{n-1,-k}-t'_{n,-k}+t'_{n,-k-1})=\\\\\n  -\\overline{\\pi} _{n-1\\flat}\\chh(\\overline{{\\mathcal O}(-k)})+\n  \\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k)})+\n  -\\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k-1)}).\n\\end{multline}\nApplying the induction hypothesis we get\n\\begin{displaymath}\n  t'_{n,-k-1}=-\\chh(\\overline{\\pi _{n-1\\ast}{\\mathcal O}(-k)}')+\n  \\chh(\\overline{\\pi _{n\\ast}{\\mathcal O}(-k)}')-\\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k-1)}). \n\\end{displaymath}\nUsing again that the exact sequence \\eqref{eq:38} is orthogonally\nsplit, we deduce that\n\\begin{displaymath}\n  t'_{n,-k-1}=\\chh(\\overline{\\pi _{n\\ast}{\\mathcal O}(-k-1)}')-\\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k-1)}). \n\\end{displaymath}\ncompleting the inductive step and proving the Theorem \\ref{thm:7} and\ntherefore Theorem \\ref{thm:5}.\n\\end{proof}\n\nOnce we have proved that the formal properties of an analytic torsion\ntheory imply the arithmetic Riemann-Roch theorem we can compute\neasily the characteristic numbers $t_{n,k}$ for the homogeneous\nanalytic torsion and all $n\\ge 0$ and $k\\in {\\mathbb Z}$. By the Theorem\n\\ref{thm:5} they satisfy\n\\begin{displaymath}\n  \\amap(t_{n,k})=\\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})-\\\n  \\pi_{n\\ast}(\\chh(\\overline{{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})\\widehat{\\Td}(\\overline{\\pi}_{n})).\n\\end{displaymath}\nObserve it is enough to compute $t_{n,k}$ for $k\\geq -n$, since the self-duality of the homogenous analytic torsion \\cite[Thm. 9.12]{BurgosFreixasLitcanu:GenAnTor} immediately yields the relation\n\\begin{displaymath}\n\tt_{n,k}=(-1)^{n}t_{n,-k-n-1}.\n\\end{displaymath}\nTherefore, from now on we restrict to this range of values of $k$.\n\nWe first compute\n$\\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})^{(1)}$. For $-n\\leq k\\leq -1$ this quantity vanishes:\n\\begin{displaymath}\n\t \\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})^{(1)}=0,\\quad -n\\leq k\\leq -1.\n\\end{displaymath}\nSuppose now $k\\geq 0$. Using that the volume\nform $1\/n!\\omega _{FS}^{n}$ is given, in a coordinate patch, by\n\\begin{displaymath}\n  \\mu =\\left(\\frac{i}{2\\pi }\\right)^{n}\\frac{\\dd z_{1}\\land \\dd \\overline\n    z_{1}\\land\\dots \\land \\dd z_{n}\\land \\dd \\overline\n    z_{n}}{(1+\\sum_{i=1}^{n}z_{i}\\overline z_{i})^{n+1}},\n\\end{displaymath}\nit is easy to see that the basis $\\{x_{0}^{a_{0}}\\dots\nx_{n}^{a_{n}}\\}_{a_{0}+\\dots+a_{n}=k}$ is orthonormal and satisfies\n\\begin{displaymath}\n  \\|x_{0}^{a_{0}}\\dots\nx_{n}^{a_{n}}\\|^{2}_{L^{2}}=\\frac{a_{0}!\\dots a_{n}!}{(k+n)!}.\n\\end{displaymath}\nTherefore\n\\begin{displaymath}\n  \\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})^{(1)}=\n  \\sum_{a_{0}+\\dots+a_{n}=k}-\\left(\\frac{1}{2}\\right)\n  \\log\\left(\\frac{a_{0}!\\dots a_{n}!}{(k+n)!}\\right).\n\\end{displaymath}\n\n\n\nTo compute\n$\\pi_{n\\ast}(\\chh(\\overline{{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})\\widehat{\\Td}(\\overline{\\pi}_{n}))$\nwe follow \\cite{GilletSoule:ATAT}, where the case $k=0$ is\nconsidered. Let $\\alpha _{n,k}$ be the coefficient of $x^{n+1}$\nin the power series\n\\begin{math}\ne^{kx}\\left(\\frac{x}{1-e^{-x}}\\right)^{n+1}  \n\\end{math}\nand let $\\beta _{n,k}$ be the coefficient of $x^{n}$ in the power series\n\\begin{math}\n  \\int_{0}^{1}\\frac{\\phi(t)-\\phi(0)}{t}\\dd t, \n\\end{math}\nwhere\n\\begin{displaymath}\n  \\phi (t)=e^{kx}\\left(\\frac{1}{tx}-\\frac{e^{-tx}}{1-e^{-tx}}\\right)\n  \\left(\\frac{x}{1-e^{-x}}\\right)^{n+1}.\n\\end{displaymath}\nThen, by a slight modification of the argument in \\cite [Proposition\n2.2.2 \\& 2.2.3]{GilletSoule:ATAT}, we derive\n\\begin{displaymath}\n  \\pi\n  _{n\\ast}(\\chh(\\overline{{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})\n  \\widehat{\\Td}(\\overline{\\pi}_{n}))^{(1)}=\\frac{1}{2}   \n  \\amap\\left(\\alpha _{n,k}\\sum_{p=1}^{n}\\sum_{j=1}^{p}\\frac{1}{j}+\\beta\n    _{n,k}\\right).\n\\end{displaymath}\nThe factor $(1\/2)$ appears from the different normalization used here (see\n\\cite[Theorem 3.33]{BurgosKramerKuehn:accavb}). We thus have to determine the coefficients $\\alpha_{n,k}$ and $\\beta_{n,k}$. The numbers $\\alpha_{n,k}$ can be obtained similarly to \\cite[Eq. (27)]{GilletSoule:ATAT}. We obtain\n\\begin{displaymath}\n\t\\alpha_{n,k}=\\begin{cases}\n\t\t0\t&\\text{for } -n\\leq k\\leq -1,\\\\\n\t\t\\binom{k+n}{n}\t&\\text{for }k\\geq 0.\n\t\\end{cases}\n\\end{displaymath}\nThe values of $\\beta_{n,k}$ are expressed in terms of some secondary Todd numbers, that in turn are determined by a generating series:\n\\begin{displaymath}\n\t\\beta_{n,k}=\\sum_{j=0}^{n}\\widetilde{\\Td}_{n-j}\\frac{k^{j}}{j!},\n\\end{displaymath}\nwhere the $\\widetilde{\\Td}_{m}$ are given by the equality of generating series\n\\begin{equation}\\label{eq:todd_numbers}\n\t\\sum_{m\\geq 0}\\frac{\\widetilde{\\Td}_{m}}{m+1}T^{m+1}=\n\t\\sum_{m\\geq 1}\\frac{\\zeta(-(2m-1))}{2m-1}\\frac{y^{2m}}{(2m)!},\\quad T=1-e^{-y}.\n\\end{equation}\nHere $\\zeta$ stands for the Riemann zeta function. The result is a direct consequence of the definition of $\\beta_{n,k}$ and the computation of the numbers $\\beta_{n,0}$ in \\cite[Prop. 2.2.3 \\& Lemma 2.4.3]{GilletSoule:ATAT}. \n\nWe summarize these computations for the principal characteristic numbers $t_{n,k}$, $-n\\leq k\\leq 0$, since it is particularly pleasant.\n\\begin{proposition}\nThe principal characteristic numbers $t_{n,k}$ are given by\n\\begin{displaymath}\n\tt_{n,k}=\\begin{cases}\n\t\t-\\frac{1}{2}\\sum_{p=1}^{n}\\sum_{j=1}^{p}\\frac{1}{j},\t&\\text{for }k=0,\\\\\t\n\t\t\\\\\n\t-\\frac{1}{2}\\sum_{j=0}^{n}\\widetilde{\\Td}_{n-j}\\frac{k^{j}}{j!},\t\t&\\text{for } -n\\leq k\\leq -1,\n\\end{cases}\n\\end{displaymath}\nwhere the sequence of numbers $\\widetilde{\\Td}_{m}$, $m\\geq 0$, is determined by \\eqref{eq:todd_numbers}. \n\\end{proposition}\n\n\n\\newcommand{\\noopsort}[1]{} \\newcommand{\\printfirst}[2]{#1}\n  \\newcommand{\\singleletter}[1]{#1} \\newcommand{\\switchargs}[2]{#2#1}\n  \\def$'${$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}