diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdusw" "b/data_all_eng_slimpj/shuffled/split2/finalzzdusw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdusw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{s:intro}\n\nHydrogen peroxide (\\ho) is a well-studied system because of its\nunusual properties, particularly the almost freely rotating\nOH moieties.\nThe role of \\ho\\ in the chemistry of the Earth's atmosphere \\cite{74Davis.H2O2,03ViMaMi.H2O2, 13AlAbBeBo.H2O2} as well as of the Martian atmosphere \\cite{04EnBeGr.HOOH,12EnGrLe.HOOH} has been widely acknowledged. It has also\nrecently been detected in the interstellar medium \\cite{11BePaLi.H2O2}.\nThe vibration-rotation spectra of hydrogen peroxide has attracted\nsignificant attention, both experimental\nand theoretical. For example it has been used\n as a benchmark system with large amplitude motion for\ntesting different variational nuclear motion codes \\cite{93.BrCarrin.HOOH,00CaHaxx.HOOH,00Luckhaus.HOOOH,01ChMaGu.HOOH,02YuMuck.HOOH,02Mladenovic.HOOH,03LiGuxx.HOOH,09CaHaBo.h2o2,11CaShBo.H2O2}.\n\n\nVery recently Ma{\\l}yszek and Koput \\cite{Koput3} presented a highly accurate \\ai\\\npotential energy surface (PES) of HOOH which was shown to reproduce the known vibrational band origins\nwith the average accuracy of 1 \\cm. Other \\ai\\ PESs of HOOH were reported by Harding \\cite{91Harding.HOOH},\nKuhn \\etal\\cite{99Kuhn.HOOH}, Senent \\etal \\cite{00Senent.HOOH}, and Koput \\etal\\cite{Koput1}.\n\n\nHydrogen peroxide has three important properties from the viewpoint of\nvariational calculations. Firstly, the large amplitude motion\nof the OH internal rotors that has already been mentioned. Secondly, its\nrelatively low dissociation energy of about 17,000 \\cm\\\nhas made \\ho\\ a benchmark tetratomic molecule for experimental study\nof the dissociation process \\cite{Rizo}. Thirdly, \\ho\\ is a tetratomic\nsystem where variational calculations can really aid the analysis\nof spectra.\n\nFor triatomic molecules, accurate\ncalculation of the rotation-vibration levels to high\naccuracy using variational nuclear motion\nmethods has become routine \\cite{jt309,jt512,12HuScTaLe.CO2}. For tetratomic\nmolecules this process is just beginning; it is natural for initial\nhigh accuracy studies to focus on molecules with large\namplitude motion such as ammonia \\cite{Huang:Schwenke:Lee:2010:I,Huang:Schwenke:Lee:2010:II} and hydrogen\nperoxide.\n\nThe advantages of using variational calculations to assign\nvibration-rotation spectra of triatomic molecules has been\ndemonstrated for several molecules. Initial studies focused\non \\hp\\ \\cite{jt80,jt102,jt193} and water\n\\cite{jt200,jt218}, systems for which the use of variational\ncalculations to analyse spectra is now the accepted procedure.\nIn particular, spectra involving hot molecules, and hence high rotational\nstates, and large amplitude motion,\nsuch as \\hp\\ on Jupiter \\cite{jt80} and water on the Sun\n\\cite{jt200}, assignments using the traditional, effective\nHamiltonian approach are almost impossible.\n\nA significant advantage of variational calculations over effective\nHamiltonian techniques is the automatic allowance for accidental\nresonances between vibrations. Whereas for most triatomic molecules such\nresonances become significant at fairly high vibrational\nenergies, for tetratomic molecules accidental resonances\ncan even make the analysis of low-lying vibrational states intractable using effective Hamiltonians. \\ho\\ is a good example of this situation. Although \\ho\\ spectral lines\nare strong and were first observed more than seventy years ago\nwith spectrometers much less sophisticated then those available\nnowadays \\cite{41ZuGi.H2O2,50Giguer.H2O2}, the analysis of experimental spectra involving high $J$ transitions for \\ho\\ is only complete up to\n2000 \\cm\\ \\cite{perin1,perin2,perin3}, significantly lower\nin frequency than transitions to the OH stretching fundamentals.\nOne reason for this is the\ncomplication of the analysis by accidental resonances.\nAccurate variational calculations on \\ho\\ offer a way out of this impasse.\n\n\n\nRecent advances in variational calculations suggest that they can be\nused for systems larger than triatomic. High accuracy variational\ncalculations of the spectra and line lists for tetratomic molecules\nsuch as ammonia \\cite{jt466,jt500,Huang:Schwenke:Lee:2010:II} have been\nperformed.\n These NH$_3$ line lists have been used both for the assignment\nof transitions involving higher vibrational states \\cite{12SuBrHu.NH3}, hot rovibrational spectra involving high $J$ levels \\cite{jt508}\nas well as for correcting and improving the analysis\nof more standard transitions \\cite{Huang:Schwenke:Lee:2010:II,jt543}. Numerical\ncalculations of wavefunctions for high $J$ states of tetratomic molecules\nare possible not only because modern computers have the ability to diagonalise\nlarger matrices but also because, as illustrated below, the accuracy\nof calculations employing\napproximate kinetic energy operators \\cite{multimode,trove-paper}\nbecomes comparable with those using an exact kinetic energy approach \\cite{jt312}. While high $J$ calculations within the exact kinetic\nenergy approach are still\ncomputationally challenging for tetratomic molecules,\ncalculations with $J \\sim 50$ are feasible with approaches such as\nTROVE \\cite{trove-paper}. Furthermore, the possibility of calculating \\ai\\\ndipole moment surfaces of extremely high accuracy \\cite{jt509}\nenhances the value of using variational calculations since they can also be\nused to\ncreate line lists.\nThese factors raise the possibility of creating accurate line lists for\n\\ho. However, the presence of the large amplitude, torsional motion\nof the two OH fragments in \\ho\\ complicates the problem. This requires an appropriate\nnuclear motion programme for calculation of the\nrovibrational energy levels by solving the corresponding Schr\\\"odinger\nequation; this programme should be able to compute high $J$ levels\nwithin the limitations of the modern computers.\n\nIn this paper we compute high accuracy rovibrational energy\nlevels going to high $J$ for\n\\ho\\ using the \\ai\\ PES due to\nMa{\\l}yszek and Koput \\cite{Koput3}.\n To do this we\ntest two nuclear motion programmes:\nthe exact kinetic energy (EKE)\nprogramme WAVR4 \\cite{jt339} and approximate kinetic energy programme TROVE\n\\cite{trove-paper}. It is shown that use of\nTROVE allows us to calculate very high $J$ energy levels which are\nin excellent agreement with observation.\n The paper is organised as follows.\nSection~\\ref{s:method} describes the modifications of the TROVE programme\nnecessary to make it suitable for the calculation of spectra\nof such a nonrigid molecule.\nSection~\\ref{s:results} describes the details of computations performed.\nSection~\\ref{s:concl} presents our results which is followed by the concluding section\nwhich discusses prospects for further work on this system.\n\n\n\n\\section{Methods of calculation }\n\\label{s:method}\n\n\nThe accuracy of a calculation of rovibrational energy levels depends\nfirst of all on the accuracy of the potential energy surface (PES) used\nas input to the nuclear motion Schr\\\"odinger equation. Until recently the most\naccurate PES for \\ho\\ was the one due to Koput \\etal\\cite{Koput1} which gave a\ntypical discrepancy between theory and experiment for vibrational band\norigins of about 10 \\cm\\ \\cite{Koput2}. However,\ntwo of us\\cite{Koput3} recently determined a very accurate PES computed using the explicitly correlated coupled-cluster method [CCSD(T)-F12] method,\n\\cite{CC,CC2} in the F12b form \\cite{F12a} as implemented in\nthe MOLPRO package \\cite{molpro}.\nVarious correlation-consistent basis sets were used for various parts of the PES, the largest being aug-cc-pV7Z. The CCSD(T)-F12 results were augmented with the Born-Oppenheimer diagonal, higher-order valence-electron correlation, relativistic, and core-electron correlation corrections. The 1762 \\ai\\ points obtained were fitted to the functional form\n\\begin{equation}\n\\label{e:koput}\nV(q_1,q_2,q_3,q_4,q_5,q_6) = \\sum_{ijklmn} c_{ijklmn} q_1^{i} q_2^{j}q_3^{k} q_4^{l} q_5^{m} \\cos{n q_6}\n\\end{equation}\nwhere $q_{i}$ ($i = 1,2,3$) are the Simons-Parr-Finlan stretching OO and OH coordinates \\cite{73SimParrFin.SPF} $q_1= (R-R_{\\rm e})\/R $ and\n$q_i = (r_i - r_{\\rm e})\/r $ ($i=1,2$), $q_4 = \\theta_1 - \\theta_{\\rm e} $ and $q_5 = \\theta_2 - \\theta_{\\rm e} $ are the two OOH bending coordinates, $q_6 = \\tau$ is the torsional angle $\\angle$HOOH (see Fig.~\\ref{f:tau}),\nand $R_{\\rm e}$, $r_{\\rm e}$, and $\\theta_{\\rm e}$ are the\ncorresponding equilibrium values. The\nexpansion coefficients $c_{ijklmn}$ used in this work are\ngiven in the supplementary material \\cite{supl} to this\narticle (see also Ref. \\citenum{Koput3}).\n\nRecent calculations\\cite{Koput3} using this PES gave, for the 30 observed vibrational band origins of \\ho, a standard deviation for the observed minus calculated (obs $-$ calc) wavenumbers of about 1 \\cm, an order of \nmagnitude improvement over the previous results.\\cite{Koput2} An \\ai\\ line list with this accuracy could be useful for a number of applications. \nHowever, for most applications it is also necessary to accurately compute highly excited rotational levels. This is done in this work. Before looking at high $J$ rotational levels, we reconsidered the $J=0$ results of Ma\\l yszek and Koput \\cite{Koput3} using both EKE programme WAVR4 \\cite{jt339} and approximate kinetic energy programme TROVE \\cite{trove-paper}.\n\n\nDiatom-diatom HO--OH coordinates were employed in the programme WAVR4; these coordinates were one of those used to consider acetylene -- vinylidene isomerisation \\cite{jt346}. The calculations used a discrete variable\nrepresentations (DVR) based on a grid of 10 radial functions for each OH coordinate\nand 18 radial functions for the OO coordinate. The parameters used for\nOH stretch Morse-oscillator like functions were $r_{\\rm e} = 0.91$ \\AA,\n$\\omega_{\\rm e} = 2500$ \\cm\\ and $D_{\\rm e} = 35000$ \\cm, and $r_{\\rm e} =1.53$ \\AA,\n$\\omega_{\\rm e} = 1500$ \\cm\\ and $D_{\\rm e} = 45 000$ \\cm\\ for the OO stretch.\nThe bending basis set consists of coupled angular functions\n\\cite{jt339} defined by $j^{\\rm max}=l^{\\rm max} =22$ and $k^{\\rm max}$ =\n12. The resulting energy levels with $J$ = 0 were within\n0.1 \\cm\\ of the previous calculations \\cite{Koput3}, see Table~\\ref{tab:J1}.\nHowever for WAVR4 calculations of the same accuracy for levels with $J = 1$\nrequire about 10 times more computer time. This is a consequence\nof the $J$ -- $K$ coupling used in the EKE procedure. This coupling\nis essential for the linear HCCH system \\cite{jt346} and very floppy\nmolecules \\cite{jt312}, but not for \\ho.\nThe use of WAVR4 to calculate energies of high $J$ levels is\ncomputationally unrealistic at present and we note that indeed corresponding\nstudies on acetylene have thus far been confined to low $J$ values\n\\cite{jt479}.\n\nTROVE is a computer suite for rovibrational calculations of energies and intensities for molecules of (at least in principle) arbitrary structures. TROVE uses a multilevel contraction scheme for constructing the rovibrational basis set. The primitive basis functions are given by products of six 1-dimensional (1D) functions $\\phi_i(\\xi_i)$, where $\\xi_i$ represents one of the six internal coordinates. For HOOH we choose $\\xi_1$, $\\xi_2$, and $\\xi_3$ to be the linearized versions of the three stretching internal displacements\n$R-R_{\\rm e}$, $r_1-r_{\\rm e}$, and $r_2-r_{\\rm e}$, respectively,\n$\\xi_{4}$ and $\\xi_{5}$\nare the linearized versions of the two bending\ndisplacements $\\theta_1-\\theta_{\\rm e}$\nand $\\theta_2-\\theta_{\\rm e}$; $\\xi_6$ is\nthe torsional coordinate $\\tau$, see Fig.~\\ref{f:tau}.\n\nThe kinetic energy operator in TROVE is given by an\nexpansion in terms of the five coordinates $\\xi_{i}$,\nrepresenting the rigid modes $i=1\\ldots 5$. The\npotential energy function is also expanded but using three Morse-type\nexpansion variables $1-\\exp(-a_i\\xi_i)$ ($i=1,2,3$) and two bending\ncoordinates $\\xi_4$ and $\\xi_5$.\nHere $a_1 = 2.2$ \\AA$^{-1}$, $a_2=a_3=2.3$ \\AA$^{-1}$\nwere selected to match closely the shape of the \\ai\\\nPES along the stretching modes. In the present work we\nemploy 6th and 8th order expansions to represent,\nrespectively, the kinetic energy operator and potential\nenergy function.\n\nThe rovibrational motion of the non-rigid molecule HOOH is\nbest represented by the extended $C_{\\rm 2h}^{+}$(M)\nmolecular symmetry group \\cite{Hoxxxx.HOOH}, which is\nisomorphic to $D_{\\rm 2h}$(M) as well as to the extended\ngroup $G(4)$(EM).\\cite{BJ} As explained in detail by Bunker\nand Jensen\\cite{BJ}, the extended group is needed to describe\nthe torsional splitting due both the \\textit{cis}-\nand \\textit{trans}-tunnelings. In the present work we use\nthe $D_{\\rm 2h}$(M) group to classify the symmetry of the HOOH states.\nThis group is given by the eight irreducible\nrepresentations $A_{\\rm g}, A_{\\rm u}, B_{\\rm 1g},\nB_{\\rm 1u}, B_{\\rm 2g}, B_{\\rm 2u}, B_{\\rm 3g}, B_{\\rm 3u}$.\nIn order to account for the extended symmetry properties of the\nfloppy HOOH molecules an extended range for torsion motion,\nfrom 0$^{\\circ}$ to 720$^\\circ$\nwas introduced into TROVE. In this representation\n$\\tau = 0$ and 720$^\\circ$ correspond to the \\textit{cis} barrier, while at $\\tau=360^{\\circ}$ the molecule has the \\textit{trans} configuration.\n\n\n\nTROVE's primitive basis functions, $\\phi_{v_i}^{(i)}$ ($i=1\\ldots 6$),\nare generated numerically by solving six 1D vibrational Schr\\\"{o}dinger\nequations for each vibrational mode $i$ employing the\nNumerov-Cooley method~\\cite{Numerov,Cooley}. The corresponding\nreduced 1D Hamiltonian operators $H^{\\rm 1D}_i$\n($i=1\\ldots 6$) are obtained by freezing the five remaining modes at\nthe corresponding equilibriums. The integration ranges are\nselected to be large enough to accommodate all basis functions\nrequired (see below for the discussion of the basis set sizes).\nThe torsional functions are obtained initially on the\nrange $\\tau=0\\ldots 360^{\\circ}$ and transform according\nwith the $C_{\\rm 2h}$(M) group. A very fine grid of 30,000 points\nand the quadruple numerical precision [real(16)] was used for\ngenerating the eigenfunctions of the corresponding Schr\\\"{o}dinger\nequation in order to resolve the $trans$ splittings up to $v_6=42$.\nThe wavefunctions are then extended to $\\tau=360\\ldots 720^\\circ$\nthrough the $+\/-$ reflection of the $C_{\\rm 2h}$(M) values and\nclassified according to $D_{\\rm 2h}$(M). The extended primitive\ntorsion functions are able to account for the torsional splitting\ndue to both the \\textit{cis} and \\textit{trans} tunneling.\n\nThe 6D primitive basis functions are then formed from different\nproducts of the 1D functions $\\phi_{v_i}^{(i)}$. The size of the basis set\nis controlled in TROVE by the so-called polyad number $P$, which in the present case is given by\n\\begin{equation}\\label{e:polyad}\n P = 4 v_1 + 8 (v_2+v_3) + 8 (v_4 + v_5) + v_6 \\le P_{\\rm max},\n\\end{equation}\nwhere $v_i$, $i=1\\ldots 6$ are the local mode quantum numbers\ncorresponding to the primitive functions $\\phi_{v_i}^{(i)}(\\xi_{i})$.\nThe primitive basis set is then processed through a number of\ncontractions, as described in detail previously \\cite{jt466},\nto give a final basis set in the $J=0$ representation.\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{HOOH.eps}\n\\vspace*{0.5cm}\n\\caption{\\label{f:opt:geom} The internal coordinates for the HOOH molecule. }\n\\label{f:tau}\n\\end{center}\n\\end{figure}\n\n\n\n\nCalculations using programme TROVE started with a search for a basis\nset and operator expansions which would give results close to\nthe EKE ones. The final values of the basis set parameters were\nchosen to try to meet two conflicting requirements: the best possible\nconvergence and a compact enough calculation to\nallow high $J$ energy levels to be computed. The\nfinal basis set parameters were the following: the maximum polyad\nnumber $P_{\\rm max}=42$, which also corresponds to the highest excitation of the torsional mode $v_6$.\nFor O-O stretch the maximal number was 8, for OH stretches - 8 and for\nthe bending modes - 10. These parameters control the size of the basis\nset used in the TROVE calculations according with Eq.~(\\ref{e:polyad}). For the $J=0$ levels\nthey give good agreement with the previous studies, see Table~\\ref{tab:J0}.\n\n\\section{Results}\n\\label{s:results}\n\nThe updated version of TROVE was used to calculate excited\nrotational levels for $J$ up to 35, the highest assigned thus far experimentally.\nInitial calculations were performed with the equilibrium distances and\nangles obtained \\ai\\ in Ref. \\citenum{Koput3}. In this case\nthe discrepancies between theory and experiment increased\nquadratically with increasing $J$: the $J = 1$ levels\nwere calculated with an accuracy around 0.001 \\cm, but those for $J = 35$\ndiffer from experiment by about 1 \\cm.\n\nWe therefore chose to adjust the equilibrium parameters $R_{\\rm e}$\nand $r_{\\rm e}$ to better reproduce the experimental values. Only\nvery small changes were needed to make the $J = 35$ levels accurate to\nabout 0.03 \\cm\\ for the ground vibrational state. In particular, the\noriginal value of $R_{\\rm e}$ of 1.45539378 \\AA\\ was shifted to\n1.45577728 \\AA\\ and $r_{\\rm e} = 0.96252476$ \\AA\\ moved to 0.96253006 \\AA.\nThe rotational structure\nwithin the excited vibrational states is of similar accuracy, meaning\nthat these levels are essentially shifted just by the discrepancy in\nthe vibrational band origin. In practice this geometry shift not only\nmeant that low $J$ energy levels were reproduced with an accuracy of 1\n\\cm, reproducing the accuracy of the vibrational band origin, but\nalso resulted in pseudo-resonance artifacts. To illustrate\nthis consider the interaction between the ground vibrational state and\nthe low-lying $v_4 = 1$ torsional vibrational state, which lies about\n2 \\cm\\ too low in the calculations. This results is an artificial\ncloseness, and interaction, between levels with the same $J$ and\n$K_a= 8$ for $v=0$ and $K_a = 6$ for $v_4 = 1$. The resulting shift\nin the energy levels is significant; it grows with $J$ and reaches\nabout 1 \\cm\\ at $J$ = 30. We call this a pseudo-resonance artifact\nsince no such interaction is seen in the experimentally-determined energy levels.\n\nThere are several ways to avoid this artificial pseudo-resonances. One\nwould be to fit the PES to experimental data, which would remove this\nartificial near-degeneracy. This is likely to be a topic of future\nwork. An alternative possibility, which is already available within\nTROVE \\cite{jt466}, is to simply adjust the calculated values of the\nvibrational band origins given by the $J = 0$ calculation to the\nobserved ones prior to their use in calculations of the $J>0$ levels.\nThis option, which is not available in EKE codes which couple the\nbending basis with the rotational functions \\cite{jt14}, not only\nshifts the energies, it also rearranges the matrix elements so that the\nartificial resonances disappear. With this adjusted calculation the\nenergy levels vary smoothly with the increasing $J$ and $K_a$ quantum\nnumbers, see Tables \\ref{tab:J1}, \\ref{tab:J30} and \\ref{tab:J35}, as one would expect\n\\cite{jt205} from purely \\ai\\ levels.\n\nOne other problem remained when comparing our rotationally excited\nenergy levels with the observed ones. This concerned rotational\nlevels with the quantum number $K_a = 1$ which did not behave as\nlevels associated with other values of $K_a$. The error for the\ntwo levels with $K_a = 1$ increases disproportionately to that of\nother levels as $J$ increases. This error was about 0.1 \\cm\\ for $J$ =\n30. A series of test calculations revealed the reason for such\ndiscrepant behaviour of the levels with $K_a = 1$. It transpires that a\nsmall change in the height of the torsional barrier, which is strongly\ninfluenced by the linear expansion coefficient $c_{000001}$, does not affect other $K_a$ levels, but significantly influences only those with $K_a$ = 1.\nVarying this expansion coefficient can both increase and decrease the splitting of the $K_a$ = 1 doublet. As this splitting is overestimated in calculations\nusing the \\ai\\ value of $c_{000001}$, its reduction by about 1~\\%\\ from\nthe \\ai\\ value of 0.00487 to 0.00483 $E_h$\nresults in roughly a fourfold improvement of the obs -- calc value for the $K_a=1$ levels. This change affects the value of\nthe ground-state torsional splitting of 11 \\cm\\ and also the values of the other torsional energy levels, all of which move significantly closer to the observed values, than the purely \\ai\\ levels given in the Table~\\ref{tab:J0}. In\nparticular, this small adjustment improves the calculated ground-state splitting to 11.4 \\cm\\ and the first torsional level to 255.2 \\cm.\nThus adjusting $c_{000001}$\nnot only improves significantly the values of levels with $K_a=1$ levels,\n it\nimproves the overall agreement with experiment for the band origins. The\nunderlying reason for this is that the expansion coefficient $c_{000001}$\ncontrols the height of the torsional barrier.\n\nThese minor adjustments result in very accurate values for rotational\nenergy levels a sample of which are presented in Tables~\\ref{tab:J1},\n\\ref{tab:J30} and \\ref{tab:J35}. A more comprehensive set of energy\nlevels is given in the supplementary material \\cite{supl}. From these\ntables one can see that the discrepancy between observed and\ncalculated energy values increases both gently and smoothly with\nrotational quantum number $J$. Such calculations therefore provide an\nexcellent starting point for assigning high $J$ transitions both\nwithin the ground state and to excited vibrational states, as the\ndensity of observed transitions is orders of magnitude smaller than the accuracy of calculations.\n\n\\section{Conclusions}\n\\label{s:concl}\n\nWe present results of \\ai\\ and slightly adjusted \\ai\\ calculations for\nthe vibrational and rovibrational energy levels of the \\ho\\ molecule.\nUse of the accurate \\ai\\ PES calculated by Ma\\l yszek and Koput\n\\cite{Koput3} reproduces the known vibrational band origin with a\nstandard deviation of about 1 \\cm. The use of programme TROVE\n\\cite{Trove} for the nuclear motion calculations allowed us compute\nhigh rotational levels up to $J=35$. Indeed,\nenergy levels with $J=50$\ncould be calculated on a high-end workstation, and the accuracy of prediction will be very\nhigh - better than 0.5 \\cm. However, we have not yet performed such calculations,\nas no comparison with experimental values is currently possible.\nExperimentally derived energy levels up to $J=35$ are compared with our\ncalculations. These are reproduced with an unprecedented accuracy of\n0.001 \\cm\\ for the levels up to $J=10$ and 0.02 \\cm\\ for all the known\nlevels above this. Variational calculations using this\nslightly adjusted \\ai\\ PES\nresults in very smooth variation in the discrepancies between the\nobserved and calculated levels as a function of the rotational quantum\nnumbers $J$ and $K_a$. This smoothness and accuracy is the key to the\nsuccessful analysis of previously unassignable spectra\n\\cite{jt200,jt205} as, in particular, the accidental resonances, which\nseriously complicate any analysis based on the use an effective\nHamiltonian, are automatically allowed for in such calculations.\n\nThere is one other important aspect of the \\ho\\ rotation-vibration\nproblem which we should mention. The detection of extrasolar planets and,\nin particular, our ability to probe the molecular composition of these\nbodies using spectroscopy \\cite{jt400}, has led to demand for\naccurate, comprehensive line lists over an extended range of both\ntemperature and wavelength for all species of possible importance in\nexoplanet atmospheres \\cite{jt528}. The accuracy\nof the calculations presented here and, especially their\nability to reliably predict highly excited rotational levels which\nare of increasing importance at higher temperatures, suggests\nthat the present work will provide an excellent starting point\nfor the calculation of a comprehensive line list for \\ho. In this\nwe will be following the recent work of Bowman and co-workers who have\ncomputed similar line lists for somewhat more rigid hydrocarbon\nsystems \\cite{09WaScSh.CH4,12CaShBo.C2H4}.\n\n\\section*{Acknowledgment}\nThis work was performed as part of ERC Advanced Investigator Project 267219.\nWe also thank the Russian Fund for Fundamental Studies\nfor their support for aspects of this project.\n\n\\newpage\n\n\n\\begin{table}\n\\caption{ Calculated and observed energy levels, in \\cm, for $J=0$ using WAVR4, TROVE and\npublished by Ma{\\l}yszek and Koput (MK) \\cite{Koput3}; ``tr-shift'' results are computed with an adjusted height\nfor the torsional barrier. The observed values are taken from Ref.~\\protect\\onlinecite{perin2,perin3,perin4,95PeVaFl.H2O2}.}\n\\label{tab:J0}\n\\begin{tabular}{cccccccrrrrr}\n\\hline\n\\hline\n$v_1$ & $v_2$ & $v_3$ & $v_4$ & $v_5$ & $v_6$ & Sym& Obs & WAVR4 & TROVE &tr-shift & MK\\\\\n\\hline\n\n0 & 0 & 0 & 1 & 0 & 0 & $A_{\\rm g}$ & 254.550 & 256.406 & 256.419 & 255.490 & 255.43\\\\\n 0 & 0 & 0 & 2 & 0 & 0 & $A_{\\rm g}$ & 569.743 & 570.334 & 570.251 & 570.690 & 570.45 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 & $A_{\\rm g}$ & 865.939 & 865.547 & 865.652 & 865.468 & 866.02\\\\\n 0 & 0 & 0 & 3 & 0 & 0 & $A_{\\rm g}$ & 1000.882 & 1001.227 &1001.073 & 1002.493 & 1001.92 \\\\\n 0 & 0 & 0 & 0 & 0 & 1 & $B_{\\rm u}$ & 1264.583 & 1264.819 &1265.121 & 1264.868 & 1264.54\\\\\n 0 & 0 & 0 & 1 & 0 & 1 & $B_{\\rm u}$ & 1504.872 & 1505.977 &1506.283 & 1505.634 & \\\\\n 0 & 0 & 0 & 2 & 0 & 1 & $B_{\\rm u}$ & 1853.634 & 1853.949 &1854.424 & 1855.305 & \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & $A_{\\rm u}$ & 11.437 & 11.014 & 10.997 & 11.289 & 11.28 \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & $A_{\\rm u}$ & 370.893 & 371.247 & 371.203 & 371.478& 371.32 \\\\\n 0 & 0 & 0 & 2 & 0 & 0 & $A_{\\rm u}$ & 776.122 & 776.465 & 776.320 & 777.336 & 776.93 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 & $A_{\\rm u}$ & 877.934 & 877.094 & 877.200 & 877.303 & \\\\\n 0 & 0 & 0 & 0 & 0 & 1 & $B_{\\rm g}$ & 1285.121 & 1284.889 &1285.249 & 1285.457& \\\\\n 0 & 0 & 0 & 1 & 0 & 1 & $B_{\\rm g}$ & 1648.367 & 1648.553 &1649.012 & 1649.485& \\\\\n 0 & 0 & 0 & 2 & 0 & 1 & $B_{\\rm g}$ & 2072.404 & 2072.384 &2072.949 & 2074.231& \\\\\n\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe vibrational ground state (left hand column) and the (000 100 $A_{\\rm g}$) state - (right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from Ref.~\\protect\\onlinecite{perin3}.}\n\\label{tab:J1}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n1 & 0 & 1 & 1.71154 & 1.71152 & 0.00002 & 256.255 & 256.255 & 0.000 \\\\\n1 & 1 & 1 & 10.90677 & 10.9068 & 0.0000 & 265.427 & 265.427 & 0.000 \\\\\n1 & 1 & 0 & 10.9426 & 10.9426 & 0.0000 & 265.474 & 265.475 & 0.001 \\\\\n & & & & & & & & \\\\\n3 & 0 & 3 & 10.2683 & 10.2682 & 0.0001 & 264.777 & 264.777 & 0.000 \\\\\n3 & 1 & 3 & 19.374 & 19.374 & 0.000 & 273.830 & 273.831 & -0.001 \\\\\n3 & 1 & 2 & 19.589 & 19.589 & 0.000 & 274.119 & 274.117 & 0.002 \\\\\n3 & 2 & 2 & 47.115 & 47.115 & 0.000 & 301.555 & 301.556 & 0.001 \\\\\n3 & 2 & 1 & 47.115 & 47.115 & 0.000 & 301.556 & 301.557 & 0.001 \\\\\n3 & 3 & 1 & 93.155 & 93.155 & 0.000 & 347.509 & 347.512 & 0.003 \\\\\n3 & 3 & 0 & 93.155 & 93.155 & 0.000 & 347.509 & 347.512 & 0.003 \\\\\n & & & & & & & & \\\\\n5 & 0 & 5 & 25.6667 & 25.6664 & 0.0003 & 280.113 & 280.112 & 0.001 \\\\\n5 & 1 & 5 & 34.613 & 34.613 & 0.000 & 288.954 & 288.954 & 0.000 \\\\\n5 & 1 & 4 & 35.151 & 35.150 & 0.001 & 289.671 & 289.671 & 0.000 \\\\\n5 & 2 & 4 & 62.513 & 62.513 & 0.000 & 316.893 & 316.893 & 0.000 \\\\\n5 & 2 & 3 & 62.517 & 62.517 & 0.000 & 316.899 & 316.899 & 0.000 \\\\\n5 & 3 & 3 & 108.551 & 108.551 & 0.000 & 362.845 & 362.847 & 0.002 \\\\\n5 & 3 & 2 & 108.551 & 108.551 & 0.000 & 362.845 & 362.847 & 0.002 \\\\\n5 & 4 & 2 & 172.968 & 172.968 & 0.000 & 427.143 & 427.146 & 0.003 \\\\\n5 & 4 & 1 & 172.968 & 172.968 & 0.000 & 427.143 & 427.146 & 0.003 \\\\\n5 & 5 & 1 & 255.733 & 255.733 & 0.000 & 509.757 & 509.764 & 0.007 \\\\\n5 & 5 & 0 & 255.733 & 255.733 & 0.000 & 509.757 & 509.764 & 0.007 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe (000 200 $A_{\\rm g}$) (left hand column) and the (000 300 $A_{\\rm g}$) state - right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n1 & 0 & 1 & 571.448 & 571.449 & -0.001 & 1002.584 & 1002.584 & 0.000 \\\\\n1 & 1 & 1 & 580.550 & 580.549 & 0.001 & 1011.664 & 1011.614 & 0.050 \\\\\n1 & 1 & 0 & 580.577 & 580.576 & 0.001 & 1011.678 & 1011.628 & 0.050 \\\\\n & & & & & & & & \\\\\n3 & 0 & 3 & 579.976 & 579.975 & 0.001 & 1011.098 & 1011.097 & 0.001 \\\\\n3 & 1 & 3 & 589.009 & 589.007 & 0.002 & 1020.143 & 1020.093 & 0.050 \\\\\n3 & 1 & 2 & 589.174 & 589.172 & 0.002 & 1020.225 & 1020.176 & 0.049 \\\\\n3 & 2 & 2 & 616.427 & 616.427 & 0.000 & 1047.246 & 1047.244 & 0.002 \\\\\n3 & 2 & 1 & 616.427 & 616.427 & 0.000 & 1047.246 & 1047.244 & 0.002 \\\\\n3 & 3 & 1 & 661.974 & 661.975 & 0.001 & 1092.462 & 1092.411 & 0.051 \\\\\n3 & 3 & 0 & 661.974 & 661.975 & 0.001 & 1092.462 & 1092.411 & 0.051 \\\\\n & & & & & & & & \\\\\n5 & 0 & 5 & 595.324 & 595.324 & 0.000 & 1026.421 & 1026.021 & 0.008 \\\\\n5 & 1 & 5 & 604.235 & 604.232 & 0.003 & 1035.404 & 1035.354 & 0.050 \\\\\n5 & 1 & 4 & 604.645 & 604.644 & 0.001 & 1035.610 & 1035.561 & 0.049 \\\\\n5 & 2 & 4 & 631.773 & 631.773 & 0.000 & 1062.566 & 1062.565 & 0.001 \\\\\n5 & 2 & 3 & 631.775 & 631.775 & 0.000 & 1062.566 & 1062.565 & 0.001 \\\\\n5 & 3 & 3 & 677.317 & 677.315 & 0.002 & 1107.779 & 1107.729 & 0.050 \\\\\n5 & 3 & 2 & 677.317 & 677.315 & 0.002 & 1107.779 & 1107.729 & 0.050 \\\\\n5 & 4 & 2 & 741.046 & 741.044 & 0.002 & 1170.935 & 1170.930 & 0.005 \\\\\n5 & 4 & 1 & 741.046 & 741.044 & 0.002 & 1170.935 & 1170.930 & 0.005 \\\\\n5 & 5 & 1 & 822.934 & 822.930 & 0.004 & 1252.191 & 1252.140 & 0.051 \\\\\n5 & 5 & 0 & 822.934 & 822.930 & 0.004 & 1252.191 & 1252.140 & 0.051 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe (001 000 $A_{\\rm g}$) (left hand column) and the (000 000 $A_{\\rm u}$)\n state - right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n1 & 0 & 1 & 867.628 & 867.628 & 0.000 & 13.150 & 13.149 & 0.001 \\\\\n1 & 1 & 0 & 876.815 & 876.816 & 0.001 & 22.337 & 22.337 & 0.000 \\\\\n1 & 1 & 0 & 876.851 & 876.850 & 0.001 & 22.369 & 21.368 & 0.001 \\\\\n & & & & & & & & \\\\\n3 & 0 & 3 & 876.074 & 875.073 & 0.001 & 21.712 & 21.711 & 0.001 \\\\\n3 & 1 & 3 & 885.171 & 884.171 & 0.000 & 30.821 & 30.820 & 0.001 \\\\\n3 & 1 & 2 & 885.386 & 885.387 & -0.001 & 31.009 & 31.009 & 0.000 \\\\\n3 & 2 & 2 & 912.886 & 912.885 & 0.001 & 58.518 & 58.518 & 0.000 \\\\\n3 & 2 & 1 & 912.887 & 912.887 & 0.000 & 58.518 & 58.518 & 0.000 \\\\\n3 & 3 & 1 & 958.884 & 958.887 & -0.003 & 104.508 & 104.508 & 0.000 \\\\\n3 & 3 & 0 & 958.884 & 958.887 & -0.003 & 104.508 & 104.508 & 0.000 \\\\\n & & & & & & & & \\\\\n5 & 0 & 5 & 891.273 & 891.2712 & 0.002 & 37.121 & 37.120 & 0.001 \\\\\n5 & 1 & 5 & 900.210 & 900.209 & 0.001 & 46.089 & 46.088 & 0.001 \\\\\n5 & 1 & 4 & 900.749 & 900.748 & 0.001 & 46.561 & 46.560 & 0.001 \\\\\n5 & 2 & 4 & 928.085 & 928.085 & 0.000 & 73.926 & 73.926 & 0.000 \\\\\n5 & 2 & 3 & 928.089 & 928.088 & 0.001 & 73.929 & 73.928 & 0.001 \\\\\n5 & 3 & 3 & 974.080 & 974.082 & -0.002 & 119.913 & 119.913 & 0.000 \\\\\n5 & 3 & 2 & 974.080 & 974.082 & -0.002 & 119.913 & 119.913 & 0.000 \\\\\n5 & 4 & 2 & 1038.438 & 1038.440 & -0.002 & 184.260 & 184.261 & -0.001 \\\\\n5 & 5 & 1 & 1121.127 & 1121.133 & -0.006 & 266.936 & 266.938 & -0.002 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for\nthe (000 100 $A_{\\rm u}$) (left hand column) and the (000 200 $A_{\\rm u}$)\nstate - (right hand column)\n with $J$ = 1, 3 and 5. Observed energy levels taken from the Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n1 & 0 & 0 & 372.601 & 372.602 & 0.001 & 777.826 & 777.827 & -0.001 \\\\\n1 & 1 & 1 & 381.733 & 381.733 & 0.000 & 786.881 & 786.891 & -0.010 \\\\\n1 & 1 & 1 & 381.764 & 381.764 & 0.000 & 786.901 & 786.911 & -0.010 \\\\\n & & & & & & & & \\\\\n3 & 0 & 3 & 381.140 & 381.139 & 0.001 & 786.350 & 786.350 & 0.000 \\\\\n3 & 1 & 3 & 390.195 & 390.195 & 0.000 & 795.355 & 795.365 & -0.010 \\\\\n3 & 1 & 2 & 390.378 & 390.376 & 0.002 & 795.472 & 795.482 & -0.010 \\\\\n3 & 2 & 2 & 417.723 & 417.722 & 0.001 & 822.639 & 822.640 & -0.001 \\\\\n3 & 2 & 1 & 417.723 & 417.722 & 0.001 & 822.639 & 822.640 & -0.001 \\\\\n3 & 3 & 1 & 463.434 & 463.434 & 0.000 & 867.976 & 867.986 & -0.010 \\\\\n3 & 3 & 0 & 463.434 & 463.434 & 0.000 & 867.976 & 867.986 & -0.010 \\\\\n & & & & & & & & \\\\\n5 & 0 & 5 & 396.505 & 396.504 & 0.001 & 801.689 & 801.689 & 0.000 \\\\\n5 & 1 & 5 & 405.426 & 405.423 & 0.003 & 810.607 & 810.617 & -0.010 \\\\\n5 & 1 & 4 & 405.881 & 405.879 & 0.002 & 810.900 & 810.909 & -0.009 \\\\\n5 & 2 & 4 & 433.088 & 433.088 & 0.000 & 837.977 & 837.977 & 0.000 \\\\\n5 & 2 & 3 & 433.090 & 433.089 & 0.001 & 837.977 & 837.977 & 0.000 \\\\\n5 & 3 & 3 & 478.796 & 478.794 & 0.002 & 883.310 & 883.320 & -0.010 \\\\\n5 & 3 & 2 & 478.796 & 478.794 & 0.002 & 883.310 & 883.320 & -0.010 \\\\\n5 & 4 & 2 & 542.756 & 542.756 & 0.000 & 946.769 & 946.770 & 0.001 \\\\\n5 & 5 & 1 & 624.938 & 624.937 & 0.001 & 1028.289 & 1028.299 & -0.010 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for $J=30$. Results are for\nthe ground vibrational state (left hand column) and the (000 100 $A_{\\rm g}$)\nstate - (right hand column). Observed energy levels taken from the Ref.~\\protect\\onlinecite{perin3}.}\n\\label{tab:J30}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n\\hline\n30 & 0 & 30 & 789.577 & 789.581 & -0.004 & 1038.983 & 1038.925 & 0.058 \\\\\n30 & 1 & 30 & 793.053 & 793.065 & -0.012 & 1041.540 & 1041.481 & 0.039 \\\\\n30 & 1 & 29 & 809.594 & 809.565 & 0.029 & 1063.053 & 1063.007 & 0.046 \\\\\n30 & 2 & 29 & 829.292 & 829.282 & 0.010 & 1080.396 & 1080.345 & 0.051 \\\\\n30 & 2 & 28 & 832.547 & 832.521 & 0.026 & 1086.079 & 1086.036 & 0.043 \\\\\n30 & 3 & 28 & 876.030 & 876.017 & 0.013 & 1127.862 & 1127.815 & 0.047 \\\\\n30 & 3 & 27 & 876.191 & 876.174 & 0.017 & 1128.304 & 1128.262 & 0.042 \\\\\n30 & 4 & 27 & 940.027 & 940.015 & 0.012 & 1191.838 & 1191.794 & 0.044 \\\\\n30 & 4 & 26 & 940.029 & 940.013 & 0.016 & 1191.850 & 1191.807 & 0.043 \\\\\n30 & 5 & 26 & 1022.323 & 1022.311 & 0.012 & 1274.332 & 1274.293 & 0.039 \\\\\n30 & 5 & 25 & 1022.324 & 1022.312 & 0.012 & 1274.332 & 1274.293 & 0.039 \\\\\n30 & 6 & 25 & 1122.848 & 1122.837 & 0.011 & 1371.047 & 1371.024 & 0.023 \\\\\n30 & 6 & 24 & 1122.849 & 1122.838 & 0.011 & 1371.047 & 1371.024 & 0.023 \\\\\n30 & 7 & 24 & 1241.304 & 1241.294 & 0.010 & 1491.957 & 1491.933 & 0.024 \\\\\n30 & 7 & 23 & 1241.304 & 1241.294 & 0.010 & 1491.957 & 1491.933 & 0.024 \\\\\n30 & 8 & 23 & 1381.938 & 1381.916 & 0.022 & 1628.749 & 1628.731 & 0.018 \\\\\n30 & 9 & 21 & 1534.588 & 1534.578 & 0.010 & 1782.735 & 1782.739 & -0.004 \\\\\n30 & 10 & 21 & 1707.358 & 1707.349 & 0.009 & 1958.577 & 1958.558 & 0.019 \\\\\n30 & 11 & 19 & 1898.169 & 1898.159 & 0.010 & 2148.182 & 2148.186 & -0.004 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Calculated and observed energy levels, in \\cm, for $J=30$ levels of the (000 000 $A_{\\rm u}$)\nvibrational state. Observed energy levels taken from Ref.~\\protect\\onlinecite{perin3}.}\n\\begin{tabular}{cccrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c \\\\\n\\hline\n30 & 0 & 30 & 802.349 & 802.340 & 0.009 \\\\\n30 & 1 & 30 & 806.280 & 806.276 & 0.004 \\\\\n30 & 1 & 29 & 820.768 & 820.746 & 0.022 \\\\\n30 & 2 & 29 & 841.344 & 841.330 & 0.013 \\\\\n30 & 2 & 28 & 843.911 & 843.892 & 0.019 \\\\\n30 & 3 & 28 & 887.854 & 887.839 & 0.015 \\\\\n30 & 3 & 27 & 887.964 & 887.949 & 0.015 \\\\\n30 & 4 & 27 & 951.826 & 951.815 & 0.011 \\\\\n30 & 4 & 26 & 951.828 & 951.817 & 0.011 \\\\\n30 & 5 & 26 & 1034.096 & 1034.085 & 0.011 \\\\\n30 & 5 & 25 & 1034.096 & 1034.085 & 0.011 \\\\\n30 & 5 & 25 & 1134.609 & 1134.599 & 0.010 \\\\\n30 & 5 & 24 & 1253.291 & 1253.284 & 0.007 \\\\\n30 & 5 & 24 & 1390.060 & 1390.055 & 0.005 \\\\\n30 & 5 & 23 & 1544.803 & 1544.803 & 0.000 \\\\\n30 & 5 & 23 & 1717.164 & 1717.167 & -0.003 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{ Variationally calculated and observed or predicted\nusing effective Hamiltonian) energy levels, in \\cm, for $J=35$. Observed energy levels taken from the Ref.~\\protect\\onlinecite{perin3}.}\n\\label{tab:J35}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n\\hline\n35 & 0 & 35 & 1067.027 & 1067.037 & -0.010 & 1314.205 & 1314.124 & 0.081 \\\\\n35 & 1 & 35 & 1069.466 & 1069.484 & -0.018 & 1315.869 & 1315.787 & 0.082 \\\\\n35 & 1 & 34 & 1091.715 & 1091.680 & 0.035 & 1344.395 & 1344.338 & 0.057 \\\\\n35 & 2 & 34 & 1108.893 & 1108.882 & 0.011 & 1358.709 & 1358.637 & 0.052 \\\\\n35 & 2 & 33 & 1114.564 & 1114.523 & 0.041 & 1368.256 & 1368.203 & 0.053 \\\\\n35 & 3 & 33 & 1156.257 & 1156.241 & 0.016 & 1407.374 & 1407.309 & 0.065 \\\\\n35 & 3 & 32 & 1156.655 & 1156.628 & 0.027 & 1408.434 & 1408.382 & 0.052 \\\\\n35 & 4 & 32 & 1220.062 & 1220.044 & 0.018 & 1471.314 & 1471.252 & 0.062 \\\\\n35 & 4 & 31 & 1220.070 & 1220.051 & 0.019 & 1471.355 & 1471.296 & 0.059 \\\\\n35 & 5 & 31 & 1302.102 & 1302.088 & 0.014 & 1553.875 & 1553.821 & 0.054 \\\\\n35 & 5 & 30 & 1302.114 & 1302.098 & 0.016 & 1553.876 & 1555.822 & 0.054 \\\\\n35 & 6 & 30 & 1402.366 & 1402.353 & 0.013 & 1648.318 & 1648.288 & 0.030 \\\\\n35 & 6 & 29 & 1402.369 & 1402.350 & 0.019 & 1648.318 & 1648.288 & 0.030 \\\\\n35 & 7 & 29 & 1520.294 & 1520.282 & 0.012 & 1769.879 & 1769.844 & 0.035 \\\\\n35 & 8 & 28 & 1662.739 & 1662.707 & 0.032 & 1906.483 & 1906.459 & 0.024 \\\\\n35 & 9 & 27 & 1814.222 & 1814.207 & 0.015 & & & \\\\\n35 & 10 & 26 & 1986.565 & 1986.552 & 0.013 & & & \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{ Variationally calculated and observed or predicted using effective Hamiltonian) energy levels, in \\cm, for $J=35$}\n\\begin{tabular}{cccrrlrrl}\n\\hline\n\\hline\n$J$ & $K_a$ & $K_c$ & Obs & Calc & o-c & Obs & Calc & o-c \\\\\n\\hline\n35 & 0 & 35 & 1933.163 & 1932.118 & 0.045 & 1080.587 & 1080.574 & 0.013 \\\\\n35 & 1 & 35 & 1936.042 & 1935.996 & 0.046 & 1083.472 & 1083.465 & 0.007 \\\\\n35 & 1 & 34 & 1955.496 & 1955.415 & 0.081 u & 1102.961 & 1102.929 & 0.032 \\\\\n35 & 2 & 34 & 1973.810 & 1973.743 & 0.067 u & 1121.270 & 1121.247 & 0.023 \\\\\n35 & 2 & 33 & 1978.296 & 1978.228 & 0.068 u & 1125.774 & 1125.743 & 0.031 \\\\\n35 & 3 & 33 & 2020.807 & 2020.749 & 0.058 u & 1168.268 & 1168.245 & 0.023 \\\\\n35 & 3 & 32 & 2021.078 & 2021.020 & 0.058 u & 1168.540 & 1168.517 & 0.023 \\\\\n35 & 4 & 32 & 2084.733 & 2084.691 & 0.042 u & 1232.076 & 1232.060 & 0.016 \\\\\n35 & 4 & 31 & 2084.738 & 2084.768 & 0.042 u & 1232.081 & 1232.055 & 0.026 \\\\\n35 & 5 & 31 & 2165.503 & 2165.420 & 0.083 u & 1314.156 & 1314.138 & 0.018 \\\\\n35 & 5 & 30 & 2165.503 & 2165.420 & 0.083 u & 1314.156 & 1314.138 & 0.018 \\\\\n & & & & & & 1414.475 & 1414.459 & 0.016 \\\\\n & & & & & & 1532.940 & 1532.928 & 0.012 \\\\\n & & & & & & 1669.447 & 1669.438 & 0.009 \\\\\n & & & & & & 1823.841 & 1823.839 & 0.002 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\bibliographystyle{rsc}\n\n\\providecommand*{\\mcitethebibliography}{\\thebibliography}\n\\csname @ifundefined\\endcsname{endmcitethebibliography}\n{\\let\\endmcitethebibliography\\endthebibliography}{}\n\\begin{mcitethebibliography}{69}\n\\providecommand*{\\natexlab}[1]{#1}\n\\providecommand*{\\mciteSetBstSublistMode}[1]{}\n\\providecommand*{\\mciteSetBstMaxWidthForm}[2]{}\n\\providecommand*{\\mciteBstWouldAddEndPuncttrue}\n {\\def\\EndOfBibitem{\\unskip.}}\n\\providecommand*{\\mciteBstWouldAddEndPunctfalse}\n {\\let\\EndOfBibitem\\relax}\n\\providecommand*{\\mciteSetBstMidEndSepPunct}[3]{}\n\\providecommand*{\\mciteSetBstSublistLabelBeginEnd}[3]{}\n\\providecommand*{\\EndOfBibitem}{}\n\\mciteSetBstSublistMode{f}\n\\mciteSetBstMaxWidthForm{subitem}\n{(\\emph{\\alph{mcitesubitemcount}})}\n\\mciteSetBstSublistLabelBeginEnd{\\mcitemaxwidthsubitemform\\space}\n{\\relax}{\\relax}\n\n\\bibitem[Davis(1974)]{74Davis.H2O2}\nD.~Davis, \\emph{Can. 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Recently there has been another resurgence of interest in the community to revisit the Witten diagrams.\n The renewed interest is partly due to the curious appearance of these diagrams in the Mellin bootstrap method \\cite{Gopakumar:2016wkt,Gopakumar:2016cpb,Gopakumar:2018xqi}, where the tree-level exchange Witten diagrams (modulo certain ambiguities in adding contact terms) are used as an expansion basis for conformal correlators. Relatedly, the usual conformal blocks admit a natural AdS bulk description in terms of a variation of the exchange Witten diagrams. They are the so-called geodesic Witten diagrams where the integration regions of the cubic vertex points are restricted to the AdS geodesics connecting the boundary operators \\cite{Hijano:2015zsa}.\\footnote{See also \\cite{Rastelli:2017ecj,Goncalves:2018fwx} for generalization of geodesic Witten diagrams to CFTs with conformal boundaries and defects.} Moreover, one is also motivated to revisit these diagrams because of the remarkable simplicity recently discovered in the holographic one-half BPS four-point functions \\cite{Rastelli:2016nze,Alday:2017xua,Aprile:2017bgs,Aprile:2017xsp,Rastelli:2017udc,Alday:2017vkk,Aprile:2017qoy,Rastelli:2017ymc,Zhou:2017zaw,Zhou:2018ofp,Aprile:2018efk,Caron-Huot:2018kta,Alday:2018pdi}.\\footnote{Modern methods have been recently invented to efficiently compute holographic correlators at the tree level. See \\cite{Rastelli:2016nze,Rastelli:2017udc,Rastelli:2017ymc,Zhou:2017zaw,Zhou:2018ofp} for bootstrap-like methods of computing holographic correlators which does not require the detailed knowledge of the effective Lagrangian. See also \\cite{Arutyunov:2017dti,Arutyunov:2018neq,Arutyunov:2018tvn} for an improved version of the original algorithm and \\cite{Caron-Huot:2018kta} for a method based on the inversion formula \\cite{Caron-Huot:2017vep} and crossing symmetry.} In particular an interesting $SO(10,2)$ symmetry \\cite{Caron-Huot:2018kta} was shown to exist in the general formula for one-half BPS four-point functions from tree-level IIB supergravity in $AdS_5\\times S^5$ \\cite{Rastelli:2016nze,Rastelli:2017udc}, unifiying the correlators of all Kaluza-Klein modes into a single ten-dimensional object. The simplicity in these results requires remarkable conspiracy of individual Witten diagrams, and some of its aspects still remain to be better understood. Finally, one is led to study Witten diagrams from the reconstruction of AdS physics using CFT principles \\cite{Heemskerk:2009pn}. Witten diagrams emerge as solutions to the crossing equation, at tree level \\cite{Heemskerk:2009pn,Alday:2017gde,Li:2017lmh} as well as at loop level \\cite{Aharony:2016dwx}. \n \n However, some basic properties of Witten diagrams still remain to be better understood, even at tree level. In particular, the following seemingly simple problem still appears to lack a satisfactory solution: how do we perform the conformal block decomposition of a tree-level exchange Witten diagram in the crossed channel? In the crossed channel an exchange Witten diagram is known to decompose into two towers of double-trace conformal blocks.\\footnote{Here we assume that the external conformal dimensions are generic such that the spectra of the two towers of double-trace operators do not overlap. In the degenerate case, both conformal blocks and their derivative with respect to the conformal dimension will appear in the expansion. We will have more discussions on this point later in the paper.} However obtaining the crossed channel OPE coefficients turns out to be very non-trivial, and no method is currently available to extract efficiently {\\it all} the coefficients. This should be contrasted with the decomposition in the direct channel, where it contains a single-trace conformal block and infinitely many double-trace blocks with bounded spins. The problem in the direct channel is much easier and can be solved using a variety of methods. For example one can obtain all the decomposition coefficients in closed forms using the split representation of propagators \\cite{Costa:2014kfa}, or the geodesic Witten diagram techniques \\cite{Hijano:2015zsa}.\\footnote{In Section \\ref{Secdirectchan} we will offer another approach to obtain these coefficients from studying the contact diagrams related to the exchange diagram.} \n \n \n\n \n A number of recent papers have appeared that revisit this problem, and the methods have both advantages and disadvantages. In \\cite{Liu:2018jhs,Cardona:2018dov} methods based on the Lorentizian inversion formula \\cite{Caron-Huot:2017vep} (see also \\cite{Simmons-Duffin:2017nub,Kravchuk:2018htv}) are introduced, which allow one to extract the crossed channel CFT data for operators with spins greater than the spin of the exchanged single-trace operator. However the inversion formula is not valid for lower spins due to the Regge behavior of the Witten diagrams, and one therefore cannot use this method to probe the rest of the operators. This difficulty is absent in Mellin space \\cite{Mack:2009mi,Penedones:2010ue}. The OPE coefficients of the double-trace operators with the leading conformal twist can be obtained by taking the residue of the Mellin amplitude at the leading double-trace pole, and then projecting the residue into continuous Hahn polynomials of different spins \\cite{Costa:2014kfa,Gopakumar:2016cpb,Sleight:2018epi,Sleight:2018ryu,Gopakumar:2018xqi} -- analogues to projecting flat space amplitudes into partial waves using Gegenbaur polynomials. But in this approach one encounters a different difficulty in obtaining the OPE coefficients of double-trace operators with sub-leading twists. The residue of the Mellin amplitude at a sub-leading double-trace pole receives contributions from both the double-trace primary operators, as well as the conformal descendants of the double-trace operators whose twists are smaller. Therefore there is a mixing of contributions between the two. The degeneracy problem must be first solved in order to extract the OPE coefficients. One method of disentangling the contributions was suggested in \\cite{Sleight:2018epi,Sleight:2018ryu}. \nIn this method one first acts on the correlator with a quartic differential operator and then repeats the Mellin space analysis of performing the projections on the residue. The differential operator can be chosen such that its kernel contains the conformal blocks of the leading double-trace twist \\cite{Alday:2016njk}, the sub-leading double-trace operators therefore become leading in the new correlator. Unfortunately, the complexity of this algorithm quickly grows upon increasing the twists of the double-trace operators. Applying this method to low orders also reveals few general patterns.\n\nIn this paper, we will attack the problem from a different angle. The main result of our analysis is a set of simple constraining linear relations satisfied by the crossed channel decomposition coefficients. These linear relations will lead us to a recursive algorithm for solving the coefficients. In our analysis we emphasize the pivotal role played by the contact Witten diagrams, and highlight the importance of an ``equation of motion'' operator which relates an exchange Witten diagram to a sum of contact Witten diagrams. More precisely, the equation of motion operator is given by the quadratic conformal Casimir operator in the exchange channel with a constant shift, and is closely related to the equation of motion for the bulk-to-bulk propagator. The direct consequence of this relation between these two types of diagrams is that the direct channel decomposition coefficients of an exchange Witten diagram are completely fixed in terms of the coefficients of the related contact Witten diagrams. This relation has important implications in the crossed channel too. Using properties of conformal blocks \\cite{Dolan:2011dv}\\footnote{See also \\cite{Karateev:2017jgd}.}, we find that the equation of motion operator admits simple actions on conformal blocks. In one dimension, the action of this operator on a conformal block with dimension $\\Delta$ produces three conformal blocks with new dimensions $\\Delta-1$, $\\Delta$ and $\\Delta+1$. In higher dimensions, the three-term relation becomes a five-term one. The action of the operator on a conformal block with dimension $\\Delta$ and spin $\\ell$ contains the original conformal block, as well as four other conformal blocks with shifted quantum numbers $(\\Delta\\mp1,\\ell\\pm1)$, $(\\Delta\\pm1,\\ell\\mp1)$. Moreover, the coefficients of the three-term and five-term relations vanish for conformal blocks of unphysical double-trace operators\\footnote{These are the operators with conformal dimensions $\\Delta=\\Delta_1+\\Delta_2+\\ell-1, \\Delta_3+\\Delta_4+\\ell-1$ or negative spins $\\ell=-1$ for $d>1$, and $\\Delta=\\Delta_1+\\Delta_2-1,\\Delta_3+\\Delta_4-1$ for $d=1$.}. These zeros of the coefficients guarantee that when we restrict $(\\Delta,\\ell)$ to be those of the double-trace operators, the spectrum is preserved after applying these relations. The simple action of the equation of motion operator on double-trace conformal blocks makes it possible to formulate a recursive algorithm for solving the crossed channel decomposition coefficients. Thanks to the equation of motion identity, these crossed channel coefficients satisfy simple linear equations with the decomposition coefficients of the contact Witten diagrams as inhomogeneous terms. We can solve the linear equations recursively, in terms of certain seed decomposition coefficients. In one dimension, the seed coefficients are just the OPE coefficients of the double-trace operators with the lowest conformal dimension. In higher dimensions, the seed coefficients are the OPE coefficients of the leading twist double-trace operators. Therefore the recursion relations give us a very efficient way to obtain OPE coefficients of sub-leading double-trace operators. \n\nThe decomposition of exchange Witten diagrams in the crossed channel is also closely related to the $6j$ symbol (or the crossing kernel) of the conformal group. In the crossed channel, a conformal partial wave is decomposed into infinitely many double-trace conformal blocks. The various decomposition coefficients of the double-trace operators can be viewed as the residues of the $6j$ symbol \\cite{Liu:2018jhs}. Because a conformal partial wave can be identified with the difference of two exchange Witten diagrams with opposite quantizations \\cite{Penedones:2007ns,Costa:2014kfa,Giombi:2018vtc}, our analysis of the Witten diagrams extends easily to conformal partial waves. \n\n\n\n\n\nThe paper is organized as follows. We start by defining the Witten diagrams in Section \\ref{SecWD} and introducing the equation of motion operator in \\ref{SecExtoCon}. In Section \\ref{CPW} we review some basic facts of conformal partial waves. In Section \\ref{SecCBdecompContact} we discuss the conformal block decomposition of contact Witten diagrams. We show that the decomposition of a generic contact diagram can always be recursively reduced to the simplest contact diagram with zero derivatives in the quartic vertex. In Section \\ref{Secdirectchan} we show how the direct channel decomposition of exchange Witten diagrams can be fixed by the decomposition of the relevant contact Witten diagrams. In Section \\ref{SecRecurcross}, we discuss the crossed channel decomposition of exchange Witten diagrams and conformal partial waves. We first outline the strategy in Section \\ref{SecStrat}. We discuss the simpler problem in $\\mathrm{CFT}_1$ in Section \\ref{1d}, and then extend the story to $\\mathrm{CFT}_d$ in Section \\ref{higherd}. We end with a brief discussion in Section \\ref{SecDiscuss}. Further technical details are relegated to the three appendices. In Appendix \\ref{appcontact} we make further comments on the contact Witten diagrams. In Appendix \\ref{appeqweight} we discuss the special case where the external conformal dimensions are degenerate. In Appendix \\ref{appseed} we discuss how to compute the seed coefficients for $AdS_2$ exchange Witten diagrams. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Witten Diagrams and Conformal Partial Waves}\n\\subsection{Witten Diagrams}\\label{SecWD}\nIn this paper we study tree-level four-point Witten diagrams, {\\it i.e.}, contact Witten diagrams and exchange Witten diagrams. We focus on scalar Witten diagrams where the external operators have zero spins and conformal dimensions $\\Delta_i$, $i=1,2,3,4$. This restricts the operator in the internal line of the exchange Witten diagram to be in the rank-$\\ell_E$ symmetric-traceless representation under the Lorentz group. We will denote the conformal dimension of the exchange operator by $\\Delta_E$. After making a choice for the cubic and quartic vertices, the Witten diagrams are built from the scalar bulk-to-boundary propagators $G_{B\\partial}^{\\Delta_i}(z,\\vec{x}_i)$, and the spin-$\\ell_E$ bulk-to-bulk propagator $\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)$. The scalar bulk-to-boundary propagator is explicitly given by \\begin{equation}\nG_{B\\partial}^{\\Delta_i}(z,\\vec{x}_i)=\\left(\\frac{z_0}{z_0^2+(\\vec{z}-\\vec{x}_i)^2}\\right)^{\\Delta_i}\n\\end{equation}\nwhere the dimension $\\Delta_i$ is associated to the scalar field mass in $AdS_{d+1}$ via $M_i^2=\\Delta_i(\\Delta_i-d)$. The spin-$\\ell_E$ bulk-to-bulk propagators are defined to satisfy the equation of motion and have vanishing divergence\n\\begin{equation}\n(\\bigtriangledown_1^2-M_E^2)\\,\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)=-g^{\\mu_1\\{\\nu_1}\\ldots g^{|\\mu_{\\ell_E}|\\nu_{\\ell_E}\\}}\\delta(z_1,z_2)+\\ldots\\;,\\label{EOMofBtoBprop}\n\\end{equation}\n\\begin{equation}\n\\bigtriangledown_1^{\\mu_1}\\,\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)=0+\\ldots\\;,\\label{divergenceless}\n\\end{equation}\nup to local source terms denoted by $\\ldots$. These terms introduce ambiguities to the exchange Witten diagrams\\footnote{These terms will only change the exchange Witten diagram by contact Witten diagrams \\cite{Costa:2014kfa}.}, but such ambiguities are not important for the propagating degrees of freedom. The squared mass of the bulk field is given by\n\\begin{equation}\nM_E^2=\\Delta_E(\\Delta_E-d)-\\ell_E\\;.\n\\end{equation}\nAn explicit example of the bulk-to-bulk propagator is given by that of a scalar field\n\\begin{equation}\\small\n\\Pi^{\\Delta_E}(z_1,z_2)=\\frac{\\Gamma(\\Delta_E)}{2\\pi^{\\frac{d}{2}}\\Gamma(\\Delta_E-\\frac{d}{2}+1)}u^{-\\Delta}{}_2F_1\\left(\\Delta_E,\\frac{2\\Delta_E-d+1}{2},2\\Delta_E-d+1,-\\frac{4}{u}\\right)\n\\end{equation}\nwhere\n\\begin{equation}\nu=\\frac{(z_1-z_2)^2}{z_{10}z_{20}}\\;.\n\\end{equation}\n\nA generic contact Witten diagram is built in terms of the bulk-to-boundary propagators only\n\\begin{equation}\\label{Wcontact}\nW^{contact}=\\int \\frac{d^{d+1}z}{z_0^{d+1}}\\prod_{i=1}^4 (\\bigtriangledown^{\\mu})^{j_i}G^{\\Delta_i}_{B\\partial}(z,x_i)\\;.\n\\end{equation}\nIn this formula, $(\\bigtriangledown^{\\mu})^{j_i}$ is schematic for the product of covariant derivatives $\\bigtriangledown^{\\mu_1}\\ldots \\bigtriangledown^{\\mu_{j_i}}$, and the indices are appropriately contracted on the RHS of (\\ref{Wcontact}). Such a contact diagram arises from a quartic AdS contact vertex\n\\begin{equation}\n(\\bigtriangledown^{\\mu})^{j_1}\\phi_1(\\bigtriangledown^{\\mu})^{j_2}\\phi_2(\\bigtriangledown^{\\mu})^{j_3}\\phi_3(\\bigtriangledown^{\\mu})^{j_4}\\phi_4\n\\end{equation}\nwhich contains a total number of $j_1+j_2+j_3+j_4$ derivatives. When $j_1=j_2=j_3=j_4=0$, we have the simplest contact diagram which is denoted in the literature as a $D$-function\n\\begin{equation}\\label{Dfunction}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int \\frac{d^{d+1}z}{z_0^{d+1}}\\prod_{i=1}^4 G^{\\Delta_i}_{B\\partial}(z,x_i)\\;.\n\\end{equation} \n\nWe also define a spin-$\\ell_E$ exchange Witten diagram (in the s-channel) as\n\\begin{equation}\\label{sWitten}\n\\begin{split}\nW^{s,\\, exchange}_{\\Delta_E,\\ell_E}={}&\\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}}\\frac{d^{d+1}z_2}{z_{20}^{d+1}}G^{\\Delta_1}_{B\\partial}(z_1,x_1)((\\bigtriangledown^{\\mu})^{\\ell_E}G^{\\Delta_2}_{B\\partial}(z_1,x_2))\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)\\\\\n{}&\\times G^{\\Delta_3}_{B\\partial}(z_2,x_3) ((\\bigtriangledown^{\\nu})^{\\ell_E}G^{\\Delta_4}_{B\\partial}(z_2,x_4))\n\\end{split}\n\\end{equation}\nwhere we assumed the cubic couplings are \n\\begin{equation}\n\\phi_1\\bigtriangledown^{\\mu_1}\\ldots\\bigtriangledown^{\\mu_{\\ell_E}}\\phi_2 h_{\\mu_1\\ldots\\mu_{\\ell_E}}\\;,\\quad\\text{and}\\quad \\phi_1\\bigtriangledown^{\\mu_1}\\ldots\\bigtriangledown^{\\mu_{\\ell_E}}\\phi_2 h_{\\mu_1\\ldots\\mu_{\\ell_E}}\\;.\n\\end{equation}\nOther distributions of the derivatives in the cubic vertices can be obtained from the above choice by using integration by parts. Because of (\\ref{EOMofBtoBprop}) and (\\ref{divergenceless}), the different choices of the cubic vertices will only affect $W^{s,\\, exchange}$ by a finite number of contact diagrams.\n\nThe main goal of this paper is to study the conformal block decomposition of these diagrams. From the Mellin representation of Witten diagrams \\cite{Penedones:2010ue,Paulos:2011ie,Fitzpatrick:2011ia,Costa:2012cb}, it is clear that a contact Witten diagram (\\ref{Wcontact}) decomposes only into double-trace operators. For example, in the s-channel the decomposition reads\\footnote{In this paper we will abuse the terminology by calling the coefficients in front of the conformal blocks, such as $a^{12}_{n,J}$, $a^{34}_{n,J}$, the ``OPE coefficients''. }\n\\begin{equation}\\label{Wcontactins}\nW^{contact}(x_i)=\\sum_{J=0}^{J_{\\rm max}}\\sum_{n=0}^\\infty a^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{J_{\\rm max}}\\sum_{n=0}^\\infty a^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nNotice that the spin of the conformal blocks has a finite support $0\\leq J\\leq J_{\\rm max}\\leq j_1+j_2+j_3+j_4$. \n\nDecomposing an exchange Witten diagram into the direct channel, one finds a single-trace conformal block and infinitely many double-trace blocks\n\\begin{equation}\\label{Wexchangeins}\nW^{s,\\,exchange}_{\\Delta_E,\\ell_E}=A\\, g^{(s)}_{\\Delta_E,\\ell_E}(x_i) +\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty A^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty A^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nAgain, the support of spins is finite. In contrast, when we decompose an exchange Witten diagram into the crossed channel ({\\it i.e.}, the t-channel and u-channel), we find only double-trace operators of which the spins are unbounded\n\\begin{equation}\\label{Wexchangeint}\nW^{s,\\,exchange}_{\\Delta_E,\\ell_E}=\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty B^{14}_{n,J} g^{(t)}_{\\Delta_1+\\Delta_4+2n+J,J}(x_i)+\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty B^{23}_{n,J} g^{(t)}_{\\Delta_2+\\Delta_3+2n+J,J}(x_i)\\;.\n\\end{equation}\nThe above discussion is for {\\it generic} external conformal dimensions. When $\\Delta_1+\\Delta_2-\\Delta_3-\\Delta_4\\in 2\\mathbb{Z}$, we will also encounter derivative conformal blocks $\\partial_\\Delta g^{(s)}_\\Delta(x_i)$ in the s-channel.\\footnote{\\label{FNderivativeblock}We can understand this fact from just large $N$ counting. For concreteness we use the counting of 4d $\\mathcal{N}=4$ SYM, then the tree level Witten diagrams are all of order $\\mathcal{O}(1\/N^2)$. If $\\Delta_1+\\Delta_2- \\Delta_3-\\Delta_4\\neq 2\\mathbb{Z}$, there is no order $\\mathcal{O}(1)$ overlap between the double-trace spectra of operators $:O_1\\square^{n_{12}}\\partial^{\\ell_{12}}O_2:$ and operators $:O_3\\square^{n_{34}}\\partial^{\\ell_{34}}O_4:$. The double trace operators $:O_1\\square^{n_{12}}\\partial^{\\ell_{12}}O_2:$ therefore can only appear in the $O_3\\times O_4$ OPE with a suppression power of $1\/N^2$ (the same for $:O_3\\square^{n_{34}}\\partial^{\\ell_{34}}O_4:$ to appear in the $O_1\\times O_2$ OPE). The conformal dimensions of the double-trace operators are corrected at order $\\mathcal{O}(1\/N^2)$. But their effect is invisible in the tree diagrams because the correction is of order $\\mathcal{O}(1\/N^4)$ due to the $\\mathcal{O}(1\/N^2)$ suppression in the OPE coefficients. On the other hand, when $\\Delta_1+\\Delta_2- \\Delta_3-\\Delta_4$ is an even integer (which we can further assume to be non negative), the double-trace operators with twists $\\tau\\geq\\Delta_1+\\Delta_2$ appear in both OPEs with $\\mathcal{O}(1)$ coefficients. The correction due to the anomalous dimensions of these operators are therefore now visible. We can view the appearance of derivative double-trace blocks $\\partial g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}$ as expanding the anomalous dimension in $g^{(s)}_{\\Delta_1+\\Delta_2+2n+J+\\frac{1}{N^2}\\gamma_{n,J},J}$ to $\\mathcal{O}(1\/N^2)$.\n } Similarly, when $\\Delta_1+\\Delta_4-\\Delta_2-\\Delta_3$ or $\\Delta_1+\\Delta_3-\\Delta_2-\\Delta_4$ is an even integer, there will be derivative conformal blocks in the t or u-channel. \n\nFor $d=1$, the decomposition of Witten diagrams has the same qualitative features, except that there is no spin. In the s-channel decomposition, we have\n\\begin{equation}\\label{Wcontactins1d}\nW^{contact}(x_i)=\\sum_{n=0}^\\infty a^{12}_{n} g^{(s)}_{\\Delta_1+\\Delta_2+2n}(x_i)+\\sum_{n=0}^\\infty a^{34}_{n} g^{(s)}_{\\Delta_3+\\Delta_4+2n}(x_i)\\;,\n\\end{equation}\n\\begin{equation}\\label{Wexchangeins1d}\nW^{s,exchange}(x_i)=Ag^{(s)}_\\Delta(x_i)+\\sum_{n=0}^\\infty A^{12}_{n} g^{(s)}_{\\Delta_1+\\Delta_2+2n}(x_i)+\\sum_{n=0}^\\infty A^{34}_{n} g^{(s)}_{\\Delta_3+\\Delta_4+2n}(x_i)\\;,\n\\end{equation}\nand only double-trace operators with one ``parity'' will show up, {\\it i.e.}, $\\Delta_1+\\Delta_2+n$ and $\\Delta_3+\\Delta_4+n$ with $n$ even. In the t-channel decomposition of the exchange Witten diagram,\n\\begin{equation}\\label{Wexchangeint1d}\nW^{s,\\,exchange}_{\\Delta_E}=\\sum_{n=0}^\\infty B^{14}_{n} g^{(t)}_{\\Delta_1+\\Delta_4+n}(x_i)+\\sum_{n=0}^\\infty B^{23}_{n} g^{(t)}_{\\Delta_2+\\Delta_3+n}(x_i)\\;,\n\\end{equation}\ndouble-trace operators of both parities will appear, {\\it i.e.}, $n\\in \\mathbb{Z}$.\n\n\\subsection{Relating Exchange Diagrams to Contact Diagrams}\\label{SecExtoCon}\nThe exchange Witten diagrams are related to the contact Witten diagrams by the a second order differential operator, as a result of the fact that the bulk-to-bulk propagators are Green's functions in AdS. To see this explicitly, let us first focus on the $z_1$ integral inside\n the s-channel exchange Witten diagram (\\ref{sWitten})\\begin{equation}\nI^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}(x_1,x_2;z_2)=\\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}}G^{\\Delta_1}_{B\\partial}(z_1,x_1)((\\bigtriangledown^{\\mu})^{\\ell_E}G^{\\Delta_2}_{B\\partial}(z_1,x_2))\\Pi^{\\Delta_E}_{\\mu_1\\ldots \\mu_{\\ell_E},\\nu_1\\ldots \\nu_{\\ell_E}}(z_1,z_2)\\;.\n\\end{equation}\nThis integral is manifestly invariant under $SO(d,2)$, and therefore satisfies the identity \n\\begin{equation}\n(\\mathbf{L}_1+\\mathbf{L}_2+\\mathfrak{L}_{z_2})_{AB}\\,I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}(x_1,x_2;z_2)=0\\;.\n\\end{equation}\nHere $\\mathbf{L}_1$ and $\\mathbf{L}_2$ are the conformal generators which act on $x_1$ and $x_2$, and $\\mathfrak{L}_{z_2}$ is the $AdS_{d+1}$ isometry generator which acts on a spin-$\\ell_E$ field at $z_2$. Using this identity twice, we obtain the following action of the conformal Casimir operator with respect to $x_1$ and $x_2$\n\\begin{equation}\n-\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2\\,I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}=-\\frac{1}{2}\\mathfrak{L}_{z_2}^2\\,I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}=(\\bigtriangledown^2_{z_2}+\\ell_E(\\ell_E+d-1))I^{s,\\,exchange}_{\\nu_1\\ldots\\nu_{\\ell_E}}\\;.\n\\end{equation}\nIn the second equality, we have used that the conformal Casimir is equal to the Laplacian up to a constant shift $\\ell_E(\\ell_E+d-1)$ \\cite{Pilch:1984xx}. We now apply the equation of motion (\\ref{EOMofBtoBprop}) to get rid of $\\bigtriangledown^2_{z_2}$ that acts on the bulk-to-bulk propagator, and perform the $z_2$ integral. We find the expression gives the following relation between an exchange diagram and a sum of contact Witten diagrams\n\\begin{equation}\\label{EOMWexWcon}\n\\left[\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}\\right]W^{s,\\, exchange}_{\\Delta_E,\\ell_E}=\\sum_I c_IW^{contact}_I\\;.\n\\end{equation}\nHere $\\sum_I c_IW^{contact}_I$ is a collection of contact Witten diagrams, obtained by replacing the bulk-to-bulk propagator with the RHS of (\\ref{EOMofBtoBprop}).\\footnote{One might worry about the contact term ambiguities in defining an exchange Witten diagram, introduced by the ``\\ldots'' in (\\ref{EOMofBtoBprop}) and (\\ref{divergenceless}). What is the action of the operator $\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}$ on a contact Witten diagram? It turns out that the action of the equation of motion operator on a contact Witten can again be expressed as the linear combination of finitely many contact Witten diagrams, as we will show at the end of Appendix \\ref{appcontact}. Therefore the relation (\\ref{EOMWexWcon}) holds for exchange Witten diagrams independent of the choice of contact terms. \\label{footnoteEOMonContact}} $C^{(2)}_{\\Delta_E,\\ell_E}$ is the eigenvalue of the quadratic conformal Casimir for an operator with dimension $\\Delta_E$ and spin $\\ell_E$\n\\begin{equation}\nC^{(2)}_{\\Delta_E,\\ell_E}=\\Delta_E(\\Delta_E-d)+\\ell_E(\\ell_E+d-2)\\;.\n\\end{equation}\n\nLet us work out how the operator on the LHS acts as a differential operator in terms of the cross ratios. It is standard to extract a kinematic factor from the four-point function\n\\begin{equation}\\label{GUV}\n\\langle O_1(x_1)\\ldots O_4(x_4) \\rangle\\equiv G(x_i)=\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}{2}}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}{2}}}\\left(\\frac{x_{14}^2}{x_{24}^2}\\right)^{a}\\left(\\frac{x_{14}^2}{x_{13}^2}\\right)^{b}\\mathcal{G}(U,V)\n\\end{equation}\nsuch that it becomes a function of the conformal cross ratios $U$ and $V$\n\\begin{equation}\nU=\\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}\\;,\\quad V=\\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}\\;.\n\\end{equation}\nWe have also defined\n\\begin{equation}\\small\na=\\frac{\\Delta_2-\\Delta_1}{2}\\;,\\quad b=\\frac{\\Delta_3-\\Delta_4}{2}\\;.\n\\end{equation}\nThe action of $\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+C^{(2)}_{\\Delta_E,\\ell_E}$ on a function of $x_i$ defines an operator acting on $\\mathcal{G}(U,V)$. Let us denote this operator as $\\mathbf{EOM}^{(s)}$, then (\\ref{EOMWexWcon}) becomes\n\\begin{equation}\\label{EOMWexWconcrossratio}\n\\mathbf{EOM}^{(s)}[\\mathcal{W}^{s,\\, exchange}_{\\Delta_E,\\ell_E}(U,V)]=\\sum_I c_I\\mathcal{W}^{contact}_I(U,V)\\;.\n\\end{equation}\n To give an explicit expression for this operator, it is convenient to make a change of variables for the conformal cross ratios\n\\begin{equation}\nU=z\\bar{z}\\;,\\quad V=(1-z)(1-\\bar{z})\\;,\n\\end{equation}\nand define a second order differential operator\n\\begin{equation}\n\\mathbf{D}_{z}(a,b)=(1-z)z^2\\frac{d^2}{dz^2}-(1+a+b)z^2\\frac{d}{dz}-abz\\;.\n\\end{equation}\nThe s-channel equation of motion operator can be written as\n\\begin{equation}\\label{EOMszzb}\n\\mathbf{EOM}^{(s)}[\\mathcal{G}(z,\\bar{z})]=-2\\mathbf{\\Delta}_\\epsilon(a,b)[\\mathcal{G}(z,\\bar{z})]+C^{(2)}_{\\Delta_E,\\ell_E}\\, \\mathcal{G}(z,\\bar{z})\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathbf{\\Delta}_\\epsilon(a,b)=\\mathbf{D}_z(a,b)+\\mathbf{D}_{\\bar{z}}(a,b)+2\\epsilon \\frac{z\\bar{z}}{z-\\bar{z}}\\bigg((1-z)\\frac{d}{dz}-(1-\\bar{z})\\frac{d}{d\\bar{z}}\\bigg)\\;,\n\\end{equation}\nand we have defined\n\\begin{equation}\n\\epsilon=\\frac{d}{2}-1\\;.\n\\end{equation}\nNote that the s-channel conformal blocks are eigenfunctions of this differential operator \\cite{Dolan:2003hv} \\footnote{In this paper we slightly abuse the notation to let $g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})$ also denote the conformal block as a function of the cross ratios where a kinematic factor of $x_{ij}^2$ is extracted from $g^{(s)}_{\\Delta,\\ell}(x_i)$ according to (\\ref{GUV}). The meaning should be clear from their different arguments and the context.}\n\\begin{equation}\n\\mathbf{EOM}^{(s)}[g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})]=(C^{(2)}_{\\Delta,\\ell}-C^{(2)}_{\\Delta_E,\\ell_E})g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})\\;.\n\\end{equation}\nIn particular, the operator annihilates the single-trace block $g^{(s)}_{\\Delta_E,\\ell_E}(z,\\bar{z})$.\n\n\nFor $d=1$, the bulk space is $AdS_2$ and only scalar fields propagate in the internal line of the exchange Witten diagrams. Acting with the Casimir operator on the scalar exchange Witten diagram, we get the $D$-function (\\ref{Dfunction})\n\\begin{equation}\n\\left[\\frac{1}{2}(\\mathbf{L}_1+\\mathbf{L}_2)^2+\\Delta_E(\\Delta_E-1)\\right]W^{s,\\,exchange}_{\\Delta_E}=D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}\\;.\n\\end{equation}\nMoreover, because there is only one cross ratio in one dimension\n\\begin{equation}\nz=\\frac{x_{12}x_{34}}{x_{13}x_{24}}\\;,\n\\end{equation}\nthe operator $\\mathbf{EOM}^{(s)}$ is a differential operator of $z$, and is given by\n\\begin{equation}\n\\mathbf{EOM}^{(s)}=-\\mathbf{D}_{z}(2a,2b)+\\Delta_E(\\Delta_E-1)\\;.\n\\end{equation}\n\n\\subsection{Conformal Partial Waves}\\label{CPW}\nIt is sometimes useful to think of the conformal block decomposition of conformal correlators as arising from a more primitive formula, in terms of the so-called conformal partial waves $\\Psi_{\\Delta,J}^{(s)}(x_i)$. We review in this section some basic properties of conformal partial waves for reader's convenience.\n\n\nThe conformal partial wave takes the form as the sum of the conformal blocks and its shadow block with $\\Delta\\to\\widetilde{\\Delta}=d-\\Delta$\n\\begin{equation}\\label{CPWdef}\n\\Psi_{\\Delta,J}^{(s)}(x_i)=K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}g^{(s)}_{\\Delta,J}(x_i)+K^{\\Delta_1,\\Delta_2}_{\\Delta,J}g^{(s)}_{\\widetilde{\\Delta},J}(x_i)\n\\end{equation}\nwhere the coefficients are given by\n\\begin{equation}\nK^{\\Delta_1,\\Delta_2}_{\\Delta,J}=\\left(-\\frac{1}{2}\\right)^J\\frac{\\pi^{\\frac{d}{2}}\\Gamma(\\Delta-\\frac{d}{2})\\Gamma(\\Delta+J-1)\\Gamma(\\frac{\\widetilde{\\Delta}+\\Delta_1-\\Delta_2+J}{2})\\Gamma(\\frac{\\widetilde{\\Delta}+\\Delta_2-\\Delta_1+J}{2})}{\\Gamma(\\Delta-1)\\Gamma(d-\\Delta+J)\\Gamma(\\frac{\\Delta+\\Delta_1-\\Delta_2+J}{2})\\Gamma(\\frac{\\Delta+\\Delta_2-\\Delta_1+J}{2})}\\;.\n\\end{equation}\nConformal partial waves with all integer spins $J$ and unphysical complex dimensions $\\Delta=\\frac{d}{2}+i \\nu$, $\\nu\\geq0$, form a complete set of functions \\cite{Dobrev:1977qv}, and are usually referred to as the principal series representation. In terms of conformal partial waves, a four-point correlation function can be written as a contour integral \\cite{Dobrev:1975ru}\\footnote{More precisely, for $d=1$ the principal series is also supplemented by a discrete series with $\\Delta=2n$, $n=1,2,\\ldots$. Here we assume that the spectral function $\\rho(\\Delta)$ does not contain physical poles at $\\Delta=2n$. Then after closing the contour and taking the residues, poles from the conformal partial wave will precisely cancel the contribution from the discrete series.}\n\\begin{equation}\n\\begin{split}\nG(x_i)={}&\\sum_{J=0}^{\\infty}\\int_{\\frac{d}{2}}^{\\frac{d}{2}+i\\infty} \\frac{d\\Delta}{2\\pi i}\\,\\rho(\\Delta,J)\\,\\Psi_{\\Delta,J}^{(s)}(x_i)\\\\\n={}&\\sum_{J=0}^{\\infty}\\int_{\\frac{d}{2}}^{\\frac{d}{2}+i\\infty} \\frac{d\\Delta}{2\\pi i}\\,\\rho(\\Delta,J)\\,(K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}g^{(s)}_{\\Delta,J}(x_i)+K^{\\Delta_1,\\Delta_2}_{\\Delta,J}g^{(s)}_{\\widetilde{\\Delta},J}(x_i))\n\\end{split}\n\\end{equation}\nUsing the symmetry that \n\\begin{equation}\n\\rho(\\Delta,J)\\,K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}=\\rho(\\widetilde{\\Delta},J)K^{\\Delta_1,\\Delta_2}_{\\widetilde{\\Delta},J}\\;,\n\\end{equation}\nwe can eliminate the shadow blocks from the above integral and rewrite it as\n\\begin{equation}\nG(x_i)=\\sum_{J=0}^{\\infty}\\int_{\\frac{d}{2}-i\\infty}^{\\frac{d}{2}+i\\infty} \\frac{d\\Delta}{2\\pi i}\\,\\rho(\\Delta,J)\\, K^{\\Delta_3,\\Delta_4}_{\\widetilde{\\Delta},J}g^{(s)}_{\\Delta,J}(x_i)\\;.\n\\end{equation}\nAfter closing the contour to the right and picking up the poles, we get the usual conformal block decomposition.\n\nThe combination (\\ref{CPWdef}) of the conformal block with its shadow is special because it makes the conformal partial wave a single-valued function in Euclidean space ({\\it i.e.}, when $\\bar{z}=z^*$). By contrast, each individual conformal block is not\\footnote{For example this can be explicitly seen from the 2d conformal block\n\\begin{equation}\ng^{(s)}_{\\Delta,J}(z,\\bar{z})=\\frac{k_{\\Delta-J}(z)k_{\\Delta+J}(\\bar{z})+k_{\\Delta+J}(z)k_{\\Delta-J}(\\bar{z})}{1+\\delta_{J,0}}\\;,\\quad k_{\\beta}(z)=z^{\\beta\/2}{}_2F_1(\\beta\/2+a,\\beta\/2+b,\\beta;z)\\;.\n\\end{equation}\nAround $z=1$ and $\\bar{z}=1$, we can use the property of ${}_2F_1$ to write the conformal block as\n\\begin{equation}\ng^{(s)}_{\\Delta,J}(z,\\bar{z})=f_{1}(z,\\bar{z})(z-1)^{-a-b}(\\bar{z}-1)^{-a-b}+f_{2}(z,\\bar{z})+f_{3}(z,\\bar{z})(z-1)^{-a-b}+f_4(z,\\bar{z})(\\bar{z}-1)^{-a-b}\n\\end{equation}\nwhere $f_i$ are regular at $z=\\bar{z}=1$. In the Euclidean regime $\\bar{z}=z^*$, the first two terms are single-valued while the latter two terms fail to be.\n\n\n}. We can most easily see this single-valuedness of the conformal partial wave from its integral representation \\cite{Ferrara:1972xe,Ferrara:1973vz,Ferrara:1972uq,Ferrara:1972ay,SimmonsDuffin:2012uy}\n\\begin{equation}\\label{CPWintegralrep}\n\\Psi_{\\Delta,J}^{(s)}(x_i)=\\int d^dx^5\\langle O_1(x_1)O_2(x_2)O_5^{\\mu_1\\ldots\\mu_J}(x_5)\\rangle \\langle \\widetilde{O}_{5,\\mu_1\\ldots\\mu_J}(x_5)O_3(x_3)O_4(x_4)\\rangle\\;,\n\\end{equation}\nwhere $\\widetilde{O}_{5,\\mu_1\\ldots\\mu_J}$ is the shadow operator of $O_5$. This integral is manifestly single-valued in Euclidean space. \n\nThe above integral representation can also be lifted into the AdS space \\cite{Costa:2014kfa}\n\\begin{equation}\\label{AdSlift}\n\\begin{split}\n\\Psi_{\\Delta,J}^{(s)}(x_i)\\propto{}& \\int d^d x_5 \\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}} G^{\\Delta_1}_{B\\partial}(z_1,x_1)(\\bigtriangledown^\\mu)^JG^{\\Delta_2}_{B\\partial}(z_1,x_2)\\Pi^{\\Delta}_{\\mu_1\\ldots\\mu_J}{}^{\\rho_1\\ldots\\rho_J}(z_1,x_5)\\\\\n{}& \\times \\int \\frac{d^{d+1}z_2}{z_{20}^{d+1}} \\Pi^{d-\\Delta}_{\\nu_1\\ldots\\nu_J,\\rho_1\\ldots\\rho_J}(z_2,x_5) G^{\\Delta_3}_{B\\partial}(z_2,x_3)(\\bigtriangledown^\\nu)^J G^{\\Delta_4}_{B\\partial}(z_2,x_4)\n\\end{split}\n\\end{equation}\nwhere $\\Pi^{\\Delta}_{\\mu_1\\ldots\\mu_J}{}^{\\rho_1\\ldots\\rho_J}(z,x_5)$ is the spin-$J$ bulk-to-boundary propagator. It is clear that the first AdS integral over $z_1$ gives the three-point function $\\langle O_1(x_1)O_2(x_2)O_5^{\\mu_1\\ldots\\mu_J}(x_5)\\rangle$ and the second AdS integral over $z_2$ gives $\\langle \\widetilde{O}_{5,\\mu_1\\ldots\\mu_J}(x_5)O_3(x_3)O_4(x_4)\\rangle$. Further performing the $x_5$ integral therefore reproduces the conformal partial wave $\\Psi_{\\Delta,J}^{(s)}(x_i)$ in the integral representation (\\ref{CPWintegralrep}).\n\n\nThe above lift of the conformal block into AdS is closely related to the split representation of AdS propagators \\cite{Costa:2014kfa}. It is convenient to first define the AdS harmonic function\n\\begin{equation}\\label{Omegasplit}\n\\Omega^{\\Delta,J}_{\\mu_1\\ldots\\mu_J,\\nu_1\\ldots\\nu_J}(z_1,z_2)=-\\frac{(\\Delta-\\frac{d}{2})^2}{\\pi J!(\\frac{d}{2}-1)_J}\\int d^dx_5 \\Pi^{\\Delta}_{\\mu_1\\ldots\\mu_J}{}^{\\rho_1\\ldots\\rho_J}(z_1,x_5)\\Pi^{d-\\Delta}_{\\nu_1\\ldots\\nu_J,\\rho_1\\ldots\\rho_J}(z_2,x_5)\n\\end{equation}\nwhich splits a function of two bulk points into a product of two bulk-to-boundary propagators with a common integrated boundary point. \nThe bulk-to-bulk propagator can then be expanded in terms of these AdS harmonic functions\n\\begin{equation}\\label{PiinOmega}\n\\Pi^{\\Delta,J}_{\\mu_1\\ldots\\mu_J,\\nu_1\\ldots\\nu_J}(z_1,z_2)=\\sum_{l=0}^J\\int d\\Delta' \\,a_\\ell(\\Delta') (\\bigtriangledown_1^\\mu)^{J-\\ell}(\\bigtriangledown_2^\\nu)^{J-\\ell}\\Omega^{\\Delta',\\ell}_{\\mu_1\\ldots\\mu_l,\\nu_1\\ldots\\nu_l}(z_1,z_2)\\;\n\\end{equation}\nwhere explicit coefficients $a_\\ell(\\Delta')$ can be found in \\cite{Costa:2014kfa}. The above split representation of the AdS harmonic function (\\ref{Omegasplit}) gives the split representation of the bulk-to-bulk propagator.\n\nRelatedly, a conformal partial wave can also be represented in AdS as the difference of two exchange Witten diagrams with opposite quantizations\n\\begin{equation}\\label{Psiasdifference}\n\\Phi^{(s)}_{\\Delta,J}(x_i)\\propto W^{s,exchange}_{\\Delta,J}(x_i)-W^{s,exchange}_{d-\\Delta,J}(x_i)\\;.\n\\end{equation}\nThis fact was pointed out, {\\it e.g.}, in \\cite{Penedones:2007ns,Costa:2014kfa,Giombi:2018vtc}, and can be understood in two steps as follows. Firstly, all the double-trace conformal blocks in $W^{s,exchange}_{\\Delta,J}$ and $W^{s,exchange}_{d-\\Delta,J}$ cancel out in (\\ref{Psiasdifference}). We can see this by noting that both $W^{s,exchange}_{\\Delta,J}$ and $W^{s,exchange}_{d-\\Delta,J}$ satisfy the same equation of motion identity\n\\begin{equation}\n\\mathbf{EOM}^{(s)}[\\mathcal{W}^{s,\\, exchange}_{\\Delta_E,\\ell_E}(U,V)]=\\mathbf{EOM}^{(s)}[\\mathcal{W}^{s,\\, exchange}_{d-\\Delta_E,\\ell_E}(U,V)],\n\\end{equation}\nbecause the contact diagrams are determined by vertices which are independent of the quantization. Moreover, the double-trace conformal blocks are diagonal under the equation of motion operator, with non vanishing and quantization-independent eigenvalues. This guarantees the cancellation of double-trace blocks in the difference and implies that (\\ref{Psiasdifference}) is just the linear combination of the single-trace conformal block and its shadow. The second step is therefore to show that the linear combination is proportional to (\\ref{CPWdef}). However, this is guaranteed by single-valuedness. Because the exchange Witten diagrams are single-valued in the Euclidean regime, there is only one way, {\\it i.e.}, in the fashion of (\\ref{CPWdef}), to combine the non-single-valued conformal block and its shadow in order to achieve single-valuedness .\n\n\n\n\n\n\n\n\n\n\nAs a final comment, let us mention that a single conformal block in $d\\geq2$ {\\it cannot} be decomposed into the crossed channel in terms of conformal blocks. This was noticed in, {\\it e.g.}, \\cite{ElShowk:2011ag,Liu:2018jhs}, and follows from the non single-valuedness of conformal block. In the Euclidean regime, an s-channel conformal block is not single-valued around $z=\\bar{z}=1$ while the t-channel conformal blocks are. It is not possible to sum over infinitely many single-valued functions to obtain a non-single-valued function. On the other hand, a conformal partial wave can always be decomposed into the crossed channel in terms of double-trace conformal blocks. This follows from (\\ref{Psiasdifference}) where the conformal partial wave is written as the difference of two s-channel exchange Witten diagrams, and each exchange Witten diagram admits decomposition into double-trace conformal blocks in the crossed channel. In 1d, the above comments do not apply. There is only one cross ratio and one can explicitly show that the conformal block can be decomposed into the crossed channel as infinitely many double-trace conformal blocks.\n\n\\section{Recursion Relations in Contact Witten Diagrams}\n\\subsection{Conformal Block Decomposition of Contact Witten Diagrams}\\label{SecCBdecompContact}\nIn this subsection we focus on the conformal block decomposition of contact Witten diagrams (\\ref{Wcontact}). We organize the contact Witten diagrams in terms of the total number of covariant derivatives in the quartic vertex. We start with the simplest contact Witten diagram where there is no derivative, {\\it i.e.}, $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$ defined in (\\ref{Dfunction}). The conformal block decomposition of higher-derivative contact Witten diagrams, as we will see, can be recursively related to the decomposition of the zero-derivative contact Witten diagrams. In this section, we will assume the external conformal dimensions are generic such that we will not encounter derivative conformal blocks. The conformal block decomposition for the special cases satisfying $\\Delta_1+\\Delta_2=\\Delta_3+\\Delta_4+2m$, {\\it etc}, can be obtained from the generic case by taking the limit. We give more details of taking the limit in Appendix \\ref{appcontact}.\n\n\\subsubsection*{The decomposition the zero-derivative contact diagram}\nTo obtain the conformal block decomposition of $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$, we use a special case of the split representation, namely the split representation of the delta-function \\cite{Penedones:2010ue,Costa:2014kfa,Bekaert:2015tva}\n\\begin{equation}\n\\delta(z_1,z_2)=\\int d^dx_5\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i} \\,\\rho_\\delta(c)\\,G_{B\\partial}^{\\frac{d}{2}+c}(z_1,x_5)\\,G_{B\\partial}^{\\frac{d}{2}-c}(z_2,x_5)\\;\n\\end{equation}\nwhere \n\\begin{equation}\n\\rho_\\delta(c)=\\frac{\\Gamma(\\frac{d}{2}+c)\\Gamma(\\frac{d}{2}-c)}{2\\pi^d\\Gamma(-c)\\Gamma(c)}\\;.\n\\end{equation}\nInserting this identity into (\\ref{Dfunction}), we have \n\\begin{equation}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int d^dx_5\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i}\\,\\rho_\\delta(c)\\, D_{\\Delta_1\\,\\Delta_2\\, \\frac{d}{2}+c}(x_1,x_2,x_5)D_{ \\frac{d}{2}-c\\, \\Delta_3\\,\\Delta_4}(x_5,x_3,x_4)\n\\end{equation}\nwhere $D_{\\Delta_1\\Delta_2\\Delta_3}(x_1,x_2,x_3)$ is a three-point function\n\\begin{equation}\\label{D3function}\n\\begin{split}\nD_{\\Delta_1\\Delta_2\\Delta_3}={}&\\int \\frac{d^{d+1}z}{z_0^{d+1}}\\prod_{i=1}^3G^{\\Delta_i}_{B\\partial}(z,x_i)=\\frac{a_{\\Delta_1\\Delta_2\\Delta_3}}{x_{12}^{\\Delta_1+\\Delta_2-\\Delta_3}x_{13}^{\\Delta_1+\\Delta_3-\\Delta_2}x_{23}^{\\Delta_2+\\Delta_3-\\Delta_1}}\\;,\\\\\na_{\\Delta_1\\Delta_2\\Delta_3}={}& \\frac{\\pi^{\\frac{d}{2}}\\Gamma(\\frac{\\Delta_1+\\Delta_2-\\Delta_3}{2})\\Gamma(\\frac{\\Delta_1+\\Delta_3-\\Delta_2}{2})\\Gamma(\\frac{\\Delta_2+\\Delta_3-\\Delta_1}{2})}{2\\Gamma(\\Delta_1)\\Gamma(\\Delta_2)\\Gamma(\\Delta_3)}\\Gamma(\\frac{\\Delta_1+\\Delta_2+\\Delta_3-d}{2})\\;.\n\\end{split}\n\\end{equation}\nWe can integrate out $x_5$ using (\\ref{CPWintegralrep}), and obtain the conformal partial wave decomposition of the $D$-function\n\\begin{equation}\\label{DfunctioninCPW}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i}\\,\\rho_D(c)\\,\\Psi^{(s)}_{\\frac{d}{2}+c,0}(x_i)\\;.\n\\end{equation}\nThe spectral density is\n\\begin{equation}\n\\rho_D(c)=\\rho_\\delta(c)\\, a_{\\Delta_1\\,\\Delta_2\\,\\frac{d}{2}+c}\\,a_{\\Delta_3\\,\\Delta_4\\,\\frac{d}{2}-c}\\;.\n\\end{equation}\nWe can use the shadow symmetry in (\\ref{DfunctioninCPW})to write it as a spectral representation with respect to the conformal blocks \n\\begin{equation}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\int_{-i\\infty}^{i\\infty} \\frac{dc}{2\\pi i}\\,2\\,\\rho_D(c)\\, K^{\\Delta_3,\\Delta_4}_{\\frac{d}{2}-c,0}g^{(s)}_{\\frac{d}{2}+c,0}(x_i)\\;.\n\\end{equation}\nBy closing the contour to the right and taking the residues, we arrive at the conformal block decomposition of $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$\n\\begin{equation}\\label{Dfunctioningxi}\nD_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}=\\sum_{n=0}^\\infty a^{12}_{n,0} g^{(s)}_{\\Delta_1+\\Delta_2+2n,0}(x_i)+\\sum_{n=0}^\\infty a^{34}_{n,0} g^{(s)}_{\\Delta_3+\\Delta_4+2n,0}(x_i)\\;\n\\end{equation}\nwhere \n\\begin{equation}\\label{Dfunctiondecomcoe}\\small\n\\begin{split}\na^{12}_{n,0}={}&\\frac{\\pi ^{d\/2} (-1)^{-n} \\Gamma (n+\\Delta_1) \\Gamma (n+\\Delta_2) \\Gamma \\left(-\\frac{d}{2}+n+\\Delta_1+\\Delta_2\\right) \\Gamma \\left(\\frac{-d+2 n+\\Delta_1+\\Delta_2+\\Delta_3+\\Delta_4}{2}\\right)}{2 n! \\Gamma (\\Delta_1) \\Gamma (\\Delta_2) \\Gamma (\\Delta_3) \\Gamma (\\Delta_4) \\Gamma (2 n+\\Delta_1+\\Delta_2)}\\\\\n{}&\\times\\frac{\\Gamma \\left(\\frac{2 n+\\Delta_1+\\Delta_2+\\Delta_3-\\Delta_4}{2}\\right) \\Gamma \\left(\\frac{2 n+\\Delta_1+\\Delta_2-\\Delta_3+\\Delta_4}{2}\\right) \\Gamma \\left(\\frac{-2 n-\\Delta_1-\\Delta_2+\\Delta_3+\\Delta_4}{2}\\right)}{ \\Gamma \\left(-\\frac{d}{2}+2 n+\\Delta_1+\\Delta_2\\right)}\\;, \n\\end{split}\n\\end{equation}\nand $a^{34}_{n,0}$ can be obtained from $a^{12}_{n,0}$ by replacing $\\Delta_1$, $\\Delta_2$ with $\\Delta_3$, $\\Delta_4$.\n\n\\subsubsection*{The decomposition of higher-derivative contact diagrams}\nNow let us consider a general contact Witten diagram (\\ref{Wcontact}) with derivatives in the quartic vertex. We first notice that a generic contact Witten diagram $W^{contact}$ can always be written as a linear combination of finitely many building blocks \\cite{Penedones:2010ue}\n\\begin{equation}\\label{Dnijdef}\nD^{\\{n_{ij}\\}}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)\\equiv\\prod_{i1$. \n \n \n\\subsection{One Dimension}\\label{1d}\nLet us consider an $AdS_2$ scalar exchange Witten diagram in the t-channel \n\\begin{equation}\nW^{t,exchange}_{\\Delta_E}=\\int \\frac{d^{d+1}z_1}{z_{10}^{d+1}}\\frac{d^{d+1}z_2}{z_{20}^{d+1}}G^{\\Delta_1}_{B\\partial}(z_1,x_1)G^{\\Delta_4}_{B\\partial}(z_1,x_4)\\Pi^{\\Delta_E}(z_1,z_2)G^{\\Delta_2}_{B\\partial}(z_2,x_2)G^{\\Delta_3}_{B\\partial}(z_2,x_3)\\;.\n\\end{equation}\nAs we reviewed in Section \\ref{SecExtoCon}, the diagram $W^{t,exchange}_{\\Delta_E}$ satisfies the following t-channel equation of motion identity \n\\begin{equation}\\label{tchannelCasimir}\n\\left[\\frac{1}{2}(\\mathbf{L}_2+\\mathbf{L}_3)^{AB}(\\mathbf{L}_2+\\mathbf{L}_3)_{AB}+M_E^2\\right]W^{t,exchange}_{\\Delta_E}=D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}\n\\end{equation}\nwhere $M_E^2=\\Delta_E(\\Delta_E-1)$. It is convenient to extract a kinematic factor in the convention of (\\ref{GUV}), so that we work with functions of the cross ratio $z$\n\\begin{equation}\n G(x_i)=\\frac{1}{(x_{12}^2)^{\\frac{\\Delta_1+\\Delta_2}{2}}(x_{34}^2)^{\\frac{\\Delta_3+\\Delta_4}{2}}}\\left(\\frac{x_{14}^2}{x_{24}^2}\\right)^{a}\\left(\\frac{x_{14}^2}{x_{13}^2}\\right)^{b}\\mathcal{G}(z)\\;.\n\\end{equation}\nThe equation of motion identity (\\ref{tchannelCasimir}) then becomes\n\\begin{equation}\\label{EOMtWexeqWcon}\n\\mathbf{EOM}^{(t)}[\\mathcal{W}^{t,exchange}](z)=\\mathcal{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(z)\n\\end{equation}\nwhere $\\mathcal{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(z)$ is $D_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(x_i)$ after stripping off the kinematic factor. It is straightforward to work out the action of the differential operator $\\mathbf{EOM}^{(t)}$ on a generic function $\\mathcal{G}(z)$. The action takes the following form\n\\begin{equation}\n\\begin{split}\n\\mathbf{EOM}^{(t)}[\\mathcal{G}]{}&=\\frac{1}{2}\\mathbf{D}_z(2a,2b)[\\mathcal{G}(z)]-\\mathbf{f}_0(2a,2b)[\\mathbf{D}_z(2a,2b)[\\mathcal{G}(z)]]+\\Delta_E(\\Delta_E-1)\\mathcal{G}(z)\\\\\n{}&-\\left(1-\\sum_{i=1}^4\\Delta_i\\right)\\mathbf{f}_1(2a,2b)[\\mathcal{G}(z)]-(\\Delta_1+\\Delta_2)(\\Delta_3+\\Delta_4)\\mathbf{f}_0(2a,2b)[\\mathcal{G}(z)]\\\\\n{}&- \\left(2ab+\\frac{1}{2}\\sum_{i=1}^4\\Delta_i(\\Delta_i-1)\\right)\\mathcal{G}(z)\\;.\n\\end{split}\n\\end{equation}\nIn the above expression, we have additionally defined two operators\n\\begin{equation}\n\\mathbf{f}_0(a,b)=\\frac{1}{z}-\\frac{1}{2}\\;, \\quad \\mathbf{f}_1(a,b)=(1-z)\\frac{d}{dz}-\\frac{1}{2}(a+b)\\;.\n\\end{equation}\n\nLet us now consider the action of $\\mathbf{EOM}^{(t)}$ when $\\mathcal{G}(z)$ is an s-channel conformal block. To proceed, the following properties of $\\mathbf{f}_0$ and $\\mathbf{f}_1$ \\cite{Dolan:2011dv} will be useful to us\\footnote{Let us manifest the normalization by writing down the explicit expression for the 1d conformal block\n\\begin{equation}\ng^{(s)}_\\Delta(z)=z^\\Delta{}_2F_1(\\Delta+\\Delta_2-\\Delta_1,\\Delta+\\Delta_3-\\Delta_4;2\\Delta;z)\\;.\n\\end{equation}}\n\\begin{equation}\n\\begin{split}\n\\mathbf{f}_0(a,b)[g_\\Delta^{(s)}(z)]={}&g^{(s)}_{\\Delta-1}(z)+\\alpha_\\Delta(a,b) g^{(s)}_{\\Delta}(z)+\\beta_\\Delta(a,b) g^{(s)}_{\\Delta+1}(z)\\;,\\\\\n\\mathbf{f}_1(a,b)[g^{(s)}_\\Delta(z)]={}&g^{(s)}_{\\Delta-1}(z)+\\alpha_\\Delta(a,b) g^{(s)}_{\\Delta}(z)-(\\Delta-1)\\beta_\\Delta(a,b) g^{(s)}_{\\Delta+1}(z)\\;\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\n\\begin{split}\n\\alpha_\\Delta(a,b)={}&-\\frac{ab}{2\\Delta(\\Delta-1)}\\;,\\\\\n\\beta_\\Delta(a,b)={}&\\frac{(\\Delta+a)(\\Delta+b)(\\Delta-a)(\\Delta-b)}{4\\Delta^2(2\\Delta-1)(2\\Delta+1)}\\;.\n\\end{split}\n\\end{equation}\nUsing these two relations and the Casimir equation\n\\begin{equation}\n\\mathbf{D}_z(2a,2b)[g^{(s)}_{\\Delta}(z)]=\\Delta(\\Delta-1)g^{(s)}_{\\Delta}(z)\\;,\n\\end{equation}\nwe find the following three-term recursion relation for $\\mathbf{EOM}^{(t)}$ acting on an s-channel conformal block\n\\begin{equation}\\label{crosseomong}\n\\mathbf{EOM}^{(t)}[g^{(s)}_{\\Delta}(z)]=\\mu\\,g^{(s)}_{\\Delta-1}(z)+\\nu\\,g^{(s)}_{\\Delta}(z)+\\rho\\,g^{(s)}_{\\Delta+1}(z)\\;.\n\\end{equation}\nThe recursion coefficients are given by\n\\begin{equation}\n\\begin{split}\n\\mu={}&-(\\Delta -\\Delta_1-\\Delta_2) (\\Delta -\\Delta_3-\\Delta_4)\\;,\\\\\n\\nu={}&\\frac{2 a b (\\Delta_1+\\Delta_2-1) (\\Delta_3+\\Delta_4-1)}{(\\Delta -1) \\Delta }+(\\Delta_E -1) \\Delta_E+\\frac{1}{2} (\\Delta -1) \\Delta\\\\\n{}&-\\frac{1}{2} \\left((\\Delta_1-1)\\Delta_1+(\\Delta_2-1)\\Delta_2+(\\Delta_3-1)\\Delta_3+(\\Delta_4-1)\\Delta_4\\right)\\;,\\\\\n\\rho={}&-\\frac{(\\Delta -2 a) (2 a+\\Delta ) (\\Delta -2 b) (2 b+\\Delta ) (\\Delta +\\Delta_1+\\Delta_2-1) (\\Delta +\\Delta_3+\\Delta_4-1)}{4 \\Delta ^2 (2 \\Delta -1) (2 \\Delta +1)}\\;.\n\\end{split}\n\\end{equation}\n\nLet us take the conformal dimension $\\Delta$ to be the dimensions of the double-trace operators $\\Delta_1+\\Delta_2+n$. The recursion relation now reads\n\\begin{equation}\\label{recurblock1d}\n\\mathbf{EOM}^{(t)}[g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)]=\\mu^{12}_n\\,g^{(s)}_{\\Delta_1+\\Delta_2+n-1}(z)+\\nu^{12}_n\\,g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\rho^{12}_n\\,g^{(s)}_{\\Delta_1+\\Delta_2+n+1}(z)\n\\end{equation}\nwhere $\\mu^{12}_n$, $\\nu^{12}_n$, $\\rho^{12}_n$ are $\\mu$, $\\nu$, $\\rho$ with $\\Delta=\\Delta_1+\\Delta_2+n$. From the explicit expression of $\\mu^{12}_n$ we can see that when $n=0$,\n\\begin{equation}\n\\mu^{12}_0=0\\;,\n\\end{equation}\nleaving only two double-trace blocks on the RHS of the recursion relation. The conformal block with dimension $\\Delta_1+\\Delta_2-1$ (which is not part of the double-trace spectrum) will not be generated. Therefore the action of $\\mathbf{EOM}^{(t)}$ preserves the double-trace spectrum $\\Delta_1+\\Delta_2+n$ with $n\\geq0$ and $n\\in \\mathbb{Z}$. A similar recursion relation also exists for double-trace operators with dimensions $\\Delta_3+\\Delta_4+n$, and can be obtained from the above relation by replacing $\\Delta_1$, $\\Delta_2$ with $\\Delta_3$, $\\Delta_4$.\n\nThese recursion relations give us an efficient way to compute the crossed channel decomposition. For simplicity, we will assume that the external conformal dimensions are generic such that no derivative conformal blocks appear. Special cases, such as $\\Delta_i=\\Delta_\\phi$, can be obtained from the generic case by taking a limit, and will be discussed in Appendix \\ref{appeqweight}. Let us insert in (\\ref{EOMtWexeqWcon}) the s-channel decomposition of the t-channel exchange diagram $\\mathcal{W}^{t,exchange}$ \n\\begin{equation}\\label{texchange1dB12B34}\n\\mathcal{W}^{t,exchange}(z)=\\sum_{n=0}^\\infty B^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\sum_{n=0}^\\infty B^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+n}(z)\\;.\n\\end{equation}\nThe relation (\\ref{recurblock1d}) for $g^{(s)}_{\\Delta_1+\\Delta_2+n}$ and its counterpart for $g^{(s)}_{\\Delta_3+\\Delta_4+n}$ allows us to express the action of $\\mathbf{EOM}^{(t)}$ on $\\mathcal{W}^{t,exchange}(z)$ again in terms of the sum of double-trace conformal blocks. By the identity (\\ref{EOMtWexeqWcon}), this decomposition should be equal to the decomposition (\\ref{Dfunctioningxi}) for the contact diagram $\\mathcal{D} _{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}$\\footnote{The decomposition result (\\ref{Dfunctioningxi}) also applies to the $AdS_2$ contact diagram. We just need to set $d=1$, and identify $g^{(s)}_{\\Delta_1+\\Delta_2+2n,0}$, $g^{(s)}_{\\Delta_3+\\Delta_4+2n,0}$ with the 1d conformal blocks $g^{(s)}_{\\Delta_1+\\Delta_2+2n}$, $g^{(s)}_{\\Delta_3+\\Delta_4+2n}$.} which takes the following form\n\\begin{equation}\n\\mathcal{D}_{\\Delta_1\\Delta_2\\Delta_3\\Delta_4}(z)=\\sum_{n=0}^\\infty a^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+2n}(z)+\\sum_{n=0}^\\infty a^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+2n}(z)\\;.\n\\end{equation}\nWe therefore arrive at the following linear recursion relations among the crossed channel decomposition coefficients\n\\begin{equation}\\label{recureq12inhomo}\n\\rho^{12}_{n-1}B^{12}_{n-1}+\\nu^{12}_{n}B^{12}_{n}+\\mu^{12}_{n+1}B^{12}_{n+1}=\n\\begin{cases}\na^{12}_{\\frac{n}{2}}\\;,\\quad n\\text{ even}\\;,\\\\\n0\\;,\\quad n\\text{ odd}\\;,\n\\end{cases}\\quad \\quad B^{12}_{-1}\\equiv 0\\;,\n\\end{equation}\n\\begin{equation}\\label{recureq34inhomo}\n\\rho^{34}_{n-1}B^{34}_{n-1}+\\nu^{34}_{n}B^{34}_{n}+\\mu^{34}_{n+1}B^{34}_{n+1}=\n\\begin{cases}\na^{34}_{\\frac{n}{2}}\\;,\\quad n\\text{ even}\\;,\\\\\n0\\;,\\quad n\\text{ odd}\\;,\n\\end{cases}\\quad \\quad B^{34}_{-1}\\equiv 0\\;.\n\\end{equation}\nIt is straightforward to solve this relation. As we have commented already, when $n=0$ the three-term relations reduce to two-term relations with only $B^{12}_0$, $B^{12}_1$ and $B^{34}_0$, $B^{34}_1$. Once the seed coefficients $B^{12}_0$, $B^{34}_0$ are determined, the sub-leading coefficients $B^{12}_n$, $B^{34}_n$ with $n\\geq 1$ can be recursively computed from the above three-term relations. In Appendix \\ref{appseed}, we will show how to compute the seed coefficients. \n\n\n\\subsubsection*{Homogeneous Solution: the Wilson Polynomial}\nThe linear recursion equations (\\ref{recureq12inhomo}) and (\\ref{recureq34inhomo}) are inhomogeneous, and their solution gives the s-channel decomposition of a t-channel exchange Witten diagram. On the other hand, it is also interesting to consider the homogenous recursion equations for which the RHS' are zero\n\\begin{equation}\\label{recureq12homo}\n\\rho^{12}_{n-1}\\widetilde{B}^{12}_{n-1}+\\nu^{12}_{n}\\widetilde{B}^{12}_{n}+\\mu^{12}_{n+1}\\widetilde{B}^{12}_{n+1}=0\\;,\n\\end{equation}\n\\begin{equation}\\label{recureq34homo}\n\\rho^{34}_{n-1}\\widetilde{B}^{34}_{n-1}+\\nu^{34}_{n}\\widetilde{B}^{34}_{n}+\\mu^{34}_{n+1}\\widetilde{B}^{34}_{n+1}=0\\;.\n\\end{equation}\n Such homogenous equations arise from a function $\\mathcal{G}(z)$ that satisfies\n\\begin{equation}\n\\mathbf{EOM}^{(t)}[\\mathcal{G}](z)=0\\;,\n\\end{equation}\nand admits an s-channel decomposition in terms of double-trace conformal blocks\n\\begin{equation}\n\\mathcal{G}(z)=\\sum_{n=0}^\\infty \\widetilde{B}^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\sum_{n=0}^\\infty \\widetilde{B}^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+n}(z)\\;.\n\\end{equation}\nIn one dimension, such $\\mathcal{G}(z)$ can be either the t-channel conformal block $g^{(t)}_{\\Delta_E}(z)$, or the conformal partial wave $\\Psi^{(t)}_{\\Delta_E}(z)$. The solution to the homogenous recursion equations (\\ref{recureq12homo}), (\\ref{recureq34homo}) gives the crossed channel decomposition of the conformal block or partial wave (as we will see, their decomposition coefficients are only different by an overall factor). We will also see in the next subsection an extension of the story to $d>1$. There the solution to the homogenous recursion equations has to be interpreted as the crossed channel decomposition of a conformal partial wave. This is because Euclidean single-valuedness dictates that a single conformal block cannot be expressed in the crossed channel as infinitely many double-trace blocks, as we have already commented on at the end of Section \\ref{CPW}.\n \nLet us now solve the homogeneous equation (\\ref{recureq12homo}). The solution to (\\ref{recureq34homo}) can be obtained by simply replacing $\\Delta_1$, $\\Delta_2$ with $\\Delta_3$, $\\Delta_4$. Notice that a three-term recursion relation like (\\ref{recureq12homo}) is characteristic of systems of orthogonal polynomials. Indeed, after making the change of variables \n\\begin{equation}\n\\mathfrak{a}=\\frac{1}{2}+\\Delta_1-\\Delta_4\\;,\\quad \\mathfrak{b}=\\frac{1}{2}+\\Delta_2-\\Delta_3\\;,\\quad \\mathfrak{c}=-\\frac{1}{2}+\\Delta_1+\\Delta_4\\;,\\quad \\mathfrak{d}=-\\frac{1}{2}+\\Delta_2+\\Delta_3\\;, \n\\end{equation}\nand writing $\\widetilde{B}^{12}_n$ as\n\\begin{equation}\n\\widetilde{B}^{12}_n=-\\frac{\\Gamma (\\mathfrak{a}+\\mathfrak{c}+n) \\Gamma (\\mathfrak{a}+\\mathfrak{d}+n) \\Gamma (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+n-1)}{\\Gamma (n+1) \\Gamma (\\mathfrak{a}+\\mathfrak{c}) \\Gamma (\\mathfrak{a}+\\mathfrak{d}) \\Gamma (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-1)}p_n\\;,\n\\end{equation}\nwe find that the recursion relation (\\ref{recureq12homo}) can be cast into the following form\n\\begin{equation}\\label{Wilsonrecur}\n-\\left(\\mathfrak{a}^2+x^2\\right)p_n(x)=\\mathcal{A}_n(p_{n+1}(x)-p_n(x))+\\mathcal{B}_n(p_{n-1}(x)-p_n(x))\n\\end{equation}\nwhere\n\\begin{equation}\nx=i\\left(\\Delta_E-\\frac{1}{2}\\right)\\;,\n\\end{equation}\n\\begin{equation}\\label{Wilsonrecurcoe}\n\\begin{split}\n\\mathcal{A}_n={}&\\frac{(\\mathfrak{a}+\\mathfrak{b}+n) (\\mathfrak{a}+\\mathfrak{c}+n) (\\mathfrak{a}+\\mathfrak{d}+n) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+n-1)}{(\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-1) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n)}\\;,\\\\\n\\mathcal{B}_n={}&\\frac{n (\\mathfrak{b}+\\mathfrak{c}+n-1) (\\mathfrak{b}+\\mathfrak{d}+n-1) (\\mathfrak{c}+\\mathfrak{d}+n-1)}{(\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-2) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+2 n-1)}\\;.\n\\end{split}\n\\end{equation}\nThis is precisely the recursion relation that defines the Wilson polynomial \\cite{Wilson1977,Wilson:1980aa}. Since the solution to the recursion relation is unique up to an overall rescaling, we can set $p_0(x)=1$ for convenience. Then $p_n(x)$ can be expressed compactly as a ${}_4F_3$ function \\cite{Wilson1977,Wilson:1980aa}\n\\begin{equation}\\label{Wilsonpolyn}\np_n(x;\\mathfrak{a},\\mathfrak{b},\\mathfrak{c},\\mathfrak{d})={}_4F_3\\left(\\left.\\begin{array}{c}-n,n+\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}-1, \\mathfrak{a}+ix,\\mathfrak{a}-ix\\\\\\mathfrak{a}+\\mathfrak{b},\\mathfrak{a}+\\mathfrak{c},\\mathfrak{a}+\\mathfrak{d}\\end{array}\\right.;1\\right)\\;.\n\\end{equation}\nThis function $p_n(x)$ is a polynomial in $x^2$ of degree $n$, as is already clear from the recursion relation (\\ref{Wilsonrecur}). \n\nNow let us use this solution in the s-channel decomposition of a 1d t-channel conformal block. The decomposition takes the following form\n\\begin{equation}\\label{tconfblockdecomp}\n\\begin{split}\n\\frac{z^{\\Delta_1+\\Delta_2}}{(1-z)^{\\Delta_2+\\Delta_3}}g^{(t)}_{\\Delta_E}(z)={}&\\frac{z^{\\Delta_1+\\Delta_2}}{(1-z)^{\\Delta_2+\\Delta_3}}{}_2F_1(\\Delta_E-\\Delta_3+\\Delta_2,\\Delta_E+\\Delta_1-\\Delta_4,2\\Delta_E,1-z)\\\\\n={}&\\sum_{n=0}^{\\infty}b^{12}_n g^{(s)}_{\\Delta_1+\\Delta_2+n}(z)+\\sum_{n=0}^{\\infty}b^{34}_n g^{(s)}_{\\Delta_3+\\Delta_4+n}(z)\\;,\n\\end{split}\n\\end{equation}\nand one can check that the LHS of (\\ref{tconfblockdecomp}) is annihilated by the equation of motion operator $\\mathbf{EOM}^{(t)}$. The action on the RHS thus gives rise to the recursion equations (\\ref{recureq12homo}) and (\\ref{recureq34homo}). The above solution to the homogenous equations (\\ref{recureq12homo}) then determines the ratios $b^{12}_n\/b^{12}_0$ of the decomposition coefficients to be\n\\begin{equation}\\label{b12ratio}\n\\frac{b^{12}_n}{b^{12}_0}=\\frac{\\Gamma (n+1) (\\mathfrak{a}+\\mathfrak{b}+\\mathfrak{c}+\\mathfrak{d}+n-1)_n}{(\\mathfrak{a}+\\mathfrak{c})_n (\\mathfrak{a}+\\mathfrak{d})_n}p_n(x;\\mathfrak{a},\\mathfrak{b},\\mathfrak{c},\\mathfrak{d})\\;.\n\\end{equation}\nBy further comparing the two sides of (\\ref{tconfblockdecomp}) expanded at $z=1^-$, we can also determine the leading coefficient\n\\begin{equation}\nb^{12}_0=\\frac{\\Gamma (2 \\Delta_E) \\Gamma (-\\Delta_1-\\Delta_2+\\Delta_3+\\Delta_4)}{\\Gamma (-\\Delta_1+\\Delta_4+\\Delta_E) \\Gamma (-\\Delta_2+\\Delta_3+\\Delta_E)}\\;,\n\\end{equation}\nand thereby obtaining the crossed channel decomposition. Similarly, $b^{34}_n$ can also be obtained by solving recursion relations. Its solution is simply the solution for $b^{12}_n$ with $\\Delta_1$, $\\Delta_2$ replaced by $\\Delta_3$, $\\Delta_4$. The above crossed channel decomposition coefficients can also be obtained from the alpha space techniques \\cite{Hogervorst:2017sfd}.\n\nNote that the coefficient ratio (\\ref{b12ratio}) depends on $\\Delta_E$ only via the shadow symmetric combination $x^2$. This implies that the decomposition coefficients for the shadow conformal block have the same ratio (\\ref{b12ratio}). Furthermore, since the conformal partial wave (\\ref{CPWdef}) is just the linear combination of the conformal block and its shadow, its crossed channel decomposition coefficients will also have the same ratio.\n\n\n\\subsection{$d>1$ Dimensions}\\label{higherd}\nThe story for $d>1$ is similar to what we have seen for $d=1$. We will organize this subsection as the follows. We start by obtaining the action of the t-channel equation of motion operator on a generic conformal block. We will find the result can be expressed as the linear combination of five conformal blocks with shifted dimensions and spins. Then we restrict the quantum numbers of the conformal blocks to those of the double-trace operators which appear in the crossed channel decomposition. The spectra of these operators are preserved by the recursion relation. This gives us an efficient algorithm for recursively solving the crossed channel decomposition coefficients. We can consider two different problems depending on whether the recursion equations are homogenous or inhomogeneous. The latter corresponds to the s-channel decomposition of a t-channel exchange Witten diagram, while the former corresponds to the decomposition of an t-channel conformal partial wave. We also consider the special case of equal external weights, where our results imply recursion relations for the anomalous dimensions.\n\n\n\\subsubsection*{Action of the t-channel equation of motion operator}\nConsider a t-channel exchange Witten diagram with dimension $\\Delta_E$ and spin $\\ell_E$. The t-channel equation of motion give the identity \n\\begin{equation}\\label{higherdtchEOM}\n\\left(\\frac{1}{2}(\\mathbf{L}_2+\\mathbf{L}_3)^2+C^{(2)}_{\\Delta_E,\\ell_E}\\right)W^{t,exchange}_{\\Delta_E,\\ell_E}=\\sum_I c_I W^{contact}_I\\;.\n\\end{equation}\nThe LHS of the identity defines a second order differential operator $\\mathbf{EOM}^{(t)}$ which acts on $\\mathcal{G}(z,\\bar{z})$\n\\begin{equation}\\small\n\\begin{split}\n{}&\\mathbf{EOM}^{(t)}[\\mathcal{G}(z,\\bar{z})]=\\mathbf{\\Delta}_\\epsilon(a,b)[\\mathcal{G}(z,\\bar{z})]-\\mathbf{F}_0(a,b)[\\mathbf{\\Delta}_\\epsilon(a,b)[\\mathcal{G}(z,\\bar{z})]]+M_E^2\\, \\mathcal{G}(z,\\bar{z})\\\\\n{}&+\\mathbf{F}_2(a,b)[\\mathcal{G}(z,\\bar{z})]+\\left(\\Delta_1+\\Delta_2+\\Delta_3+\\Delta_4-2+2\\epsilon\\right)\\mathbf{F}_1(a,b)[\\mathcal{G}(z,\\bar{z})]\\\\\n{}&-\\frac{1}{2}(\\Delta_1+\\Delta_2)(\\Delta_3+\\Delta_4)\\mathbf{F}_0(a,b)[\\mathcal{G}(z,\\bar{z})]\\\\\n{}&+\\left(2+\\epsilon(\\Delta_1+\\Delta_2+\\Delta_3+\\Delta_4)-\\frac{(\\Delta_1-1)^2+(\\Delta_2-1)^2+(\\Delta_3-1)^2+(\\Delta_4-1)^2}{2}-2ab\\right)\\mathcal{G}(z,\\bar{z})\n\\end{split}\n\\end{equation}\nwhere $\\mathbf{F}_i$ is a differential operator of order $i$\n\\begin{equation}\n\\begin{split}\n\\mathbf{F}_0(a,b)={}&\\frac{1}{z}+\\frac{1}{\\bar{z}}-1\\;,\\\\\n\\mathbf{F}_1(a,b)={}&(1-z)\\frac{\\partial}{\\partial z}+(1-\\bar{z})\\frac{\\partial}{\\partial \\bar{z}}\\;,\\\\\n\\mathbf{F}_2(a,b)={}&\\frac{z-\\bar{z}}{z\\bar{z}}(\\mathbf{D}_z(a,b)-\\mathbf{D}_{\\bar{z}}(a,b))\\;.\n\\end{split}\n\\end{equation}\nHere it will turn out to be more convenient to use the normalization of \\cite{Dolan:2011dv} for the conformal blocks, which are denoted by $F_{\\lambda_1,\\lambda_2}(z,\\bar{z})$ \n\\begin{equation}\n\\begin{split}\nF_{\\lambda_1,\\lambda_2}(z,\\bar{z})={}&\\mathcal{N}_{\\epsilon,\\ell}g^{(s)}_{\\Delta,\\ell}(z,\\bar{z})\\;,\\\\\n\\mathcal{N}_{\\epsilon,\\ell}={}&\\frac{(\\epsilon)_\\ell}{(2\\epsilon)_\\ell}\\;,\\\\\n\\lambda_1=\\frac{\\Delta+\\ell}{2}\\;,{}&\\quad \\lambda_2=\\frac{\\Delta-\\ell}{2}\\;.\n\\end{split}\n\\end{equation}\nThe $\\mathbf{F}_i$ operators have the following recursion relations on conformal blocks \\cite{Dolan:2011dv}\n\\begin{equation}\\label{Ffivetermrecur}\n\\mathbf{F}_i(a,b)[\\mathcal{F}_{\\lambda_1,\\lambda_2}]=r_i F_{\\lambda_1,\\lambda_2-1}+s_iF_{\\lambda_1-1,\\lambda_2}+t_iF_{\\lambda_1+1,\\lambda_2}+u_iF_{\\lambda_1,\\lambda_2+1}+w_iF_{\\lambda_1,\\lambda_2}\\;,\n\\end{equation}\nwith various coefficients defined as follows. The coefficients with label $i=0$ are\n\\begin{equation}\n\\begin{split}\n{}&r_0=\\frac{\\lambda_1-\\lambda_2+2 \\epsilon }{\\lambda_1-\\lambda_2+\\epsilon }\\;,\\quad s_0=\\frac{\\lambda_1-\\lambda_2}{\\lambda_1-\\lambda_2+\\epsilon }\\;,\\\\\n{}&t_0=\\frac{(\\lambda_1+\\lambda_2-1) (\\lambda_1+\\lambda_2-2 \\epsilon ) (\\lambda_1-\\lambda_2+2 \\epsilon )}{(\\lambda_1+\\lambda_2-\\epsilon -1) (\\lambda_1+\\lambda_2-\\epsilon ) (\\lambda_1-\\lambda_2+\\epsilon )}\\beta_{\\lambda_1}(a,b)\\;,\\\\\n{}&u_0=\\frac{(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-1) (\\lambda_1+\\lambda_2-2 \\epsilon )}{(\\lambda_1+\\lambda_2-\\epsilon -1) (\\lambda_1+\\lambda_2-\\epsilon ) (\\lambda_1-\\lambda_2+\\epsilon )}\\beta_{\\lambda_2-\\epsilon}(a,b)\\;,\\\\\n{}&w_0=-\\frac{(c_{\\lambda_1\\lambda_2}+2\\epsilon)}{2\\lambda_1(\\lambda_1-1)(\\lambda_2-\\epsilon)(\\lambda_2-1-\\epsilon)} ab\\;.\n\\end{split}\n\\end{equation}\nIn terms of the $i=0$ coefficients, the higher $i$ coefficients are given by\\footnote{There are two typos in (4.29) of \\cite{Dolan:2011dv}. The first typo is in the expression for $s_2$ and is corrected below. The other one is in $s_3$, and the correct expression should be\n\\begin{equation}\ns_3=(\\lambda_1+\\epsilon)(\\lambda_1-\\lambda_2+2\\epsilon)(\\lambda_1+\\lambda_2-1)s_0\\;.\n\\end{equation}}\n\\begin{equation}\n\\begin{split}\n{}&r_1=\\lambda_2 r_0\\;,\\quad s_1=(\\lambda_1+\\epsilon)s_0\\;,\\quad t_1=-(\\lambda_1-1-\\epsilon)t_0\\;,\\quad u_1=-(\\lambda_2-1-2\\epsilon)u_0\\;,\\\\\n{}& w_1=(1+\\epsilon)w_0\\;,\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n{}& r_2=(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-1)r_0\\;,\\quad s_2= -(\\lambda_1+\\lambda_2-1) (\\lambda_1-\\lambda_2+2\\epsilon )s_0\\;,\\\\\n{}& t_2=-(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-2 \\epsilon -1)t_0\\;,\\quad u_2=(\\lambda_1-\\lambda_2+2 \\epsilon ) (\\lambda_1+\\lambda_2-2 \\epsilon -1)u_0\\;,\\\\\n{}& w_2=-\\frac{(\\lambda_1-\\lambda_2) (\\lambda_1+\\lambda_2-1) (\\lambda_1-\\lambda_2+2 \\epsilon ) (\\lambda_1+\\lambda_2-2 \\epsilon -1)}{2\\lambda_1 (\\lambda_1-1) (\\lambda_2-\\epsilon ) (\\lambda_2-\\epsilon -1)}ab\\;.\n\\end{split}\n\\end{equation}\nUsing the recursion relations (\\ref{Ffivetermrecur}) and the Casimir equation\n\\begin{equation}\n2\\mathbf{\\Delta}_\\epsilon(a,b)[F_{\\lambda_1,\\lambda_2}]=C^{(2)}_{\\Delta,\\ell}F_{\\lambda_1,\\lambda_2}\\;,\n\\end{equation}\n the action of $\\mathbf{EOM}^{(t)}$ on a conformal block $F_{\\lambda_1,\\lambda_2}$ can be expressed as the linear combination of five conformal blocks with shifted dimensions and spins\n\\begin{equation}\\label{EOMrecuranyd}\n\\mathbf{EOM}^{(t)}[\\mathcal{F}_{\\lambda_1,\\lambda_2}]=\\mathfrak{R}\\, F_{\\lambda_1,\\lambda_2-1}+\\mathfrak{S}\\,F_{\\lambda_1-1,\\lambda_2}+\\mathfrak{T}\\,F_{\\lambda_1+1,\\lambda_2}+\\mathfrak{U}\\,F_{\\lambda_1,\\lambda_2+1}+\\mathfrak{W}\\,F_{\\lambda_1,\\lambda_2}\\;.\n\\end{equation}\nThe coefficients are determined to be\\footnote{Here we can appreciate the advantage of using the normalization of \\cite{Dolan:2011dv}: the coefficients $\\mathfrak{S}$ and $\\mathfrak{U}$ are simply related to $\\mathfrak{R}$ and $\\mathfrak{T}$ with $\\ell\\to-\\ell-2\\epsilon$. Otherwise there will be an additional factor which depends on $\\epsilon$ and $\\ell$. }\n\\begin{equation}\\label{EOMrecuranydcoe}\n\\begin{split}\n\\mathfrak{R}={}&-\\frac{(\\ell +2 \\epsilon ) (-\\Delta +\\Delta_1+\\Delta_2+\\ell ) (-\\Delta +\\Delta_3+\\Delta_4+\\ell )}{2 (\\ell +\\epsilon )}\\,\\\\\n\\mathfrak{T}={}&-\\frac{(2 a+\\Delta +\\ell ) (-2 a+\\Delta +\\ell ) (2 b+\\Delta +\\ell ) (-2 b+\\Delta +\\ell )}{32 (\\Delta -\\epsilon ) (\\Delta +\\ell )^2 (\\Delta -\\epsilon -1) }\\\\\n{}&\\times \\frac{(\\Delta +\\Delta_1+\\Delta_2+\\ell -2 \\epsilon -2) (\\Delta +\\Delta_3+\\Delta_4+\\ell -2 \\epsilon -2)(\\Delta -1) (\\Delta -2 \\epsilon )(\\ell +2 \\epsilon)}{(\\Delta +\\ell -1)(\\Delta +\\ell +1) (\\ell +\\epsilon )}\\,\\\\\n\\mathfrak{S}={}&\\mathfrak{R}\\big|_{\\ell\\to-\\ell-2\\epsilon}\\;,\\\\\n\\mathfrak{U}={}&\\mathfrak{T}\\big|_{\\ell\\to-\\ell-2\\epsilon}\\;.\\\\\n\\mathfrak{W}={}&C^{(2)}_{\\Delta_E,\\ell_E}+\\frac{C^{(2)}_{\\Delta,\\ell}-\\sum_{i=1}^4C^{(2)}_{\\Delta_i,\\ell_i=0}}{2}+\\frac{2ab(\\Delta_1+\\Delta_2-2\\epsilon-2)(\\Delta_3+\\Delta_4-2\\epsilon-2)(C^{(2)}_{\\Delta,\\ell}+4\\epsilon)}{(\\Delta +\\ell -2) (\\Delta +\\ell ) (-\\Delta +\\ell +2 \\epsilon ) (-\\Delta +\\ell +2 \\epsilon +2)}\n\\end{split}\n\\end{equation}\nNote that the above recursion relation has the following desirable features similar to those of (\\ref{crosseomong}). We first look at the factor $\\mathfrak{R}$ multiplying the conformal block $F_{\\lambda_1,\\lambda_2-1}$ which has conformal dimension $\\Delta-1$ and twist $\\Delta-\\ell$. This factor vanishes $\\mathfrak{R}$ when the conformal block $F_{\\lambda_1,\\lambda_2}$ has the minimal twist for a double-trace operator formed with $\\mathcal{O}_1$ and $\\mathcal{O}_2$ or with $\\mathcal{O}_3$ and $\\mathcal{O}_4$, {\\it i.e.}, \n\\begin{equation}\n\\mathfrak{R}=0\\;,\\quad\\text{when}\\quad\\Delta=\\ell+\\Delta_1+\\Delta_2\\;,\\quad \\text{or}\\quad \\Delta=\\ell+\\Delta_3+\\Delta_4\\;.\n\\end{equation}\nLet us also notice that the coefficients $\\mathfrak{S}$ and $\\mathfrak{U}$, which multiply conformal blocks with shifted spin $\\ell-1$, both contain a factor $\\ell$. It implies that when the spin of $F_{\\lambda_1,\\lambda_2}$ is zero, both $\\mathfrak{S}$ and $\\mathfrak{U}$ vanish and no blocks with negative spins will be generated on the RHS. Moreover, the twists of conformal blocks are always shifted by an even integer. It is therefore not hard to see that these properties of the coefficients guarantee that the double-trace spectra, labelled by $\\{\\Delta,\\ell\\}$ with\n\\begin{equation}\n\\Delta=\\Delta_1+\\Delta_2+2n+\\ell\\quad \\text{and}\\quad \\Delta=\\Delta_3+\\Delta_4+2n+\\ell\\;,\\quad n,\\ell=0,1,2,\\ldots,\n\\end{equation}\nare preserved by the recursion relation (\\ref{EOMrecuranyd}). \n\n\\subsubsection*{Reduction to 1d}\nWe can show that the recursion relation in $d>1$ reduces to the 1d recursion relation (\\ref{crosseomong}) in an appropriate limit. Setting $\\epsilon=-\\frac{1}{2}$, $\\ell_E=0$, $\\ell=0$ and restricting to $z=\\bar{z}$ we find from (\\ref{EOMrecuranydcoe}) that\n\\begin{equation}\n\\mathfrak{S}=\\mathfrak{U}=0\\;,\\quad \\mathfrak{R}=\\mu\\;,\\quad \\mathfrak{W}=\\nu\\;,\\quad \\mathfrak{T}=\\rho\\;.\n\\end{equation}\nWe then use the following identity for conformal blocks\\footnote{In the second identity, $\\ell=1$ for the conformal block $F_{\\lambda_1\\lambda_2}$. Note that $\\ell$ appears in the Casimir eigenvalue as $\\ell(\\ell+2\\epsilon)=\\ell(\\ell-1)$, both $\\ell=0$ and $\\ell=1$ give the same 1d conformal block.} \\cite{Dolan:2011dv}\n\\begin{equation}\nF^{(-\\frac{1}{2})}_{\\frac{\\Delta}{2}\\frac{\\Delta}{2}}(z,z)=g_\\Delta(z)\\;,\\quad F^{(-\\frac{1}{2})}_{\\frac{\\Delta+1}{2}\\frac{\\Delta-1}{2}}(z,z)=g_\\Delta(z)\\;,\n\\end{equation}\nto reduce the higher dimensional conformal blocks to one dimensional blocks $g_\\Delta(z)$. We find that the 1d recursion relation (\\ref{crosseomong}) is precisely reproduced.\n\n\n\\subsubsection*{The recursive algorithm for the crossed channel decomposition}\nLet us now use the recursion relation (\\ref{EOMrecuranyd}) to formulate an algorithm for the crossed channel decomposition of an exchange Witten diagram. We start with the s-channel decomposition for $W^{t,\\,exchange}_{\\Delta_E,\\ell_E}$, with the assumption that the external dimensions $\\Delta_i$ are generic\n\\begin{equation}\nW^{t,\\,exchange}_{\\Delta_E,\\ell_E}=\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty \\underbrace{\\mathcal{B}^{12}_{n,J} \\mathcal{N}_{\\epsilon,J}}_{B^{12}_{n,J}} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\infty}\\sum_{n=0}^\\infty \\underbrace{\\mathcal{B}^{34}_{n,J}\\mathcal{N}_{\\epsilon,J}}_{B^{34}_{n,J}} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nHere $\\mathcal{N}_{\\epsilon,J}=\\frac{(\\epsilon)_J}{(2\\epsilon)_J}$ is a normalization factor such that $\\mathcal{N}_{\\epsilon,J} g^{(s)}_{\\Delta,J}=F_{\\frac{\\Delta+J}{2},\\frac{\\Delta-J}{2}}$. We insert this decomposition in (\\ref{higherdtchEOM}). Thanks to the recursion relation (\\ref{EOMrecuranyd}), the action of $\\mathbf{EOM}^{(t)}$ can be rewritten as the linear combination of double-trace blocks. The resulting expansion should be equal to the conformal block decomposition of the contact terms on the RHS\n\\begin{equation}\n\\sum_I c_I W_I^{contact}=\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty \\tilde{a}^{12}_{n,J} g^{(s)}_{\\Delta_1+\\Delta_2+2n+J,J}(x_i)+\\sum_{J=0}^{\\ell_E}\\sum_{n=0}^\\infty \\tilde{a}^{34}_{n,J} g^{(s)}_{\\Delta_3+\\Delta_4+2n+J,J}(x_i)\\;.\n\\end{equation}\nFor convenience, let us define $\\mathfrak{R}^{12}_{n,\\ell}$, $\\mathfrak{S}^{12}_{n,\\ell}$, $\\mathfrak{T}^{12}_{n,\\ell}$, $\\mathfrak{U}^{12}_{n,\\ell}$, $\\mathfrak{W}^{12}_{n,\\ell}$ to be $\\mathfrak{R}$, $\\mathfrak{S}$, $\\mathfrak{T}$, $\\mathfrak{U}$, $\\mathfrak{W}$ with $\\Delta=\\Delta_1+\\Delta_2+2n+\\ell$, and $\\mathfrak{R}^{34}_{n,\\ell}$, $\\mathfrak{S}^{34}_{n,\\ell}$, $\\mathfrak{T}^{34}_{n,\\ell}$, $\\mathfrak{U}^{34}_{n,\\ell}$, $\\mathfrak{W}^{34}_{n,\\ell}$ to be $\\mathfrak{R}$, $\\mathfrak{S}$, $\\mathfrak{T}$, $\\mathfrak{U}$, $\\mathfrak{W}$ with $\\Delta=\\Delta_3+\\Delta_4+2n+\\ell$.\nThen we arrive at the following recursion relations for the crossed channel decomposition coefficients. The recursion relation for $\\mathcal{B}^{12}_{n,\\ell}$ is\n\\begin{equation}\\label{Recur12nleqatilde12nl}\n\\begin{split}\nRecur^{12}_{n,\\ell}\\equiv\\mathfrak{R}^{12}_{n+1,\\ell-1}\\mathcal{B}^{12}_{n+1,\\ell-1}+\\mathfrak{S}^{12}_{n,\\ell+1}\\mathcal{B}^{12}_{n,\\ell+1}+\\mathfrak{T}^{12}_{n,\\ell-1}\\mathcal{B}^{12}_{n,\\ell-1}{}&\\\\\n+\\mathfrak{U}^{12}_{n-1,\\ell+1}\\mathcal{B}^{12}_{n-1,\\ell+1}+\\mathfrak{W}^{12}_{n,\\ell}\\mathcal{B}^{12}_{n,\\ell}{}&=\\tilde{a}^{12}_{n,\\ell}\\;,\n\\end{split}\n\\end{equation}\nwith $n\\geq0$, $\\ell\\geq0$, and \n\\begin{equation}\n\\mathcal{B}^{12}_{n,-1}=\\mathcal{B}^{12}_{-1,\\ell}=0\\;.\n\\end{equation}\nThe recursion relation for $\\mathcal{B}^{34}_{n,\\ell}$ takes the same form, and can be obtained by replacing 1, 2 with 3, 4.\n\nLet us discuss how we can solve these recursion relations. Without the loss of generality, we focus on the coefficients $\\mathcal{B}^{12}_{n,\\ell}$ and their recursion relations. Just as in the 1d case where we need to input the seed coefficient for the double-trace operator with minimal conformal dimension, here we need to input the OPE coefficients of the double-trace operators with minimal {\\it twist}, {\\it i.e.}, $\\mathcal{B}^{12}_{0,\\ell}$. These seed coefficients can be obtained from applying the inversion formula \\cite{Caron-Huot:2017vep} for $\\ell>\\ell_E$ \\cite{Liu:2018jhs,Cardona:2018dov}, or from Mellin space \\cite{Costa:2014kfa,Sleight:2018epi,Sleight:2018ryu}. After inputting the seed coefficients, the coefficients $\\mathcal{B}^{12}_{n,\\ell}$ of operators with higher twists are uniquely determined. More precisely, the equation \n\\begin{equation}\\label{eqn12nllgeq1}\nRecur^{12}_{n-1,\\ell+1}=\\tilde{a}^{12}_{n-1,\\ell+1}\\;,\\quad n\\geq1\\;, \\ell\\geq0\\;,\n\\end{equation}\ndetermines the coefficient $\\mathcal{B}^{12}_{n,\\ell}$ in terms of $\\mathcal{B}^{12}_{n',\\ell}$ with $n'1$. \n\n\nBefore we end this subsection let us consider a special case where the t-channel conformal partial wave has $\\ell_E=0$, and the external weights are degenerate $\\Delta_i=\\Delta_\\phi$. In this case the solution to (\\ref{Recur12nleq0nl}) and (\\ref{Recur34nleq0nl}) has the interpretation of anomalous dimensions (see Appendix \\ref{appeqweight}). The seed coefficients for the scalar exchange was obtained first in \\cite{Giombi:2018vtc}\\footnote{A straightforward exercise is to check that the expressions satisfy the relations (\\ref{simpleratio12}), (\\ref{simpleratio34}).}\n\\begin{equation}\n\\begin{split}\n\\tilde{\\mathcal{B}}^{12}_{0,\\ell}={}&\\frac{4 \\Gamma (\\Delta_E) \\Gamma \\left(\\frac{d-\\Delta_E}{2}\\right)^2 \\Gamma (d+\\ell -2) \\Gamma (\\ell +\\Delta_\\phi )^2 \\Gamma (\\ell +2 \\Delta_\\phi -1)}{\\Gamma (d-1) \\Gamma \\left(\\frac{\\Delta_E}{2}\\right)^2 \\Gamma (\\Delta_\\phi )^2 \\Gamma (\\ell +1) \\Gamma \\left(\\frac{d}{2}-\\Delta_E\\right) \\Gamma \\left(\\frac{d}{2}+\\ell -1\\right) \\Gamma (2 \\ell +2 \\Delta_\\phi -1)}\\\\\n{}&\\times p_\\ell\\left[i\\left(\\frac{\\Delta_E}{2}-\\frac{d}{4}\\right);\\frac{d}{4},\\Delta_\\phi-\\frac{d}{4},\\Delta_\\phi-\\frac{d}{4},\\frac{d}{4}\\right]\n\\end{split}\n\\end{equation}\nwhere $p_\\ell(x;\\mathfrak{a},\\mathfrak{b},\\mathfrak{c},\\mathfrak{d})$ is the Wilson polynomial defined in (\\ref{Wilsonpolyn}),\\footnote{These Wilson polynomials also appear in the crossed channel decomposition of conformal partial waves of spinning operators \\cite{Sleight:2018ryu}.} and we will suppress its argument in the following to write it as $p_\\ell$. With this input, our method allows us to efficiently compute all the subleading coefficients. From $\\widetilde{Recur}^{12}_{0,\\ell}=0$, we find \n\\begin{equation}\\small\n\\begin{split}\n{}&\\tilde{\\mathcal{B}}^{12}_{1,\\ell}=-\\frac{2^{\\Delta_E-2 \\Delta_\\phi -2 \\ell -2}\\Gamma \\left(\\frac{\\Delta_E+1}{2}\\right) \\Gamma \\left(\\frac{d-\\Delta_E}{2}\\right)^2 \\Gamma (\\ell +\\Delta_\\phi +1) \\Gamma (\\ell +2 \\Delta_\\phi )}{\\Gamma \\left(\\frac{d}{2}\\right) \\Gamma \\left(\\frac{\\Delta_E}{2}\\right) \\Gamma (\\Delta_\\phi )^2 \\Gamma (\\ell +1) \\Gamma \\left(\\frac{d}{2}-\\Delta_E\\right) \\Gamma \\left(\\ell +\\Delta_\\phi +\\frac{3}{2}\\right)}\\\\{}&\\times\\bigg[\\frac{(\\Delta_\\phi +\\ell ) (-d+2 \\Delta_\\phi +\\ell +2) (-d+4 \\Delta_\\phi +2 \\ell )}{-d+4 \\Delta_\\phi +2 \\ell +2}p_{\\ell}+\\frac{(d+\\ell -1) (d+2 \\ell +2) (\\Delta_\\phi +\\ell +1) (2 \\Delta_\\phi +\\ell )}{(\\ell +1) (d+2 \\ell )}p_{\\ell+2}\\\\\n{}&-\\frac{(2 \\Delta_\\phi +2 \\ell +1) (C^{(2)}_{\\Delta_E,0}+\\Delta_\\phi (d+2 \\ell +2)+\\ell ^2+\\ell)}{\\ell +1}p_{\\ell+1}\\bigg]\\;.\n\\end{split}\n\\end{equation}\nSubstituting the solution into $\\widetilde{Recur}^{12}_{1,\\ell}=0$, we get an expression for $\\tilde{\\mathcal{B}}^{12}_{2,\\ell}$ which has the following form \n\\begin{equation}\n\\tilde{\\mathcal{B}}^{12}_{2,\\ell}=(\\ldots)p_{\\ell}+(\\ldots)p_{\\ell+1}+(\\ldots)p_{\\ell+2}+(\\ldots)p_{\\ell+3}+(\\ldots)p_{\\ell+4}\\;.\n\\end{equation}\nThe coefficients in this expression are more complicated and we will refrain from writing down their explicit expressions. The above two sets of subleading coefficients were computed in \\cite{Sleight:2018ryu} using a different method. We have checked that these expressions are equivalent to their results.\\footnote{We thank Massimo Taronna for providing the Mathematica notebook that contains their relevant results.} It is totally straightforward to iterate and get $\\tilde{\\mathcal{B}}^{12}_{n,\\ell}$ with higher $n$. The result is expressed as a linear combination of $p_{\\ell}$, $p_{\\ell+1}$, \\ldots $p_{\\ell+2n+2}$, and the coefficients can be efficiently computed using the recursion relations. However, in contrast to the 1d case, we have not found an obvious way to write the coefficients $\\tilde{\\mathcal{B}}^{12}_{n,\\ell}$ for any $n$ in a closed form.\n\n\n\n\\section{Discussion and Outlook}\\label{SecDiscuss}\nIn this paper we performed a systematic position space analysis of the conformal block decomposition of Witten diagrams and conformal partial waves. In our analysis we emphasized the use of the equation of motion operator and the contact Witten diagrams. These objects played important roles in the decomposition of exchange Witten diagrams, both in the direct channel and in the crossed channel. Our main finding is a recursive algorithm for computing the crossed channel decomposition coefficients of exchange Witten diagrams and conformal partial waves. This algorithm allows us to efficiently obtain the OPE coefficient of any double-trace operator with sub-leading conformal twist, in terms of the coefficients of the double-trace operators with the minimal twist. At face value, our results provide a useful tool to extract CFT data from tree-level holographic correlators for all internal spins, especially when the exchange Witten diagrams do not admit a truncated representation in terms of finitely many $D$-functions.\n\n\n\nLet us also mention other problems where our results might be useful.\n\\subsubsection*{Mellin bootstrap}\nOne use of our results is to simplify the Mellin bootstrap program. The Mellin bootstrap approach uses the crossing symmetrized exchange Witten diagrams as an expansion basis. The use of the exchange Witten diagrams introduces spurious double-trace operators in the conformal block decomposition of the correlator ansatz. The method hence derives the bootstrap conditions by requiring the vanishing of all the double-trace coefficients when summing over the physical spectrum. In the Mellin bootstrap method one obtains the double-trace coefficients of a single Witten diagram by taking residues of the Mellin amplitude at the double-trace poles, and then projecting them to different spins using continuous Hahn polynomials. As we already mentioned in the introduction, the projection at a certain pole gives only the mixed OPE coefficients between the primary double-trace operators and the descendant double-trace operators for which the primaries have smaller twists. For the Mellin bootstrap method, it is not necessary to solve the mixing problem. This is because the descendant contributions always vanish in the bootstrap equations if the conditions on the primary operators are already satisfied. Nevertheless, it might be useful to clarify the structures of the Mellin bootstrap equations by eliminating the redundant descendant contributions. Using our method, such contributions are absent automatically because we work with the primary operators only. Furthermore, because the decomposition coefficients satisfy recursion relations, it should be possible to recursively derive sub-leading bootstrap conditions from the leading ones. These sub-leading conditions are crucial for probing operators with sub-leading twists, for example in the $4-\\epsilon$ expansion \\cite{Gopakumar:2018xqi}. One should however notice that there still is a subtlety in the Mellin bootstrap method which is to fix the contact term ambiguity in the basis (see \\cite{Dey:2017fab,Gopakumar:2018xqi} for recent progress in general $d$, and \\cite{Mazac:2018ycv} for $d=1$). This issue must be addressed separately.\n\n\\subsubsection*{Analytic functionals in $\\mathrm{CFT}_1$}\nIt was pointed out in \\cite{Mazac:2018ycv} that the decomposition coefficients of the following crossing symmetric combination of $AdS_2$ Witten diagrams\\footnote{In \\cite{Mazac:2018ycv} the external operators are restricted to be identical and all have the same conformal dimension $\\Delta_\\phi$. The coefficient $\\lambda$ can be fixed such that the coefficient of $\\partial g^{(s)}_{2\\Delta_\\phi}$ is zero in the s-channel decomposition. This is one of the infinitely many equivalent choices of $\\lambda$, see \\cite{Mazac:2018ycv} for more details.}\n\\begin{equation}\nW^{s,exchange}_\\Delta+W^{t,exchange}_\\Delta+W^{u,exchange}_\\Delta+\\lambda W^{contact}\n\\end{equation}\nencode the information of the complete set of functionals for $\\mathrm{CFT}_1$ (see \\cite{Mazac:2016qev,Mazac:2018mdx} for earlier related work, also \\cite{Mazac1dinversion}). A basis of analytic functionals is given by $\\alpha_n$ and $\\beta_m$, labelled by integers $n=0,1,2,\\ldots$, $m=1,2,3,\\ldots$, and the application of the functionals to the crossing equation gives the complete set of constraints. It is sufficient to know the action of the functionals on the bootstrap vector function $F_\\Delta(z)$ defined by\n\\begin{equation}\nF_\\Delta(z)=\\frac{g^{(s)}_\\Delta(z)}{z^{2\\Delta_\\phi}}-\\frac{g^{(s)}_\\Delta(1-z)}{(1-z)^{2\\Delta_\\phi}}\\;.\n\\end{equation}\nThe action of the $n$-th functional on the function $F_\\Delta$ can be expressed as the ratio of decomposition coefficients of the Witten diagrams\n\\begin{equation}\\label{functionalaction}\n\\begin{split}\n\\alpha_n[F_\\Delta]={}&-\\frac{A_n+2B_{2n}+\\lambda a_n}{A}\\;,\\\\\n\\beta_n[F_\\Delta]={}&-\\frac{C^{(s)}_n+2C^{(t)}_{2n}+\\lambda c_n}{A}\\;.\n\\end{split}\n\\end{equation}\nWe have computed all these decomposition coefficients in this paper, and we briefly remind the reader of our notations. $A$, $A_n$, $C^{s}_n$ are respectively the coefficient of $g^{(s)}_\\Delta$, $g^{(s)}_{2\\Delta_\\phi+2n}$ and $\\partial g^{(s)}_{2\\Delta_\\phi+2n}$ of the exchange diagram $W^{s,exchange}_\\Delta$ in the s-channel\\footnote{The explicit expressions are given in (\\ref{DirectcoeA}), (\\ref{DirectcoeAn}), (\\ref{DirectcoeCns}).}; $B_{2n}$, $C^{(t)}_{2n}$ are the coefficient of $g^{(s)}_{2\\Delta_\\phi+2n}$ and $\\partial g^{(s)}_{2\\Delta_\\phi+2n}$ of the exchange diagrams $W^{t,exchange}_\\Delta$ or $W^{u,exchange}_\\Delta$ in the t and u-channel; and finally $a_n$ and $c_n$ are the coefficient of $g^{(s)}_{2\\Delta_\\phi+2n}$ and $\\partial g^{(s)}_{2\\Delta_\\phi+2n}$ of the contact diagram $W^{contact}=D_{\\Delta_\\phi\\Delta_\\phi\\Delta_\\phi\\Delta_\\phi}$. In \\cite{Mazac:2018ycv}, the actions of the functionals are constructed as contour integrals against certain weight functions. For general $\\Delta_\\phi$ and $n$, evaluating these integrals to yield explicit expressions is still technically challenging. On the other hand, our methods for computing the decomposition coefficients of Witten diagrams have no restrictions on quantum numbers.\n By exploiting the relation (\\ref{functionalaction}) between the two, our techniques therefore provide a complementary way to compute the analytic functional actions. Our results for the decomposition coefficients can be easily assembled to give the general analytic functionals for arbitrary external dimension $\\Delta_\\phi$, and recursively for all $n$, which should be particularly useful for the numeric bootstrap application using the analytic functionals \\cite{Paulos1dnumeric}.\n\n\\subsubsection*{$6j$ symbols}\nMoreover, the recursive algorithm we formulated here may provide some help to the computation of the $6j$ symbol in general dimensions. In \\cite{Liu:2018jhs}, the $6j$ symbols in $d=1,2,4$ were computed using the Lorentzian inversion formula \\cite{Caron-Huot:2017vep}, and expressed in terms of hypergeometric functions ${}_4F_3$. However evaluating $6j$ symbols in other dimensions still remains a challenging open problem.\\footnote{The $6j$ symbols also admit an Mellin Barnes integral representation \\cite{Sleight:2018ryu}. The integral representation is in general quite complicated but simplifies in certain cases.} This is due to the fact that the explicit form of the conformal partial waves is not known in odd spacetime dimensions, while in even dimensions $d>4$ the Lorentzian inversion integral does not factorize. The $6j$ symbols are intimately related to the crossed channel decomposition of conformal partial waves. More precisely the $6j$ symbol can be viewed as the spectral density function for decomposing an t-channel conformal partial wave into the s-channel conformal partial waves. By eliminating the s-channel shadow conformal block and closing the contour, the encircled poles of the $6j$ symbol correspond to the double-trace operators and their residues give the crossed channel decomposition coefficients. Since we can in principle obtain all the double-trace OPE coefficients using our recursive algorithm, once we input the coefficients of the leading twist double-trace operators ({\\it e.g.} from Mellin space), it is natural to ask the following question: knowing the poles and residues, is there a convenient way to reverse engineer the $6j$ symbol? We will not further explore this problem in this paper, but the apparent advantage of such a method is that it is insensitive to the spacetime dimensions. \n\n\\subsubsection*{Boundary conformal field theories}\nThe techniques we discussed in this paper also admit a straightforward extension to boundary conformal field theories. The simplest holographic setup for BCFT is obtained by taking a half of the $AdS_{d+1}$ space which ends on a $AdS_d$ boundary. We further require fields in $AdS_{d+1}$ to satisfy Neumann boundary condition on $AdS_d$. Two-point functions on the conformal boundary $AdS_{d+1}$ now become the simplest non-trivial objects to study. There are two types of exchange Witten diagrams for two-point functions, namely, the bulk channel exchange Witten diagram and the boundary channel exchange Witten diagrams. These exchange Witten diagrams have similar decomposition properties to those of the four-point functions, {\\it i.e.}, only double-trace operators appear in the crossed channel and both the single-trace operator and double-trace operators appear in the direct channel \\cite{Rastelli:2017ecj}. The equation of motion operators and properties of conformal blocks allow us to similarly formulate recursive algorithms for solving the crossed channel decomposition coefficients of exchange Witten diagrams. We discuss the BCFT extension in a separate publication \\cite{bcftpolyakov}, where we also use the decomposition coefficients to perform Polyakov style bootstrap \\cite{Polyakov:1974gs}.\n\n\n\nThere are also other extensions worth exploring. One is to repeat the analysis for four-point Witten diagrams with external spinning operators. Another question is whether one can also find similar recursive techniques for decomposing AdS loop diagrams. Finally, it would also be interesting to incorporate supersymmetry which presumably will further facilitate the extraction of CFT data from holographic correlators in supersymmetric backgrounds. \n\n\\acknowledgments\nI thank Dalimil Maz\\'a\\v{c} and for discussions and collaboration on a related project \\cite{bcftpolyakov}. I am grateful to Dalimil Maz\\'a\\v{c}, Wolfger Peelaers, Leonardo Rastelli and especially Massimo Taronna for carefully reading the draft and helpful comments. I also thank Rajesh Gopakumar, Eric Perlmutter, Jo\\~ao Penedones, David Simmons-Duffin, Massimo Taronna, Balt van Rees and other participants of the Bootstrap 2018 for useful conversations and comments on the work. I wish to thank the California Institute of Technology for the great hospitality during the Bootstrap 2018 workshop where part of this work was finished. This work is supported in part by NSF Grant PHY-1620628.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA \\emph{fiber graph} is a graph on the finitely many lattice points\n$\\cF\\subset\\ZZ^d$ of a polytope where two lattice points are connected\nby an edge if their difference lies in a finite set of allowed moves\n$\\cM\\subset\\ZZ^d$. The implicit\nstructure of these graphs makes them a useful tool to explore the set of lattice\npoints randomly: At the current lattice point $u\\in\\cF$, an\nelement $m\\in\\pm\\cM$ is sampled and the random walk moves along $m$ if\n$u+m\\in\\cF$ and stays at $u$ otherwise. The corresponding Markov chain\nis irreducible if the underlying fiber graph is connected and the set\n$\\cM$ is called a \\emph{Markov basis} for $\\cF$ in this case. \nThis paper investigates the \\emph{heat-bath} version of this random\nwalk: At the current lattice point $u\\in\\cF$, we sample $m\\in\\cM$ and\nmove to a random element in the integer ray $(u+\\ZZ\\cdot m)\\cap\\cF$.\nThe authors of~\\cite{Diaconis1998a} discovered that this random\nwalk can be seen as a discrete version of the \\emph{hit-and-run}\nalgorithm~\\cite{lovasz1999,Vempala2005,Lovasz2006} that has been used frequently\nto sample from all the points of a polytope -- not only from its\nlattice points. The popularity of the continuous version of\nthe hit-and-run algorithm has not spread to its discrete analog, and\nnot much is known about its mixing behaviour. One reason is that it is\nalready challenging to guarantee that all points\nin the underlying set $\\cF$ can be reached by a random walk that uses\nmoves from $\\cM$, whereas for the continuous version, a random sampling\nfrom the unit sphere suffices. However, in many situations where a\nMarkov basis is known, the heat-bath random walk is evidently fast.\nFor instance, it was shown in~\\cite{Cryan2002} that the heat-bath\nrandom walk on contingency tables mixes rapidly when the number of\ncolumns is fixed.\nTo work around the connectedness issue, a \\emph{discrete\nhit-and-run} algorithm was introduced in~\\cite{Baumert2009} for\narbitrary finite sets $\\cF\\subset\\ZZ^d$. \nAt each step in this random walk, a subordinate and unrestricted\nrandom walk starts at the current lattice point $u \\in \\cF$ and uses\nthe unit vectors to collect a set of proposals $S \\subset \\ZZ^d$. The\nrandom walk then moves from $u$ to a random point in $S \\cap \\cF$.\n\nRandom walks of the heat-bath type, such as the one presented above, have been\nstudied recently in~\\cite{Dyer2014} in a more general context. In this\npaper, we explore the mixing behaviour of heat-bath random walks on\nthe lattice points of polytopes with Markov bases. Throughout, we\nassume that a Markov basis has been found already and refer to the\nrelevant literature for their\ncomputation~\\cite{Sturmfels1996,sullivant2003,hemmecke2005,malkin2007,hara2010,Rauh2014}.\nWe call the underlying graph of the heat-bath\nrandom walk a \\emph{compressed fiber graph}\n(Definition~\\ref{d:CompressedFibergraph}) and determine in\nSection~\\ref{s:Diameter} bounds on its graph-diameter. We prove that \nfor any $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$, the\ndiameter of compressed fiber graphs on $\\{u\\in\\NN^d:\nAu=b\\}$ that use a fixed Markov bases\n$\\cM\\subset\\ker_\\ZZ(A)$ is bounded from above by a constant as $b$ varies\n(Theorem~\\ref{t:DiameterCompressedFiberGraphs}). In contrast, we \nshow that the diameter of conventional fiber graphs grow linearly\nunder a dilation of the underlying polytope\n(Remark~\\ref{r:RaysOfMatrices}). This gives\nrise to slow mixing results for conventional fiber walks as observed\nin~\\cite{windisch2015-mixing}.\nIn Section~\\ref{s:HeatBath}, we study in more detail the combinatorial\nand analytical structure of the transition matrices of heat-bath random\nwalks on lattice points and prove upper and lower bounds on their\nsecond largest eigenvalues. We also discuss how the distribution on\nthe moves $\\cM$ affects the speed of convergence\n(Example~\\ref{ex:SolvingSLEMOpti}).\nTheorem~\\ref{t:MixingofAugmentingMarkovBases} establishes with\nthe \\emph{canonical path approach} from~\\cite{Sinclair1992} an upper\nbound on the second largest eigenvalue when the Markov basis is\n\\emph{augmenting} (Definition~\\ref{d:Augmentation}) and the stationary\ndistribution is uniform. From that, we conclude fast mixing\nresults for random walks on lattice points in fixed dimension.\n\n\\subsection*{Acknowledgements}\nCS was partially supported by the US National Science Foundation (DMS\n0954865). TW gratefully acknowledges the support received from the\nGerman National Academic Foundation. \n\n\\subsection*{Conventions and Notation}\nThe natural numbers are $\\NN:=\\{0,1,2,\\ldots\\}$ and for any $N\\in\\NN$,\n$\\NN_{> N}:=\\{n\\in\\NN: n> N\\}$ and $\\NN_{\\ge N}:=\\{N\\}\\cup \\NN_{> N}$.\nFor $n\\in\\NN_{>0}$, let $[n]:=\\{1,\\ldots,n\\}$. Let $\\cM\\subset\\QQ^d$\nbe a finite set, then $\\ZZ\\cdot\\cM:=\\{\\lambda m: m\\in\\cM,\n\\lambda\\in\\ZZ\\}$ and $\\NN\\cM$ is the affine semigroup in $\\ZZ^d$\ngenerated by $\\cM$. For an integer matrix $A\\in\\ZZ^{m\\times d}$ with\ncolumns $a_1,\\dots,a_d\\in\\ZZ^m$, we write $\\NN\nA:=\\NN\\{a_1,\\ldots,a_d\\}$. A\ngraph is always undirected and can have multiple loops.\nThe distance of two nodes $u,v$ which are contained in the same\nconnected component of a graph $G$, i.e. the number of\nedges in a shortest path between $u$ and $v$ in $G$, is denoted by\n$\\dist{G}{u}{v}$. We set $\\dist{G}{u}{v}:=\\infty$ if $u$\nand $v$ are disconnected. A mass\nfunction on a finite set $\\Omega$ is a map $f:\\Omega\\to[0,1]$ such\nthat $\\sum_{\\omega\\in\\Omega}f(\\omega)=1$. A mass function $f$ on\n$\\Omega$ is \\emph{positive} if $f(\\omega)>0$ for all\n$\\omega\\in\\Omega$. A set $\\cF\\subset\\ZZ^d$ is \\emph{normal} if it there\nexists a polytope $\\cP\\subset\\QQ^d$ such that $\\cP\\cap\\ZZ^d=\\cF$. \n\n\\section{Graphs and statistics}\n\nWe first introduce the statistical framework in which this paper\nlives and recall important aspects of the interplay between graphs\nand statistics. A \\emph{random walk} on a graph $G=(V,E)$ is a map\n$\\cH:V\\times V\\to[0,1]$ such that for all $v\\in V$, $\\sum_{u\\in\nV}\\cH(v,u)=1$ and such that $\\cH(v,u)=0$ if $\\{v,u\\}\\not\\in E$. \nWhen there is no ambiguity, we represent a random walk as an\n$|V|\\times|V|$-matrix, for example when it is clear how the elements\nof $V$ are ordered. Fix a random walk $\\cH$ on $G$. Then $\\cH$ is\n\\emph{irreducible} if for all $v,u\\in V$ there exists $t\\in\\NN$ such\nthat $\\cH^t(v,u)>0$. The random\nwalk $\\cH$ is \\emph{reversible} if there exists a mass function\n$\\mu:V\\to[0,1]$ such that $\\mu(u)\\cdot \\cH(u,v)=\\mu(v)\\cdot \\cH(v,u)$ for\nall $u,v\\in V$ and \\emph{symmetric} if $\\cH$ is a symmetric map. A\nmass function $\\pi:V\\to[0,1]$ is a \\emph{stationary distribution}\nof $\\cH$ if $\\pi\\circ \\cH =\\pi$. For symmetric random walks, the uniform\ndistribution on $V$ is always a stationary distribution. \nIf $|V|=n$, then we denote the eigenvalues of $\\cH$ by\n$1=\\lambda_1(\\cH)\\ge\\lambda_2(\\cH)\\ge\\dots\\ge\\lambda_n(\\cH)\\ge -1$ and we\nwrite $\\lambda(\\cH):=\\max\\{\\lambda_2(\\cH),-\\lambda_n(\\cH)\\}$ for the\n\\emph{second largest eigenvalue modulus} of $\\cH$. Any irreducible\nrandom walk has a unique stationary\ndistribution~\\cite[Corollary~1.17]{levin2008} and $\\lambda(\\cH)\\in[0,1]$\nmeasures the convergence rate: the smaller $\\lambda(\\cH)$,\nthe faster the convergence.\n\nThe aim of this paper is to study random walks on lattice points that\nuse a set of moves. Typically, this is achieved by constructing a\ngraph on the set of lattice points as follows (compare\nto~\\cite[Section~1.3]{drton2008} and~\\cite[Chapter~5]{Sturmfels1996}).\n\n\\begin{defn}\\label{d:FiberGraphs}\nLet $\\cF\\subset\\ZZ^d$ be a finite set and $\\cM\\subset\\ZZ^d$. The graph\n$\\cF(\\cM)$ is the graph on $\\cF$ where two nodes $u,v\\in\\cF$ are adjacent\nif $u-v\\in\\cM$ or $v-u\\in\\cM$. \n\\end{defn}\n\nA normal set $\\cF\\subset\\ZZ^d$ is finite and satisfies\n$\\cF=\\conv{\\cF}\\cap\\ZZ^d$. A canonical class of normal sets that arise\nin many applications, is given by the fibers of an integer matrix:\n\n\\begin{defn}\\label{d:Fibers}\nLet $A\\in\\ZZ^{m\\times d}$ and $b\\in\\cone{A}$. The set\n$\\fiber{A}{b}:=\\{u\\in\\NN^d: Au=b\\}$ is the \\emph{$b$-fiber} of $A$.\nThe collection of all fibers of $A$ is $\\cP_A:=\\{\\fiber{A}{b}:\nb\\in\\NN A\\}$. For $\\cM\\subset\\ker_\\ZZ(A)$, the graph\n$\\fibergraph{A}{b}{\\cM}$ is a \\emph{fiber graph}.\n\\end{defn}\n\nLet $\\cF,\\cM\\subset\\ZZ^d$ be finite. If the membership in $\\cF$ can be\nverified efficiently -- for instance when $\\cF$ is given implicitly by\nlinear equations and inequalities -- then it is possible to explore\n$\\cF$ randomly using $\\cM$ as follows: At a given node $v\\in\\cF$, a\nuniform element $m\\in\\cM$ is selected. If $v+m\\in\\cM$, then the random\nwalk moves along $m$ to $v+m$ and if $v+m\\not\\in\\cM$, the we stay at\n$v$. Formally, we obtain the following random walk.\n\n\\begin{defn}\\label{d:SimpleFiberWalk}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be two finite sets. The\n\\emph{simple walk} is the random walk on $\\cF(\\cM)$ where the\nprobability to traverse between to adjacent nodes $u$ and $v$ is\n$|\\pm\\cM|^{-1}$ and the probability to stay at a node $u$ is \n$|\\{m\\in\\pm\\cM: u+m\\not\\in\\cF\\}|\\cdot|\\pm\\cM|^{-1}$.\n\\end{defn}\n\nThe simple walk is symmetric and hence the uniform distribution is a\nstationary distribution (see also~\\cite[Section~2]{windisch2015-mixing}). To\nensure convergence, the random walk has to be irreducible, that is,\nthe underlying graph has to be connected. The following definition is\na slight adaption of the generalized Markov basis as defined\nin~\\cite[Definition~1]{Rauh2014}.\n\n\\begin{defn}\\label{d:Markovbasis}\nLet $\\cP$ be a collection of finite subsets of $\\ZZ^d$. A finite set\n$\\cM\\subset\\ZZ^d$ is a \\emph{Markov basis} of $\\cP$, if for all\n$\\cF\\in\\cP$, $\\cF(\\cM)$ is a connected graph. \n\\end{defn}\n\nWe refer to~\\cite[Theorem~3.1]{Diaconis1998a} for a proof that for collections $\\cP_A$,\na finite Markov basis always exists and can be computed with tools\nfrom commutative algebra (see also~\\cite{hemmecke2005} for more on the\ncomputation of Markov bases). We now introduce a construction of\ngraphs on lattice points that also give rise to implementable random\nwalks, but whose edges have far more reach.\n\n\\begin{defn}\\label{d:CompressedFibergraph}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets. The\n\\emph{compression} of the graph $\\cF(\\cM)$ is the graph\n$\\cF^c(\\cM):=\\cF(\\ZZ\\cdot\\cM)$.\n\\end{defn}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[b]{0.45\\textwidth} \n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n \\foreach \\X in {0,1,2,3,4,5}{\n \\draw[dotted](\\X,0) -- (\\X,5);\n }\n\n \\foreach \\Y in {0,1,2,3,4,5}{\n \\draw[dotted](0,\\Y) -- (5,\\Y);\n }\n\n \n \n \n\n \n \n\n\n \\draw[thick](1,3) --(1,4);\n \\draw[thick](2,2) --(2,3);\n \\draw[thick](3,1) --(3,2);\n\n\n \\draw[thick](4,1) -- (1,4);\n \\draw[thick](3,1) -- (1,3);\n\n\n \\draw [fill=gray, opacity=0.25] (1,3) --(1,4) -- (4,1) --(3,1) --cycle; \n\n\t\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n \\foreach \\X in {0,1,2,3,4,5}{\n \\draw[dotted](\\X,0) -- (\\X,5);\n }\n\n \\foreach \\Y in {0,1,2,3,4,5}{\n \\draw[dotted](0,\\Y) -- (5,\\Y);\n }\n\n \n \n \n\n \n \n\n\n \\draw[thick](1,3) --(1,4);\n \\draw[thick](2,2) --(2,3);\n \\draw[thick](3,1) --(3,2);\n\n\n\n \\draw[thick,bend angle=40, bend right](4,1) to (1,4);\n \\draw[thick,bend angle=40, bend right](4,1) to (2,3);\n \\draw[thick,bend angle=40, bend right](3,2) to (1,4);\n \\draw[thick,bend angle=40, bend left](3,1) to (1,3);\n\n \\draw[thick] (4,1) -- (1,4);\n \\draw[thick] (3,1) -- (1,3);\n\n\n\n\n\n \\draw [fill=gray, opacity=0.25] (1,3) --(1,4) -- (4,1) --(3,1) --cycle; \n\n\t\\end{tikzpicture}\n\\end{minipage}\n\\end{minipage}\n\\begin{minipage}[b]{0.45\\textwidth} \n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n \\foreach \\X in {0,1,2,3,4,5}{\n \\draw[dotted](\\X,0) -- (\\X,5);\n }\n\n \\foreach \\Y in {0,1,2,3,4,5}{\n \\draw[dotted](0,\\Y) -- (5,\\Y);\n }\n\n \n \n \n\n \n \n\n\n \\draw[thick](1,1) to (1,4);\n \\draw[thick](2,1) to (2,3);\n \\draw[thick](3,1) to (3,2);\n\n \\draw[thick](1,1) to (4,1);\n \\draw[thick](1,2) to (3,2);\n \\draw[thick](1,3) to (2,3);\n\n\n\n \\draw [fill=gray, opacity=0.25] (1,1) --(1,4) -- (4,1) --cycle; \n\n\t\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[b]{0.45\\textwidth} \n\\centering\n\t\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,3) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (1,4) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,2) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (2,3) {};\n\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,1) {};\n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (3,2) {};\n \n\t\t\\node [fill, circle, inner sep=1.5pt](b2) at (4,1) {};\n\n\n \\foreach \\X in {0,1,2,3,4,5}{\n \\draw[dotted](\\X,0) -- (\\X,5);\n }\n\n \\foreach \\Y in {0,1,2,3,4,5}{\n \\draw[dotted](0,\\Y) -- (5,\\Y);\n }\n\n \n \n \n\n \n \n\n\n \\draw[thick](1,1) to (1,4);\n \\draw[thick](2,1) to (2,3);\n \\draw[thick](3,1) to (3,2);\n\n \\draw[thick](1,1) to (4,1);\n \\draw[thick](1,2) to (3,2);\n \\draw[thick](1,3) to (2,3);\n\n \\draw[thick,bend angle=40, bend left](1,1) to (1,4);\n \\draw[thick,bend angle=25, bend left](1,1) to (1,3);\n \\draw[thick,bend angle=25, bend left](1,2) to (1,4);\n\n \\draw[thick,bend angle=40, bend right](1,1) to (4,1);\n \\draw[thick,bend angle=25, bend right](1,1) to (3,1);\n \\draw[thick,bend angle=25, bend right](2,1) to (4,1);\n\n \\draw[thick,bend angle=25, bend right](1,2) to (3,2);\n \\draw[thick,bend angle=25, bend left](2,1) to (2,3);\n\n\n\n \\draw [fill=gray, opacity=0.25] (1,1) --(1,4) -- (4,1) --cycle; \n\n\t\\end{tikzpicture}\n \\end{minipage}\n \\end{minipage}\n\t\\caption{\\label{f:CompressedFibergraph}Compressing graphs.}\n\\end{figure}\n\nCompressing a graph $\\cF(\\cM)$ preserves its connectedness: $\\cF(\\cM)$\nis connected if and only if $\\cF^c(\\cM)$ is connected.\n\n\\section{Bounds on the diameter}\\label{s:Diameter}\n\nIn general knowledge of the diameter of the graph underlying a Markov\nchain can provide information about the mixing time. For random walks\non fiber graphs, the chains which we consider, the underlying graph\ncoincides with the fiber graph. In this section, we determine lower\nand upper bounds on the diameter of fiber graphs and their compressed\ncounterparts. For a finite set\n$\\cM\\subset\\ZZ^d$ and any norm $\\|\\cdot\\|$ on $\\RR^d$, let $\\|\\cM\\|:=\\max_{m\\in\\cM}\\|m\\|$.\n\n\\begin{lemma}\\label{l:LowerBound}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets, then\n\\begin{equation*}\n\\diam{\\cF(\\cM)}\\ge\\frac{1}{\\|\\cM\\|}\\cdot\\max\\{\\|u-v\\|: u,v\\in\\cF\\}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nIf $\\cF(\\cM)$ is not connected, then the statement holds\ntrivially, so assume that $\\cM$ is a Markov basis for $\\cF$. Let $u',v'\\in\n\\cF$ such that $\\|u'-v'\\|=\\max\\{\\|u-v\\|: u,v\\in\\cF\\}$ and let\n$m_1,\\dots,m_r\\in\\cM$ so that $u'=v'+\\sum_{i=1}^rm_i$ is a path of\nminimal length, then $\\|u'-v'\\|\\le r\\cdot\\|\\cM\\|$ and the claim follows\nfrom $\\diam{\\cF(\\cM)}\\ge\\dist{\\cF(\\cM)}{u'}{v'}=r$.\n\\end{proof}\n\n\\begin{remark}\\label{r:LatticeWidth}\nLet $\\cF\\subset\\ZZ^d$ be a normal set. For all\n$l\\in\\{-1,0,1\\}^d$ and $u,v\\in\\cF$ we have $(u-v)^Tl\\le\\|u-v\\|_1$ and\nthus $\\mathrm{width}_l(\\cF):=\\max\\{(u-v)^Tl:\nu,v\\in\\cF\\}\\le\\max\\{\\|u-v\\|_1: u,v\\in\\cF\\}$. \nSuppose that $u',v'\\in\\cF$ are such that $\\|u'-v'\\|_1=\\max\\{\\|u-v\\|_1:\nu,v\\in\\cF\\}$ and let\n$l'_i:=\\mathrm{sign}(u'_i-v'_i)$ for $i\\in[d]$, then\n\\begin{equation*}\n\\|u'-v'\\|_1= (u'-v')^T\\cdot l'\\le\\mathrm{width}_{l'}(\\cF)\\le\\max\\{\\|u-v\\|_1:\nu,v\\in\\cF\\}=\\|u'-v'\\|_1.\n\\end{equation*}\nThe \\emph{lattice width} of $\\cF$ is\n$\\mathrm{width}(\\cF):=\\min_{l\\in\\ZZ^d}\\mathrm{width}_l(\\cF)$ and thus\nLemma~\\ref{l:LowerBound} gives\n\\begin{equation*}\n\\|\\cM\\|_1\\cdot\\diam{\\cF(\\cM)}\\ge\\mathrm{width}(\\cF).\n\\end{equation*}\n\\end{remark}\n\n\\begin{defn}\nLet $\\cP$ be a collection of finite subsets of $\\ZZ^d$. A\nfinite set $\\cM\\subset\\ZZ^d$ is\n\\emph{norm-like} for $\\cP$ if there exists a constant $C\\in\\NN$ such\nthat for all $\\cF\\in\\cP$ and all $u,v\\in\\cF$, $\\dist{\\cF(\\cM)}{u}{v}\\le\nC\\cdot\\|u-v\\|$. The set $\\cM$ is $\\|\\cdot\\|$-\\emph{norm-reducing} for $\\cP$ if for\nall $\\cF\\in\\cP$ and all $u,v\\in\\cF$ there exists $m\\in\\cM$ such that\n$u+m\\in\\cF$ and $\\|u+m-v\\|<\\|u-v\\|$.\n\\end{defn}\n\nThe property of being norm-like does not depend on the norm, whereas\nbeing norm-reducing does. Norm-reducing sets are always norm-like, and\nnorm-like sets are in turn always Markov bases, but the reverse of both\nstatements is false in general (Example~\\ref{ex:NonNormLikeMB}\nand Example~\\ref{ex:NonNormReducingNormLike}). For collections $\\cP_A$\nhowever, every Markov basis is norm-like\n(Proposition~\\ref{p:EveryMBIsNormLike}).\n\n\\begin{example}\\label{ex:NonNormLikeMB}\nFor any $n\\in\\NN$, consider the normal set\n$\\cF_n:=([2]\\times[n]\\times\\{0\\})\\cup\\{(2,n,1)\\}$\nwith the Markov basis $\\{(0,1,0),(0,0,1),(-1,0,-1)\\}$. The distance\nbetween $(1,1,0)$ and $(2,1,0)$ in $\\cF_n(\\cM)$ is $2n$ and thus $\\cM$ is\nnot norm-like for $\\{\\cF_n:n\\in\\NN\\}$ (see also\nFigure~\\ref{f:NonNormLikeMB}).\n\\end{example}\n\n\\begin{example}\\label{ex:NonNormReducingNormLike}\nLet $d\\in\\NN$ and consider $A:=(1,\\dots,1)\\in\\ZZ^{1\\times d}$, then the\nset $\\cM:=\\{e_1-e_i: 2\\le i\\le d\\}$ is a Markov basis for\nthe collection $\\cP_{A}$. However, $\\cM$ is not\n$\\|\\cdot\\|_p$-norm-reducing for any $d\\ge\n3$ and any $p\\in[1,\\infty]$. For\ninstance, consider $e_2$ and $e_3$ in\n$\\fibergraph{A}{1}{\\cM}$. The only move from $\\cM$ that can be applied\non $e_2$ is $e_1-e_2$, but $\\|(e_2+e_1-e_2)-e_3)\\|_p=\\|e_2-e_3\\|_p$.\nOn the other hand,\nin the case we cannot find a move that decreases the\n$1$-norm of two nodes $u,v\\in\\fiber{A}{b}$ by $1$, we can\nfind instead two moves $m_1,m_2\\in\\cM$ such that\n$u+m_1,u+m_1+m_2\\in\\fiber{A}{b}$ and $\\|u+m_1+m_2-v\\|=\\|u-v\\|-2$.\nThus, the graph-distance of any two elements $u$ and $v$ in\n$\\fibergraph{A}{b}{\\cM}$ is at most $\\|u-v\\|_1$ and hence $\\cM$ is\nnorm-like for $\\cP_A$.\n\\end{example}\n\n\\begin{figure}[htbp]\n\\tdplotsetmaincoords{70}{110}\n\\begin{tikzpicture}[scale=0.6,tdplot_main_coords]\n\n\\def3{3}\n\\def8{8}\n\\def1{1}\n\n\\draw[thick,->] (0,0,0) -- (3+0.25,0,0) node[anchor=north east]{};\n\\draw[thick,->] (0,0,0) -- (0,8+0.25,0) node[anchor=north west]{};\n\\draw[thick,->] (0,0,0) -- (0,0,1) node[anchor=south]{};\n\n\\foreach \\X in {0,...,3\n {\\draw[dotted,color=black] (\\X,0,0) --(\\X,8,0) node[anchor=north]{};}\n\\foreach \\Y in {0,...,8} \n {\\draw[dotted,color=black] (0,\\Y,0) --(3,\\Y,0) node[anchor=north]{};}\n\n\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,1,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,2,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,3,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,4,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,5,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,6,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (1,7,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,1,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,2,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,3,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,4,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,5,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,6,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,7,0) {};\n\\node[draw,circle,inner sep=0.05cm,fill=black] () at (2,7,1) {};\n\n\n\\draw[thick] (1,1,0) -- (1,7,0) ;\n\\draw[thick] (1,7,0) -- (2,7,1) ;\n\\draw[thick] (2,7,1) -- (2,7,0) ;\n\\draw[thick] (2,7,0) -- (2,1,0) ;\n\n\\end{tikzpicture}\n\\caption{The graph from\nExample~\\ref{ex:NonNormLikeMB}}\\label{f:NonNormLikeMB}\n\\end{figure}\n\n\\begin{remark}\\label{r:NormLikeUpperDiamBound}\nLet $\\cP$ be a collection of finite subsets of $\\ZZ^d$ and\n$\\cM\\subset\\ZZ^d$ be norm-like for $\\cP$. It follows\nfrom the definition that there exists a constant $C\\in\\QQ_{\\ge 0}$ such that \nfor all $\\cF\\in\\cP$\n$$\\diam{\\cF(\\cM)}\\le C\\cdot\\max\\{\\|u-v\\|: u,v\\in\\cF\\}.$$\n\\end{remark}\n\nThe proof of our next results uses the \\emph{Graver basis}\n$\\graver{A}\\subset\\ZZ^d$ for an integer matrix $A\\in\\ZZ^{m\\times d}$\nwith $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$. We refer to\n\\cite[Chapter~3]{Loera2013} for a precise definition.\n\n\\begin{prop}\\label{p:EveryMBIsNormLike}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and\n$\\cM\\subset\\ker_\\ZZ(A)$ be a Markov basis of $\\cP_A$. Then $\\cM$ is\nnorm-like for $\\cP_A$.\n\\end{prop}\n\\begin{proof}\nLet $\\cM$ be a Markov basis for $\\cP_A$.\nThe Graver basis $\\graver{A}$ for $A$ is a finite set which is\n$\\|\\cdot\\|_1$-norm-reducing for $\\cP_A$. \nThus, define\n$C:=\\max_{g\\in\\graver{A}}\\diam{\\fibergraph{A}{Ag^+}{\\cM}}$.\nNow, pick $u,v\\in\\fiber{A}{b}$ arbitrarily and let\n$u=v+\\sum_{i=1}^r g_i$ be a walk from $u$ to $v$ in\n$\\fibergraph{A}{b}{\\graver{A}}$ of minimal length. Since the Graver\nbasis is norm-reducing for $\\fiber{A}{b}$, there always exists a path\nof length at most $\\|u-v\\|_1$ and hence $r\\le\\|u-v\\|_1$. Every $g_i$\ncan be replaced by a path in $\\fibergraph{A}{Ag_i^+}{\\cM}$ of length\nat most $C$ and these paths stay in $\\fiber{A}{b}$. This gives a path\nof length $C\\cdot r$, hence $\\dist{\\fibergraph{A}{b}{\\cM}}{u}{v}\\le\nC\\|u-v\\|_1$.\n\\end{proof}\n\n\n\n\\begin{prop}\\label{p:DiamBounds}\nLet $\\cP\\subset\\ZZ^d$ be a polytope with $\\dim(\\cP\\cap\\ZZ^d)>0$ and let $\\cM$ be a Markov\nbasis for $\\cF_i:=(i\\cdot\\cP)\\cap\\ZZ^d$ for all $i\\in\\NN$.\nThere exists a constant $C'\\in\\QQ_{>0}$ such that for all $i\\in\\NN$,\n$C'\\cdot i\\le\\diam{\\cF_i(\\cM)}$. If $\\cM$ is norm-like for $\\{\\cF_i:\ni\\in\\NN\\}$, then there exists a constant $C\\in\\QQ_{>0}$ such that \n$\\diam{\\cF_i(\\cM)}\\le C\\cdot i$ for all $i\\in\\NN$.\n\\end{prop}\n\\begin{proof}\nFor the lower bound on the diameter, it suffices to show the existence\nof $C'$ such that $C'\\cdot i\\le\\max\\{\\|u-v\\|: u,v\\in\\cF_i\\}$ for all\n$i\\in\\NN$ due to Lemma~\\ref{l:LowerBound}. Since\n$\\dim(\\cP\\cap\\ZZ^d)>0$, we can pick distinct $w,w'\\in\n\\cP\\cap\\ZZ^d$. For all $i\\in\\NN$, $i\\cdot w,i\\cdot w'\\in\\cF_i$\nand hence $i\\cdot\\|w-w'\\|\\le\\max\\{\\|u-v\\|: u,v\\in\\cF_i\\}$. \n\nTo show the upper bound, assume that $\\cM$ is norm-like. It suffices\nto show that there exists $C\\in\\QQ_{\\ge 0}$ such that $\\max\\{\\|u-v\\|:\nu,v\\in \\cF_i\\}\\le i\\cdot C$ by Remark~\\ref{r:NormLikeUpperDiamBound}.\nNow, let $v_1,\\ldots,v_r\\in\\QQ^d$ such that\n$\\cP=\\conv{v_1,\\ldots,v_r}$ and define $C:=\\max\\{\\|v_s-v_t\\|: s\\neq\nt\\}$. \nSince $\\cF_i=(i\\cdot\n\\cP)\\cap\\ZZ^d\\subset\\conv{iv_1,\\ldots,iv_r}$ for all\n$i\\in\\NN$, we have\n$\\max\\{\\|u-v\\|: u,v\\in\\cF_i\\}\\le\\max\\{\\|iv_s-iv_t\\|: s\\neq\nt\\}\\le C\\cdot i$. \n\\end{proof}\n\n\\begin{remark}\\label{r:RaysOfMatrices}\nLet $A\\in\\ZZ^{m\\times n}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and let $\\cM$ be a Markov\nbasis for $\\cP_A$. Then $\\cM$ is norm-like due to\nProposition~\\ref{p:EveryMBIsNormLike} and thus for all $b\\in\\cone{A}$\nthere exists $C,C'\\in\\QQ_{\\ge 0}$ such that \n$$i\\cdot C'\\le\\diam{\\fibergraph{A}{ib}{\\cM}}\\le i\\cdot C$$ \nfor all $i\\in\\NN$. This generalizes for instance\n\\cite[Proposition~2.10]{potka2013}\nand~\\cite[Example~4.7]{windisch2015-mixing}, where linear diameters\non a ray in $\\cone{A}$ have been observed.\nThis also implies that the construction of\nexpanders from~\\cite[Section~4]{windisch2015-mixing} works for every\nright-hand side $b\\in\\cone{A}$.\n\\end{remark}\n\n\\begin{remark}\\label{r:Mixing}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$,\n$b\\in\\cone{A}$, and let $\\cM$ be a Markov basis\nfor $\\cP_A$. Proposition~\\ref{p:DiamBounds} provides a new proof that\nthe simple walk on $(\\fibergraph{A}{ib}{\\cM})_{i\\in\\NN}$ cannot mix\nrapidly. The lower bound on the diameter from Proposition~\\ref{p:DiamBounds}\nimplies, in general, the following upper bound on the edge-expansion\n(see for example~\\cite[Proposition~1.30]{gardam2012}):\n\\begin{equation*}\nh(\\fibergraph{A}{i\\cdot b}{\\cM})\\le\n|\\cM|\\left(\\exp\\left(\\frac{\\log|\\fiber{A}{i\\cdot b}|}{D\\cdot\ni}\\right)-1\\right).\n\\end{equation*}\nIn particular, the edge-expansion cannot be bounded from below by\n$\\Omega(\\frac{1}{p(i)})_{i\\in\\NN}$ for a polynomial $p\\in\\QQ[t]$ and\nsince $(|\\fiber{A}{i\\cdot\nb}|)_{i\\in\\NN}\\in\\mathcal{O}(i^r)_{i\\in\\NN}$, the simple walk cannot\nmix rapidly. In~\\cite{windisch2015-mixing}, it was shown that the\nedge-expansion can be bounded from above by\n$\\mathcal{O}(\\frac{1}{i})_{i\\in\\NN}$, which cannot be concluded from\nthe upper expression. \n\\end{remark}\n\nWe now turn our attention to the diameter of compressed fiber graphs.\nIn particular, we want to know for which collections of normal sets is\ntheir diameter bounded. In general, compressing a fiber\ngraph does not necessarily have an effect on the diameter\n(Example~\\ref{ex:StairCase}). \n\nAlthough a low diameter is a necessary\ncondition for good mixing, it is not sufficient. For instance, let\n$G_n$ be the disjoint union of two complete graphs $K_n$ connected by\na single edge. Then $\\diam{G_n}=3$, but $h(G_n)\\le\\frac{1}{n}$ implies\nthat the simple walk does not mix rapidly. \n\n\\begin{example}\\label{ex:StairCase}\nFor any $n\\in\\NN$, let\n$\\cF_n:=\\{(0,0),(0,1),(1,1),(1,2),\\dots,(n,n)\\}\\subset\\ZZ^2$. The\nunit vectors $\\cM=\\{e_1,e_2\\}$ are a Markov basis for\n$\\{\\cF_n:n\\in\\NN\\}$. However,\n$\\cF_n^c(\\cM)=\\cF_n(\\cM)$ and thus $\\diam{\\cF^c_n(\\cM)}=\\diam{\\cF_n(\\cM)}=2n$\nis unbounded.\n\\end{example}\n\n\\begin{lemma}\\label{l:SignCompatibility}\nLet $A\\in\\ZZ^{m\\times d}$ and $z\\in\\ker_\\ZZ(A)$. There exists\n$r\\in[2d-2]$, distinct elements $g_1,\\dots,g_r\\in\\cG_A$, and\n$\\lambda_1,\\dots,\\lambda_r\\in\\NN_{>0}$ such that\n$z=\\sum_{i=1}^r\\lambda_ig_i$ and $g_i\\sqsubseteq z$ for all $i\\in[r]$\n\\end{lemma}\n\\begin{proof}\nThis is \\cite[Lemma~3.2.3]{Loera2013}, although it only becomes clear\nfrom the original proof of~\\cite[Theorem~2.1]{Sebo1990} that the\nappearing elements are all distinct.\n\\end{proof}\n\n\\begin{prop}\\label{p:DiamCompressedGraverFiberGraphs}\nLet $A\\in\\ZZ^{m\\times d}$ and\n$\\cP:=\\left\\{\\{x\\in\\ZZ^d: Ax=b, l\\le x\\le u\\}:\nl,u\\in\\ZZ^d,b\\in\\ZZ^m\\right\\}$.\nThen for all $\\cF\\in\\cP$, \n$\\diam{\\cF^c(\\graver{A})}\\le 2d-2$.\n\\end{prop}\n\\begin{proof} Let $s,t\\in\\{x\\in\\ZZ^d: Ax=b, l\\le x\\le u\\}$, then\n$s-t\\in\\ker_\\ZZ(A)$ and thus $s=t+\\sum_{i=1}^r\\lambda_ig_i$ with $r\\le\n2d-2$, $\\lambda_1,\\dots,\\lambda_r\\in\\NN_{>0}$, and distinct\n$g_1,\\dots,g_r\\in\\graver{A}$ such that $g_i\\sqsubseteq s-t$ according\nto Lemma~\\ref{l:SignCompatibility}. It's now a\nconsequence from~\\cite[Lemma~3.2.4]{Loera2013} that all intermediate\npoints $t+\\sum_{i=1}^k\\lambda_ig_i$ for $k\\le r$ are in $\\{x\\in\\ZZ^d:\nAx=b, l\\le x\\le u\\}$.\n\\end{proof}\n\n\\begin{lemma}\\label{l:DistOfScaledMoves}\nLet $\\cF\\subset\\ZZ^d$ be finite and let \n$\\cF_i:=(i\\cdot\\conv{\\cF})\\cap\\ZZ^d$ for $i\\in\\NN$. For all $u,v\\in\\cF$,\n$\\dist{\\cF^c_i(\\cM)}{iu}{iv}\\le\\dist{\\cF(\\cM)}{u}{v}$ for all\n$i\\in\\NN$.\n\\end{lemma}\n\\begin{proof}\nThe statement is trivially true if $u$ and $v$ are disconnected in\n$\\cF(\\cM)$. Thus, assume the contrary and let $u=v+\\sum_{j=1}^k m_j$\nwith $m_j\\in\\cM$ be a path in $\\cF(\\cM)$ of length\n$k=\\dist{\\cF(\\cM)}{u}{v}$ and let $i\\in\\NN$. Clearly, $i\\cdot u=i\\cdot\nv+i\\cdot\\sum_{j=1}^km_j=i\\cdot v+ \\sum_{l=1}^ki\\cdot m_j$, so it is\nleft to prove that the elements traversed by this paths are in $\\cF_i$.\nLet $l\\in[k]$, since $v+\\sum_{j=1}^lm_j\\in\\cF$, we have $i\\cdot v+\\sum_{j=1}^l\ni\\cdot m_j\\in i\\cdot \\cF\\subseteq\\cF_i$. Hence, this is a path in\n$\\cF^c_i(\\cM)$ of length $k=\\dist{\\cF(\\cM)}{u}{v}$.\n\\end{proof}\n\nWe are ready to prove that the diameter of compressed fiber graphs\ncoming from an integer matrix can be bounded for all right-hand sides\nsimultaneously.\n\n\\begin{thm}\\label{t:DiameterCompressedFiberGraphs}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and let\n$\\cM$ be a Markov basis for $\\cP_A$. There exists a constant\n$C\\in\\NN$ such that $\\diam{\\cF^c(\\cM)}\\le C$ for all $\\cF\\in\\cP_A$.\n\\end{thm}\n\\begin{proof}\nOur proof relies on basic properties of the Graver basis $\\cG_A$ of\n$A$. For any $g\\in\\graver{A}$, let $\\cF_g:=\\fiber{A}{Ag^+}$\nand let\n$K:=\\max\\{\\dist{\\cF_g(\\cM)}{g^+}{g^-}: g\\in\\graver{A}\\}$. We\nshow that the diameter of any compressed fiber graph of $A$ is\nbounded from above by $(2d-2)\\cdot K$. \nLet $b\\in\\cone{A}$ arbitrary and choose elements $u,v\\in\\fiber{A}{b}$. \nAccording to Proposition~\\ref{p:DiamCompressedGraverFiberGraphs},\nthere exists $r\\in[2d-2]$, \n$g_1,\\ldots,g_{r}\\in\\graver{A}$ and\n$\\lambda_1,\\ldots,\\lambda_{r}\\in\\ZZ$ such that\n$u=v+\\sum_{i=1}^{r}\\lambda_i g_i$, and\n$v+\\sum_{i=1}^l\\lambda_ig_i\\in\\NN^d$ for all $l\\in[r]$. According to\nLemma~\\ref{l:DistOfScaledMoves}, for any $i\\in[r]$ there are\n$m_1^i,\\ldots,m_{k_i}^i\\in\\cM$ and\n$\\alpha_1,\\ldots,\\alpha_{k_i}\\in\\ZZ$ such that\n$\\lambda_ig_i^+=\\lambda_ig_i^-+\\sum_{j=1}^{k_i}\\alpha_j m_j^i$ is a\npath in the compression of $\\fiber{A}{A\\lambda_ig_i^+}(\\cM)$ of length $k_i\\le K$. Lifting these\npaths for every $i\\in[r]$ yields a path\n$u=v+\\sum_{i=1}^{r}\\sum_{j=1}^{k_i}\\alpha_jm_j^i$\nin $\\scfibergraph{A}{b}{\\cM}$ of length $r\\cdot K\\le(2d-2)\\cdot K$.\n\\end{proof}\n\n\\section{Heat-bath random walks}\\label{s:HeatBath}\n\nIn this section, we establish the heat-bath random walk on compressed\nfiber graphs. We refer to~\\cite{Dyer2014} for a more general\nintroduction on random walks of heat-bath type. Let\n$\\cF\\subset\\ZZ^d$ be finite set. For any $u\\in\\cF$ and\n$m\\in\\ZZ^d$, the\nray in $\\cF$ through $u$ along $m$ is denoted by\n$\\polyray{u}{m}{\\cF}:=(u+m\\cdot\\ZZ)\\cap\\cF$. \nAdditionally, given a mass function\n$\\pi:\\cF\\to[0,1]$, we define\n\\begin{equation*}\n\\heatbathmove{\\pi}{\\cF}{m}(x,y):=\\begin{cases}\n\\frac{\\pi(y)}{\\pi(\\polyray{x}{m}{\\cF})}&,\\text{ if\n}y\\in\\polyray{x}{m}{\\cF}\\\\\n0&,\\text{ otherwise}\n\\end{cases}\n\\end{equation*}\nfor $x,y\\in\\cF$. For $\\cM\\subset\\ZZ^d$ and a mass function $f:\\cM\\to[0,1]$, the\n\\emph{heat-bath random walk} is \n\\begin{equation}\\label{equ:HeatBath}\n\\heatbath{\\pi}{f}{\\cF}{\\cM}=\\sum_{m\\in\\cM}f(m)\\cdot\\heatbathmove{\\pi}{\\cF}{m}.\n\\end{equation}\nThe underlying graph of the heat-bath random walk is the compression\n$\\cF^c(\\cM)$ and in this section, we assume throughout that for all\n$m\\in\\cM$ and $\\lambda\\in\\ZZ\\setminus\\{-1,1\\}$, $\\lambda\\cdot\nm\\not\\in\\cM$. Let us first recall the basic properties of this random\nwalk (compare also to~\\cite[Lemma~2.2]{Diaconis1998a}).\n\n\\begin{algorithm}[h]\n\\caption{Heat-bath random walk on compressed fiber\ngraphs}\\label{a:HeatBath}\n\\begin{algorithmic}[1]\n\\Require{$\\cF\\subset\\ZZ^d$, $\\cM\\subset\\ZZ^d$,\n$v\\in\\cF$, mass functions\n$f:\\cM\\to[0,1]$ and $\\pi:\\cF\\to[0,1]$, $r\\in\\NN$}\n\n\\Procedure{HeatBath:}{}\n\\State $v_0:=v$ \n\\State \\texttt{FOR} $s=0$; $s=s+1$, $s0\\}$ is a Markov basis for $\\cF$.\n\\end{prop}\n\\begin{proof}\nSince for any $u\\in\\cF$ and any $m\\in\\cM$,\n$\\heatbathmove{\\pi}{\\cF}{m}(u,u)>0$, there are halting states and thus\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}$ is aperiodic. By definition,\n$\\pi(x)\\heatbathmove{\\pi}{\\cF}{m}(x,y)=\\pi(y)\\heatbathmove{\\pi}{\\cF}{m}(y,x)$\nand thus $\\heatbath{\\pi}{f}{\\cF}{\\cM}$ is reversible with\nrespect to $\\pi$ and $\\pi$ is a stationary distribution. The\nstatement on the eigenvalues is exactly \\cite[Lemma~1.2]{Dyer2014}.\nLet $\\cM'=\\{m\\in\\cM: f(m)>0\\}$ and $f'=f|_{\\cM'}$, then\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}=\\heatbath{\\pi}{f'}{\\cF}{\\cM'}$ and thus the\nheat-bath random walk is irreducible if and only if $\\cM'$ is a Markov\nbasis for $\\cF$.\n\\end{proof}\n\n\\begin{remark}\\label{r:ExecutionHeatBath}\nAnalyzing the speed of convergence of random walks with second largest\neigenvalues does not take the computation time of a single transition\ninto account. From a computational point of view, the\ndifference of the simple walk and the heat-bath random walk is \nStep~4 of Algorithm~\\ref{a:HeatBath}. However, we argue that Step~4\ncan be done efficiently in many cases. For instance, a hard\nnormalizing constant of $\\pi$ cancels out. If $\\pi$ is the uniform\ndistribution, then one needs to sample uniformly from\n$\\polyray{v}{m}{\\cF}$ in Step~4, which can be done efficiently. If the\ninput of Algorithm~\\ref{a:HeatBath} is a normal set $\\cF=\\{u\\in\\ZZ^d:\nAu\\le b\\}$ that is given in $\\mathcal{H}$-representation, then the\nlength of the ray $\\polyray{v}{m}{\\cF}$ can be computed with a number\nof rounding, division, and comparing operations that is linear in the\nnumber of rows of $A$. \n\\end{remark}\n\nThere are situations in which the heat-bath random walk provides no\nspeed-up compared with the simple walk (Example~\\ref{ex:NoSpeedup}).\nIntuitively, adding more moves to the set of allowed moves should\nimprove the mixing time of the random walk. In general, however, this\nis not true for the heat-bath walk\n(Example~\\ref{ex:AddingMovesSlowingChain}).\n\n\\begin{example}\\label{ex:NoSpeedup}\nFor $n\\in\\NN$, consider the normal set\n\\begin{equation*}\n\\cF_n:=\n\\left\\{\n\\begin{bmatrix}\n0 & 1 & 1 &\\cdots& 1\\\\\n1 & 0 & 0 &\\cdots& 0\\\\\n\\end{bmatrix},\n\\begin{bmatrix}\n1 & 0 & 1 &\\cdots& 1 \\\\\n0 & 1 & 0 &\\cdots& 0\\\\\n\\end{bmatrix},\n\\ldots,\n\\begin{bmatrix}\n1 & 1 &\\cdots & 1 & 0 \\\\\n0 & 0 &\\cdots & 0 & 1 \\\\\n\\end{bmatrix}\\right\\}\\subset\\QQ^{2\\times n}.\n\\end{equation*}\nIn the language of~\\cite[Section~1.1]{drton2008}, $\\cF_n$ is precisely\nthe fiber of the $2 \\times n$ independence model where row sums are\n$(n-1,1)$ and column sums are $(1,1,\\ldots,1)$. The minimal Markov\nbasis of the independence model, often referred to as the \\emph{basic\nmoves}, is precisely the set $\\cM_n:=\\{v-u: u,v\\in\n\\cF_n\\}\\setminus\\{0\\}$. In particular, the fiber graph $\\cF_n(\\cM_n)$ is\nthe complete graph on $n$ nodes. All rays along basic moves have\nlength $2$ and thus the transition matrices of the simple random walk\nand the heat-bath random walk coincide. There are $n\\cdot(n-1)$ many\nbasic moves and the transition matrix of both random walks is\n\\begin{equation*}\n\\frac{1}{n(n-1)}\n\\begin{bmatrix}\n1 & \\dots & 1 \\\\ \n\\vdots & & \\vdots \\\\ \n1 & \\dots & 1 \\\\ \n\\end{bmatrix}\n+\n\\frac{(n(n-1)-n)}{n(n-1)}\\cdot I_n.\n\\end{equation*}\nThe second largest eigenvalue is $1-\\frac{1}{n-1}$ which implies that\nfor $n\\to\\infty$, neither the simple walk nor the heat-bath random walk\nare rapidly mixing.\n\\end{example}\n\n\n\n\\begin{example}\\label{ex:AddingMovesSlowingChain}\nLet $\\cF=[2]\\times[5]\\subset\\ZZ^2$, $\\cM=\\{e_1,e_2,2e_1+e_2\\}$, and let\n$\\pi$ be the uniform distribution on $\\cF$. Since $\\{e_2,2e_1+e_2\\}$ is\nnot a Markov basis for $\\cF$, any mass function $f:\\cM\\to[0,1]$ must\nhave $f(e_1)>0$ in order to make the corresponding heat-bath random\nwalk irreducible. Comparing the second largest eigenvalue modulus of\nheat-bath random walks that sample from $\\{e_1,e_2\\}$ and $\\cM$\nuniformly, we obtain\n\\begin{equation*}\n\\lambda\\left(\\frac{1}{2}\\heatbathmove{\\pi}{\\cF}{e_1}+\\frac{1}{2}\\heatbathmove{\\pi}{\\cF}{e_2}\\right)=\\frac{1}{2}<\n\\frac{2}{3}=\\lambda\\left(\\frac{1}{3}\\heatbathmove{\\pi}{\\cF}{e_1}+\\frac{1}{3}\\heatbathmove{\\pi}{\\cF}{e_2}+\\frac{1}{3}\\heatbathmove{\\pi}{\\cF}{2e_1+e_2}\\right).\n\\end{equation*}\nSo, adding $2e_1+e_2$ to the set of allowed moves slows the\nwalk down. This phenomenon does not appear for the simple walk on\n$\\cF$, where the second largest eigenvalue modulus improves from\n$\\approx 0.905$ to $\\approx 0.888$ when adding the move $2e_1+e_2$ to\nthe Markov basis. \n\\end{example}\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[xscale=0.5,yscale=0.5]\n\n\t\t\\node [fill, circle, inner sep=2pt](D00) at (-6,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D10) at (-5,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D20) at (-4,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D30) at (-3,0) {};\n\t\t\\node [fill, circle, inner sep=2pt](D40) at (-2,0) {};\n\n\t\t\\node [fill, circle, inner sep=2pt](D01) at (-6,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D11) at (-5,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D21) at (-4,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D31) at (-3,1) {};\n\t\t\\node [fill, circle, inner sep=2pt](D41) at (-2,1) {};\n\n\t\t\\node [fill, circle, inner sep=2pt](D02) at (-6,2) {};\n\t\t\\node [fill, 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(C22);\n\\draw[thick] (C11) -- (C32);\n\\draw[thick] (C21) -- (C42);\n\n\t\\end{tikzpicture}\n \\caption{Decomposition of the graph in\n Example~\\ref{ex:AddingMovesSlowingChain}}\\label{f:AddingMovesSlowingChain}\n\\end{figure}\n\n\n\\begin{remark}\\label{r:GlauberDynamics}\nLet $\\cF\\subset\\ZZ^d$ be finite and $\\cM=\\{m_1,\\dots,m_d\\}\\subset\\ZZ^d$ be a\nlinearly independent Markov basis of $\\cF$. If the moves are selected\nuniformly, then the heat-bath random walk on $\\cF$ coincides with the\n\\emph{Glauber dynamics} on $\\cF$. To see it, choose $u\\in\\cF$ and let\n\\begin{equation*}\n\\cF':=\\{\\lambda\\in\\ZZ^d: u+\\lambda_1m_1+\\dots+\\lambda_dm_d\\in\\cF\\}.\n\\end{equation*}\nIt is easy to check that $\\cF'$ is unique up to translation and\ndepends only on $\\cF$, $u$, and $\\cM$. Since the vectors in $\\cM$ are linearly\nindependent, every element of $\\cF$ can be represented by a\nunique choice of coefficients in $\\cF'$. Thus, the heat-bath random walk\non $\\cF$ using $\\cM$ is equivalent to the heat-bath random walk on\non $\\cF'$ using the unit vectors as moves. For any unit vector\n$e_i\\in\\ZZ^d$, the ray through an element $v\\in\\cF'$ is $\\{w\\in\\cF:\nw_j=v_j \\forall j\\neq i\\}$ and this is precisely the form desired in\nthe Glauber dynamics~\\cite[Section~3.3.2]{levin2008}.\n\\end{remark}\n\nFor the remainder of this section, we primarily focus on heat-bath\nrandom walks $\\heatbath{\\pi}{f}{\\cF}{\\cM}$ that converge to the\nuniform distribution $\\pi$ on a finite, but not necessarily normal, set $\\cF$. We particularly\naim for bounds on its second largest eigenvalue by\nmaking use of the decomposition from equation~\\ref{equ:HeatBath}. Our\nfirst observations consider its summands $\\heatbathmove{\\pi}{\\cF}{m}$\nthat can be well understood analytically\n(Proposition~\\ref{p:MoveMatrix}) and combinatorially\n(Proposition~\\ref{p:IntersectionTheorem}). \n\n\\begin{prop}\\label{p:MoveMatrix}\nLet $\\cF\\subset\\ZZ^d$ be a finite set, $m\\in\\ZZ^d$, and $\\pi:\\cF\\to[0,1]$\nbe the uniform distribution. Let $\\cR_1,\\dots,\\cR_k$ be the disjoint\nrays through $\\cF$ along $m$. Then\n\\begin{enumerate}\n\\item $\\heatbathmove{\\pi}{\\cF}{m}$ is symmetric and idempotent.\n\\item\n$\\img(\\heatbathmove{\\pi}{\\cF}{m})=\\spann{\\sum_{x\\in\\cR_1}e_x,\\sum_{x\\in\\cR_2}e_x,\\dots,\\sum_{x\\in\\cR_k}e_x}$.\n\\item $\\ker(\\heatbathmove{\\pi}{\\cF}{m})=\\bigoplus_{i=1}^k\\spann{e_x-e_y:\nx,y\\in\\cR_i, x\\neq y}$.\n\\item $\\rank(\\heatbathmove{\\pi}{\\cF}{m})=k$ and\n$\\dim\\ker(\\heatbathmove{\\pi}{\\cF}{m})=|\\cF|-k$.\n\\item The spectrum of $\\heatbathmove{\\pi}{\\cF}{m}$ is $\\{0,1\\}$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nSymmetry of $\\heatbathmove{\\pi}{\\cF}{m}$ follows from the\ndefinition. By assumption, $\\cF$ is the disjoint union of\n$\\cR_1,\\dots,\\cR_k$ and hence there exists a permutation matrix\n$S$ such that $S\\heatbathmove{\\pi}{\\cF}{m}S^T$ is a block matrix whose\nbuilding blocks are the matrices \n\\begin{equation*}\n\\frac{1}{|\\cR_i|}\\begin{bmatrix}\n1 & \\dots & 1 \\\\ \n\\vdots & & \\vdots \\\\ \n1 & \\dots & 1 \\\\ \n\\end{bmatrix}\\in\\QQ^{|\\cR_i|\\times|\\cR_i|}.\n\\end{equation*}\nThus, $\\heatbathmove{\\pi}{\\cF}{m}$ is\nidempotent and the rank of $\\heatbathmove{\\pi}{\\cF}{m}$ is $k$. A basis\nof its image and its kernel can be read off directly and\nidempotent matrices can only have the eigenvalues $0$ and~$1$. \n\\end{proof}\n\n\\begin{prop}\\label{p:IntersectionTheorem}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets,\n$\\pi:\\cF\\to[0,1]$ be the uniform distribution, and let\n$V_1,\\dots,V_c\\subseteq\\cF$ be the nodes of the connected components of\n$\\cF(\\cM)$, then\n$$\\bigcap_{m\\in\\cM}\\img(\\heatbathmove{\\pi}{\\cF}m)=\\spann{\\sum_{x\\in\nV_1}e_x,\\dots,\\sum_{x\\in V_c}e_x}.$$\n\\end{prop}\n\\begin{proof}\nIt is clear by Proposition~\\ref{p:MoveMatrix} that the set\non the right-hand side is contained in any\n$\\img(\\heatbathmove{\\pi}{\\cF}m)$ since any $V_i$ decomposes disjointly into rays\nalong $m\\in\\cM$.\nTo show the other inclusion, write $\\cM=\\{m_1,\\dots,m_k\\}$ and let for any $i\\in[k]$,\n$\\cR^i_1,\\dots,\\cR^i_{n_i}$ be the disjoint rays through $\\cF$ parallel\nto $m_i$. In particular, $\\{\\cR^i_1,\\dots,\\cR^i_{n_i}\\}$ is a\npartition of $\\cF$ for any $i\\in[k]$. Let $v\\in\\bigcap_{m\\in\\cM}\\img(\\heatbathmove{\\pi}{\\cF}m)$.\nAgain by Proposition~\\ref{p:MoveMatrix}, there exists for any\n$i\\in[k]$, $\\lambda^i_1,\\dots,\\lambda^i_{n_i}\\in\\QQ$ such that\n\\begin{equation*}\nv=\\sum_{j=1}^{n_i}\\sum_{x\\in\\cR^i_j}\\lambda_j^ie_x.\n\\end{equation*}\nNotice that if two distinct Markov moves $m_i$ and $m_{i'}$\nand two indices $j\\in[n_i]$ and $j'\\in[n_{i'}]$\nsatisfy $\\cR_{j}^i\\cap\\cR^{i'}_{j'}\\neq\\emptyset$,\nthen $\\lambda^i_j=\\lambda^{i'}_{j'}$. We show that for\nany $i\\in[k]$ and any $a\\in[c]$, $\\lambda_{j}^i=\\lambda_{j'}^i$ when\n$\\cR^i_j$ and $\\cR^i_{j'}$ are a subset of $V_a$. This implies the\nproposition. So take distinct $x,x'\\in V_a$\nand assume that $x$ and $x'$ lie on different rays of\n$m_i$ and let that be $x\\in\\cR^i_{j}$ and $x'\\in\\cR^i_{j'}$ with\n$j\\neq j'$. Since $x$ and $x'$ are in the same connected component\n$V_a$ of $\\cF(\\cM)$, let $y_{i_0},\\dots,y_{i_r}\\in\\cF$ be the nodes on a\nminimal path in $\\cF^c(\\cM)$ with $y_{i_0}=x$ and $y_{i_r}=x'$. For any $s\\in[r]$,\n$y_{i_s}$ and $y_{i_{s-1}}$ are contained in the same ray\n$\\cR^{k_s}_{t_s}$ coming from a\nMarkov move $m_{k_s}$. In particular,\n$\\cR_{k_{s-1}}^{t_{s-1}}\\cap\\cR^{k_s}_{t_s}\\neq\\emptyset$ and due to\nour observation made above\n$\\lambda_{j}^i=\\lambda_{t_1}^{k_1}=\\lambda_{t_2}^{k_2}=\\dots=\\lambda_{t_r}^{k_r}=\\lambda_{j'}^i$\nwhich finishes the proof.\n\\end{proof}\n\n\\begin{defn}\\label{d:RayMatrix}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets and\n$\\cM'\\subseteq\\cM$. Let $\\mathcal{V}$ be the set of\nconnected components of $\\cF(\\cM\\setminus\\cM')$ and\n$\\cR$ be the set of all rays through $\\cF$ along all elements\nof $\\cM'$. The \\emph{ray matrix} of $\\cF(\\cM)$ along $\\cM'$\nis\n$\\rayMat{\\cF}{\\cM}{\\cM'}:=(|R\\cap\nV|)_{R\\in\\cR,V\\in\\cV}\\in\\NN^{\\cR\\times\\cV}$.\n\\end{defn}\n\n\\begin{example}\nLet $\\cF =[3]\\times [3]$, $\\cM=\\{e_1, e_2, e_1+e_2 \\}$, and $\\cM'= \\{\ne_1, e_2\\}$. Then $\\cF(\\cM\\setminus \\cM')$ has five connected\ncomponents and the ray matrix of $\\cF(\\cM)$ along\n$\\cM'$ is \n\\[ \\rayMat{\\cF}{\\cM}{\\cM'}=\n\\begin{bmatrix}\n1 & 1 & 1 & 0 & 0 \\\\ \n0 & 1 & 1 & 1 & 0 \\\\ \n0 & 0 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 1 & 1\\\\\n0 & 1 & 1 & 1 & 0\\\\\n1 & 1 & 1 & 0 & 0\n\\end{bmatrix}.\n\\]\n\\end{example}\n\n\\begin{remark}\\label{r:RayMatUnitVecTranspose}\nLet $\\cF\\subset\\ZZ^2$, then the rays through $\\cF$ along $e_1$ are the\nconnected components of $\\cF(\\{e_1,e_2\\}\\setminus\\{e_2\\})$ and the rays through $\\cF$\nalong $e_2$ are the connected components of\n$\\cF(\\{e_1,e_2\\}\\setminus\\{e_1\\})$, thus\n$\\rayMat{\\cF}{\\cM}{e_1}=\\rayMat{\\cF}{\\cM}{e_2}^T$.\n\\end{remark}\n\n\\begin{prop}\\label{p:RaysAndCC}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets,\n$\\pi:\\cF\\to[0,1]$ be the uniform distribution, and $\\cM'\\subseteq\\cM$.\nThen\n\\begin{equation*}\n\\ker(\\rayMat{\\cF}{\\cM}{\\cM'})\\cong\n\\bigcap_{m\\in\\cM\\setminus\\cM'}\\img(\\heatbathmove{\\pi}{\\cF}{m})\n\\cap\\bigcap_{m\\in\\cM'}\\ker(\\heatbathmove{\\pi}{\\cF}{m}).\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nLet $V_1,\\dots,V_c$ be the connected components of $\\cF(\\cM\\setminus\\cM')$ and\n$\\cR_1,\\dots,\\cR_r$ be the rays along elements in $\\cM'$. \nLet $I:=\\bigcap_{m\\in\\cM\\setminus\\cM'}\\img(\\heatbathmove{\\pi}{\\cF}{m})$ and\n$K:=\\bigcap_{m\\in\\cM'}\\ker(\\heatbathmove{\\pi}{\\cF}{m})$. By\nProposition~\\ref{p:IntersectionTheorem}, any element of $I$ has the\nform $v=\\sum_{i=1}^c(\\lambda_i\\sum_{x\\in V_i}e_x)$ for\n$\\lambda_1,\\dots,\\lambda_c\\in\\QQ$. Assume additionally that\n$v\\in\\ker(\\heatbathmove{\\pi}{\\cF}{m})$ for $m\\in\\cM'$ and let\n$\\cR_{i_1},\\dots\\cR_{i_j}$ be the rays which belong to $m$,\nthen for any $k\\in[j]$,\n$0=\\sum_{x\\in\\cR_{i_k}}v_x=\\sum_{j=1}^c\\lambda_j|\\cR_{i_k}\\cap V_j|$. Put \ndifferently, a vector $\\lambda\\in\\RR^c$ is in the kernel\nof $(|\\cR_i\\cap V_j|)_{i\\in[r],j\\in[c]}$ if and only if\n$\\sum_{i=1}^c(\\lambda_i\\sum_{x\\in V_i}e_x)\\in I\\cap K$. \n\\end{proof}\n\nConditions on the kernel of the ray matrix allow us to give a lower bound on the \nsecond largest eigenvalue of the heat-bath random walk.\n\n\\begin{prop}\\label{p:KernelRayMat}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets and\n$\\pi$ be the uniform distribution. Let\n$\\cM'\\subseteq\\cM$ such that $\\ker(\\rayMat{\\cF}{\\cM}{\\cM'})\\neq\\{0\\}$, then $\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\ge\n1-\\sum_{m\\in\\cM'}f(m)$ for any mass function $f:\\cM\\to[0,1]$.\n\\end{prop}\n\\begin{proof}\nUsing the isomorphism from Proposition~\\ref{p:RaysAndCC}, we can\nchoose a non-zero $v\\in\\QQ^P$ such that\n$\\heatbathmove{\\pi}{\\cF}{m}v=v$ for all $m\\in\\cM\\setminus\\cM'$ and\n$\\heatbathmove{\\pi}{\\cF}{m}v=0$ for all $m\\in\\cM'$. In particular\n\\begin{equation*}\n\\heatbath{\\pi}{f}{\\cF}{\\cM}v=\\sum_{m\\in\\cM}f(m)\\heatbathmove{\\pi}{\\cF}{m}v=\n\\sum_{m\\in\\cM\\setminus\\cM'}f(m)\\heatbathmove{\\pi}{\\cF}{m}v=\\sum_{m\\in\\cM\\setminus\\cM}f(m)v.\n\\end{equation*}\nSince $f$ is a mass function, $1-\\sum_{m\\in\\cM'}f(m)$ is an\neigenvalue of $\\heatbath{\\pi}{f}{\\cF}{\\cM}$.\n\\end{proof}\n\n\\begin{defn}\\label{d:IntersectingRayProperty}\nLet $\\cF\\subset\\ZZ^d$ and $m,m'\\in\\ZZ^d$ not collinear. The pair\n$(m,m')$ has the \\emph{intersecting\nray property} in $\\cF$ if the following holds:\nFor any pair of rays $\\cR_1,\\cR_2$ parallel to $m$ and any pair of rays\n$\\cR_1',\\cR_2'$ parallel to $m'$ where both $\\cR_1\\cap\\cR_1'$ and\n$\\cR_2\\cap\\cR_2'$ are not empty, then\n$\\cR_1\\cap\\cR_2'\\neq\\emptyset$ implies $\\cR_1'\\cap\\cR_2\\neq\\emptyset$\nand $|\\cR_1|\\cdot|\\cR_1'|^{-1}=|\\cR_2|\\cdot|\\cR_2'|^{-1}$. For a\nfinite set $\\cM\\subset\\ZZ^d$, the graph $\\cF^c(\\cM)$ has the\n\\emph{intersecting ray property} if all $(m,m')$ have the intersecting\nray property in $\\cF$.\n\\end{defn}\n\n\\begin{example}\nThe compressed fiber graph on\n$[n_1]\\times\\cdots\\times[n_d]\\subset\\ZZ^d$ that uses the unit vectors\n$\\{e_1,\\dots,e_d\\}$ as moves has the intersecting ray property. On the\nother hand, consider $\\cF=\\{u\\in\\NN^2: u_1+u_2\\le 1\\}$ and take the\nrays $\\cR_1:=\\{(0,0),(0,1)\\}$ and $\\cR_2:=\\{(1,0)\\}$ that\nare parallel to $e_2$ and the rays $\\cR_1':=\\{(0,1)\\}$ and\n$\\cR_2':=\\{(0,0),(1,0)\\}$ that are parallel to $e_1$. Then\n$\\cR_1\\cap\\cR_1'=\\{(1,0)\\}$ and $\\cR_2\\cap\\cR_2'=\\{(0,1)\\}$, but\n$\\cR_1\\cap\\cR_2'=\\{(0,0)\\}\\neq\\emptyset$ and\n$\\cR_1'\\cap\\cR_2=\\emptyset$.\n\\end{example}\n\n\\begin{prop}\\label{p:CommutingMoveMatrices}\nLet $m,m'\\in\\ZZ^d$ not collinear and $\\cF\\subset\\ZZ^d$ be a finite set. The\nmatrices $\\heatbathmove{\\pi}{\\cF}{m}$ and $\\heatbathmove{\\pi}{\\cF}{m'}$\ncommute if and only if $(m,m')$ have the intersecting ray property in\n$\\cF$.\n\\end{prop}\n\\begin{proof}\nLet $u_1,u_2\\in\\cF$. Then \n\\begin{equation*}\n(\\heatbathmove{\\pi}{\\cF}{m}\\cdot\\heatbathmove{\\pi}{\\cF}{m'})_{u_1,u_2}=\n\\begin{cases}\n|\\polyray{u_1}{m}{\\cF}|^{-1}\\cdot|\\polyray{u_2}{m'}{\\cF}|^{-1},&\\text{ if\n}\\polyray{u_1}{m}{\\cF}\\cap\\polyray{u_2}{m'}{\\cF}\\neq\\emptyset\\\\\n0,&\\text{ otherwise}\n\\end{cases}.\n\\end{equation*}\nLet $\\cR_1:=\\polyray{u_1}{m}{\\cF}$, $\\cR_1':=\\polyray{u_1}{m'}{\\cF}$,\n$\\cR_2:=\\polyray{u_2}{m}{\\cF}$, and $\\cR_2':=\\polyray{u_2}{m'}{\\cF}$\nThus,\n$(\\heatbathmove{\\pi}{\\cF}{m}\\cdot\\heatbathmove{\\pi}{\\cF}{m'})_{u_1,u_2}=(\\heatbathmove{\\pi}{\\cF}{m'}\\cdot\\heatbathmove{\\pi}{\\cF}{m})_{u_1,u_2}$.\nIt is easy to see that the matrices commute if and only if $(m,m')$\nhave the intersecting ray property.\n\\end{proof}\n\n\\begin{lemma}\\label{l:DiagonizableCommutingMatrices}\nLet $H_1,\\dots,H_n\\in\\RR^{n\\times n}$ be pairwise\ncommuting matrices. Then any eigenvalue of $\\sum_{i=1}^nH_i$ has the\nform $\\lambda_1+\\dots+\\lambda_n$ where $\\lambda_i$ is an eigenvalue\nof~$H_i$.\n\\end{lemma}\n\\begin{proof}\nThis is a straightforward extension of the case $n=2$\nin~\\cite[Theorem~2.4.8.1]{Horn2013} and relies on the fact that\ncommuting matrices are simultaneously triangularizable.\n\\end{proof}\n\n\\begin{prop}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets and suppose\nthere exists $m\\in\\cM$ such that $(m,m')$ has the intersecting ray\nproperty in $\\cF$ for all $m'\\in\\cM':=\\cM\\setminus\\{m\\}$. Let\n$\\cV_1,\\dots,\\cV_c$ be the connected components of\n$\\cF(\\cM')$, $\\pi_i:\\cV_i\\to[0,1]$ the uniform distribution, and\n$f'=(1-f(m))^{-1}\\cdot f|_{\\cM'}$, then\n\\begin{equation*}\n\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le\nf(m)+(1-f(m))\\cdot\\max\\{\\lambda(\\heatbath{\\pi_i}{f'}{\\cV_i}{\\cM'}):i\\in\n[c]\\}.\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nLet $\\cH:=\\heatbath{\\pi}{f'}{\\cF}{\\cM'}$\nbe the heat-bath random walk on $\\cF(\\cM)$ that samples moves from\n$\\cM'$ according to $f'$, then\n$\\heatbath{\\pi}{f}{\\cF}{\\cM}=f(m)\\cdot\\heatbathmove{\\pi}{\\cF}{m}+(1-f(m))\\cdot\n\\cH$.\nBy assumption, all pairs $(m,m')$ with $m'\\in\\cM'$ have the intersecting ray\nproperty and thus the matrices $\\heatbathmove{\\pi}{\\cF}{m}$ and\n$\\cH$ commute according to\nProposition~\\ref{p:CommutingMoveMatrices}. The eigenvalues of all\ninvolved matrices are non-negative and thus\nLemma~\\ref{l:DiagonizableCommutingMatrices} implies that the second largest\neigenvalue of $\\heatbath{\\pi}{f}{\\cF}{\\cM}$ has the form\n$\\lambda+\\lambda'$ where $\\lambda\\in\\{0,f(m)\\}$ by\nProposition~\\ref{p:MoveMatrix} and where $\\lambda'$\nis an eigenvalue of $(1-f(m))\\cdot\\cH$.\nThe matrix $\\cH$ is a block matrix\nwhose building blocks are the matrices\n$\\heatbath{\\pi}{f'}{\\cV_i}{\\cM'}=\\heatbath{\\pi_i}{f'}{\\cV_i}{\\cM'}$\nand thus the statement follows.\n\\end{proof}\n\n\\begin{prop}\\label{p:UpperSLEMBound}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^k$ be finite sets. If\n$\\cF(\\cM)$ has the intersecting ray\nproperty, then $\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le 1-\\min(f)$.\n\\end{prop}\n\\begin{proof}\nLet $\\cM=\\{m_1,\\dots,m_k\\}$. The intersecting ray property and\nProposition~\\ref{p:CommutingMoveMatrices} give that\nthe matrices\n$f(m_1)\\cdot\\heatbathmove{\\pi}{\\cF}{m_i},\\dots,f(m_k)\\cdot\\heatbathmove{\\pi}{\\cF}{m_k}$\ncommute pairwise. According to Proposition~\\ref{p:MoveMatrix}, the\neigenvalues of\n$f(m_i)\\cdot\\heatbathmove{\\pi}{\\cF}{m_i}$ are $\\{0,f(m_i)\\}$.\nLemma~\\ref{l:DiagonizableCommutingMatrices} gives that the second\nlargest eigenvalue of $\\heatbath{\\pi}{f}{\\cF}{\\cM}$, which equals the\nsecond largest eigenvalue modulus since all of its eigenvalues are\nnon-negative, fulfills\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})=\\sum_{i\\in\nI}f(m_i)$ for a subset $I\\subseteq[k]$. Since\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})<1$ and $\\sum_{i=1}^kf(m_i)=1$, we\nhave $I\\neq[k]$ and the claim follows.\n\\end{proof}\n\n\\begin{prop}\\label{p:HeatBathOnHyperrectangle}\nLet $n_1,\\dots,n_d\\in\\NN_{>1}$,\n$\\cF=[n_1]\\times\\cdots\\times[n_d]$, and\n$\\cM=\\{e_1,\\dots,e_d\\}$. Then for any positive\nmass function $f:\\cM\\to[0,1]$,\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})=1-\\min(f)$. \n\\end{prop}\n\\begin{proof}\nIt is easy to verify that $\\cF^c(\\cM)$ has the intersecting ray property\nand thus Proposition~\\ref{p:UpperSLEMBound} shows\n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le 1-\\min(f)$. Assume that\n$\\min(f)=f(e_i)$. The connected\ncomponents of $\\cF^c(\\{e_1,\\dots,e_d\\}\\setminus\\{e_i\\})$ are the layers $V_j:=\\{u\\in\n\\cF: u_i=j\\}$ for any $j\\in[n_i]$ and the rays through $\\cF$ parallel are\n$\\cR_{k}:=\\{(0,k)+s\\cdot e_i: s\\in[n_i]\\}$ for\n$k=(k_1,\\dots,k_{i-1},k_{i+1},\\dots,k_d)\\in[n_1]\\times\\cdots\\times[n_{i-1}]\\times[n_{i+1}]\\times\\cdots\\times[n_d]$. In particular, any\nray intersects any connected component exactly once. Thus, the matrix\n$(|R_k\\cap V_j|)_{k,j}$ is the all-ones matrix, which has a\nnon-trivial kernel. Proposition~\\ref{p:KernelRayMat} implies \n$\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\ge 1-f(e_i)$.\n\\end{proof}\n\n\\begin{remark}\\label{r:RooksWalk}\nIn the special case $n:=n_1=\\dots=n_d$ and\n$f:\\{e_1,\\dots,e_d\\}\\to[0,1]$ the uniform distribution in\nProposition~\\ref{p:HeatBathOnHyperrectangle}, the heat-bath\nrandom walk on $[n]^d$ is known as \\emph{Rook's walk} in the\nliterature. In this case, Proposition~\\ref{p:HeatBathOnHyperrectangle}\nis exactly~\\cite[Proposition~2.3]{Kim2012}.\nIn~\\cite{Mcleman2015}, upper bounds on the mixing time of the Rook's\nwalk were obtained with \\emph{path-coupling}.\n\\end{remark}\n\nThe stationary distribution of the heat-bath random walk is\nindependent of the actual mass function on the Markov moves. \nThe problem of finding the mass function which leads to the fastest\nmixing behaviour can be formulated as the following optimization\nproblem:\n\\begin{equation}\\label{equ:SLEMOpti}\n\\arg\\min\\left\\{\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM}): f:\\cM\\to(0,1),\n\\sum_{m\\in\\cM}f(m)=1\\right\\}.\n\\end{equation}\nIt follows from\nProposition~\\ref{p:HeatBathOnHyperrectangle} that the optimal value of\n\\eqref{equ:SLEMOpti} for $\\cF=[n_1]\\times\\cdots\\times[n_d]$,\n$\\cM=\\{e_1,\\dots,e_d\\}$, and the uniform distribution $\\pi$ on $\\cF$\nis the uniform distribution on $\\cM$. Another example where the\nuniform distribution is the optimal solution to~\\eqref{equ:SLEMOpti},\nbut where the verification is more involved, is presented in\nExample~\\ref{ex:SolvingSLEMOpti}.\n\n\\begin{example}\\label{ex:SolvingSLEMOpti}\nLet $\\cF=[2]\\times[5]$ as in Example~\\ref{ex:AddingMovesSlowingChain}\nand consider $\\cM=\\{e_1,2e_1+e_2\\}$.\nWe investigate for which $\\mu\\in(0,1)$, the transition matrix\n$\\mu\\heatbathmove{\\pi}{\\cF}{e_1}+(1-\\mu)\\heatbathmove{\\pi}{\\cF}{2e_1+e_2}$ has the\nsmallest second largest eigenvalue modulus. Its characteristic polynomial in $\\QQ[\\mu,x]$ is\n$$-\\frac{1}{25}x^4(x-1)(\\mu+x-1)^6(-5x^2+5x+2\\mu^2-2\\mu)(-5x^2+5x+4\\mu^2-4\\mu)$$\nand hence its eigenvalues are \n\\begin{equation*}\n\\begin{split}\nx_1(\\mu):=1,& \\quad x_2(\\mu):=1-\\mu,\\\\\nx_3(\\mu):=\\frac{1}{2}\\left[1+\\sqrt{1+\\frac{8}{5}(\\mu^2-\\mu)}\\right],\n&\\quad\nx_4(\\mu):=\\frac{1}{2}\\left[1-\\sqrt{1+\\frac{8}{5}(\\mu^2-\\mu)}\\right],\\\\\nx_5(\\mu):=\\frac{1}{2}\\left[1+\\sqrt{1+4(\\mu^2-\\mu)}\\right],&\\quad\nx_6(\\mu):=\\frac{1}{2}\\left[1-\\sqrt{1+4(\\mu^2-\\mu)}\\right].\\\\\n\\end{split}\n\\end{equation*}\nIt is straightforward to check that $x_5(\\mu)>\\frac{1}{2}> x_6(\\mu)$, $x_3(\\mu)>\\frac{1}{2}>\nx_4(\\mu)$. Since $\\mu^2-\\mu<0$ for $u\\in(0,1)$ and \n$x_3(\\mu)\\ge x_6(\\mu)$. We can show that\n$x_4(\\mu)\\ge x_2(\\mu)$ and thus\n\\begin{equation*}\n\\lambda(\\mu\\heatbathmove{\\pi}{\\cF}{e_1}+(1-\\mu)\\heatbathmove{\\pi}{\\cF}{2e_1+e_2})=\\frac{1}{2}\\left[1+\\sqrt{1+\\frac{8}{5}(\\mu^2-\\mu)}\\right].\n\\end{equation*}\nThe fastest heat-bath random walk on $\\cF(\\cM)$ which converges to\nuniform is thus obtained for\n$\\mu=\\frac{1}{2}$, i.e. when the moves are selected uniformly. The\nsecond largest eigenvalue in this case is\n$\\frac{1}{10}(5+\\sqrt{15})\\approx 0.887$, which is larger than the\nsecond largest eigenvalue of the heat-bath walk that selects uniformly\nfrom $\\{e_1,e_2\\}$ (see Proposition~\\ref{p:HeatBathOnHyperrectangle}).\n\\end{example}\n\n\\section{Augmenting Markov bases}\n\nIt follows from our investigation in Section~\\ref{s:Diameter} that the\ndiameter of all compressed fiber graphs coming from a fixed integer matrix\n$A\\in\\ZZ^{m\\times d}$ can be bounded from above by a constant.\nHowever, Markov moves can be used twice in a minimal path which can\nmake the diameter of the compressed fiber graph larger than the size\nof the Markov basis. The next definition puts more constraints on the\nMarkov basis and postulates the existence of a path that uses every\nmove from the Markov basis at most once.\n\n\\begin{defn}\\label{d:Augmentation}\nLet $\\cF\\subset\\ZZ^d$ be a finite set and\n$\\cM=\\{m_1,\\dots,m_k\\}\\subset\\ZZ^d$. An \\emph{augmenting path} between\ndistinct $u,v\\in\\cF$ of length $r\\in\\NN$ is a path in $\\cF^c(\\cM)$ of\nthe form\n\\begin{equation*}\nu\\to u+\\lambda_{i_1}m_{i_1}\\to u+\\lambda_{i_1}m_{i_1}+\\lambda_{i_2}m_{i_2}\n\\to\\cdots \\to u+\\sum_{k=1}^r\\lambda_{i_k}m_{i_k}=v\n\\end{equation*}\nwith distinct indices $i_1,\\dots,i_r\\in[k]$. An augmenting path is\n\\emph{minimal} for $u,v\\in\\cF$ if there exists no shorter augmenting\npath between $u$ and $v$ in $\\cF^c(\\cM)$. \nA Markov basis $\\cM$ for $\\cF$ is \\emph{augmenting} if there is an\naugmenting path between any distinct nodes in $\\cF$. The\n\\emph{augmentation length} $\\auglen{\\cM}{\\cF}$ of an augmenting Markov\nbasis $\\cM$ is the maximum length of all minimal augmenting paths in\n$\\cF^c(\\cM)$.\n\\end{defn}\n\nNot every Markov basis is augmenting (see Example~\\ref{ex:StairCase}),\nbut the diameter of compressed fiber graphs that use an augmenting\nMarkov basis is at most the number of the moves. For fiber graphs\ncoming from an integer matrix, an augmenting Markov basis for all of its\nfibers can be computed (Remark~\\ref{r:GraverAugmentation}).\n\n\n\\begin{remark}\\label{r:GraverAugmentation}\nLet $A\\in\\ZZ^{m\\times d}$ with $\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and let\n$b\\in\\cone{A}$. The\nGraver basis is an augmenting Markov basis for\n$\\fiber{A}{b}$ for any $b\\in\\cone{A}$. We claim that when $A$ is\ntotally unimodular, then $\\auglen{\\graver{A}}{\\fiber{A}{b}}\\le d^2(\\rank(A)+1)$. In\nparticular, the augmentation length is independent of the right-hand\nside $b$. Let $u,v\\in\\fiber{A}{b}$ be arbitrary and for $i\\in\\NN$, let\n$l_i:=\\min\\{u_i,v_i\\}$, $w_i:=\\max\\{u_i,v_i\\}$, and\n$c_i:=\\mathrm{sign}(u_i-v_i)\\in\\{-1,0,1\\}$.\nThen $v$ is the unique optimal value of the linear integer optimization\nproblem\n\\begin{equation*}\n\\min\\{c^Tx: Ax=b, l\\le x\\le w,x\\in\\ZZ^d\\}.\n\\end{equation*}\nA \\emph{discrete steepest decent} as defined in\n\\cite[Definition~3]{loera-augmentation} using Graver moves needs at most\n$\\|c\\|_1\\cdot d\\cdot(\\rank(A)+1)\\le d^2\\cdot(\\rank(A)+1)$ many\naugmentations from $u$ to reach the optimal value $v$. We refer to \n\\cite[Corollary~8]{loera-augmentation} which ensures that every Graver move is used\nat most once. Note that in~\\cite{loera-augmentation}, $x$ is constrained to\n$x\\ge 0$ instead to $x\\ge l$, but their argument works for any lower\nbound.\n\\end{remark}\n\n\\begin{example}\\label{ex:AugmentingMarkovBasis}\nFix $d\\in\\NN$ and consider $A$ and $\\cM$ from\nExample~\\ref{ex:NonNormReducingNormLike}. We show that $\\cM$ is an\naugmenting Markov basis for $\\fiber{A}{b}$ for any $b\\in\\NN$.\nLet $u,v\\in\\fiber{A}{b}$ be distinct, then there exists $i\\in[d]$ such\nthat $u_i>v_i$ or $u_i0}$ such that $u+\\delta v\\in\\cP$. Thus,\n$\\frac{1}{\\delta}u+v\\in\\frac{1}{\\delta}\\cP$. Let $c\\in\\NN_{\\ge 1}$\nsuch that $i:=\\frac{c}{\\delta}\\in\\NN$ and \n$w:=\\frac{c}{\\delta}u\\in\\ZZ^d$. Then\n$w+c\nv=c(\\frac{1}{\\delta}u+v)\\in(i\\cdot\\cP)\\cap\\ZZ^d=\\cF_{i}$. By\nassumption, there exists an augmenting path from $w$ to $w+c v$ using\nonly $r$ elements from $\\cM$. Put differently, the element $c v$ from\n$V$ can be represented by a linear combination of $r$ vectors from $\\cM$.\nSince $v$ was chosen arbitrarily, Lemma~\\ref{l:VectorSpace} implies\n$\\dim(\\cP)=\\dim(V)\\le r$.\n\\end{proof}\n\n\\begin{remark}\nIt is a consequence from Proposition~\\ref{p:AugmentingDilatation}\nthat for any matrix $A\\in\\ZZ^{m\\times d}$ with\n$\\ker_\\ZZ(A)\\cap\\NN^d=\\{0\\}$ and an augmenting Markov\nbasis $\\cM$, there exists $\\cF\\in\\cP_A$ such that\n$\\auglen{\\cM}{\\cF}\\ge\\dim(\\ker_\\ZZ(A))$.\n\\end{remark}\n\nLet us now shortly recall the framework from~\\cite{Sinclair1992} which\nis necessary to prove our main theorem. Let $G=(V,E)$ be a graph. For\nany ordered pair of distinct nodes $(x,y)\\in V\\times V$, let\n$p_{x,y}\\subseteq E$ be a path from $x$ to $y$ in $G$ and let\n$\\Gamma:=\\{p_{x,y}: (x,y)\\in V\\times V, x\\neq y\\}$ be the collection\nof these paths, then $\\Gamma$ is \\emph{a set of canonical paths}.\nLet for any edge $e\\in E$, $\\Gamma_e:=\\{p\\in\\Gamma: e\\in p\\}$ be the\nset of paths from $\\Gamma$ that use $e$. Now, let $\\cH:V\\times V\\to[0,1]$ be a symmetric\nrandom walk on $G$ and define\n\\begin{equation*}\n\\rho(\\Gamma,\\cH):=\\frac{\\max\\{|p|:\np\\in\\Gamma\\}}{|V|}\\cdot\\max_{\\{u,v\\}\\in\nE}\\frac{|\\Gamma_{\\{u,v\\}}|}{\\cH(u,v)}.\n\\end{equation*}\nObserve that symmetry of $\\cH$ is needed to make $\\rho(\\Gamma,\\cH)$\nwell-defined. This can be used to prove the following upper bound on\nthe second largest eigenvalue.\n\n\\begin{lemma}\\label{l:CanonicalPaths}\nLet $G$ be a graph, $\\cH$ be a symmetric random walk on $G$, and\n$\\Gamma$ be a set of canonical paths in $G$. Then\n$\\lambda_2(\\cH)\\le 1-\\frac{1}{\\rho(\\Gamma,\\cH)}$.\n\\end{lemma}\n\\begin{proof}\nThe stationary distribution of $\\cH$ is the uniform distribution and\nthus the statement is a direct consequence of~\\cite[Theorem~5]{Sinclair1992},\nsince $\\rho(\\Gamma,\\cH)$ is an upper bound on the constant defined\nin~\\cite[equation~4]{Sinclair1992}.\n\\end{proof}\n\n\\begin{thm}\\label{t:MixingofAugmentingMarkovBases}\nLet $\\cF\\subset\\ZZ^d$ be finite and let\n$\\cM:=\\{m_1,\\dots,m_k\\}\\subset\\ZZ^d$ be an augmenting Markov\nbasis. Let $\\pi$ be the uniform and $f$ be a positive\ndistribution on $\\cF$ and $\\cM$ respectively. For $i\\in[k]$, let\n$r_i:=\\max\\{|\\polyray{u}{m_i}{\\cF}|:u\\in\\cF\\}$ \nand suppose that $r_1\\ge r_2\\ge\\dots\\ge r_k$. Then\n\\begin{equation*}\n\\lambda(\\heatbath{\\pi}{f}{\\cM}{\\cF})\\le1-\\frac{|\\cF|\\cdot\\min(f)}{\\auglen{\\cM}{\\cF}\\cdot\\auglen{\\cM}{\\cF}!\\cdot\n3^{\\auglen{\\cM}{\\cF}-1}\\cdot2^{|\\cM|}\\cdot r_1r_2\\cdots\nr_{\\auglen{\\cM}{\\cF}}}.\n\\end{equation*}\n\\end{thm}\n\\begin{proof}\nChoose for any distinct\n$u,v\\in\\cF$ an augmenting path $p_{u,v}$ of minimal length in\n$\\cF^c(\\cM)$ and let $\\Gamma$ be the collection of all\nthese paths. Let $u+\\mu m_k=v$ be an edge in\n$\\cF^c(\\cM)$, then our goal is to bound $|\\Gamma_{\\{u,v\\}}|$ from\nabove. Let $\\mathcal{S}:=\\{S\\subseteq[r]: |S|\\le\\auglen{\\cM}{\\cF},\nk\\in S\\}$ and take any path $p_{x,y}\\in\\Gamma_{\\{u,v\\}}$. Then there exists\n$S:=\\{i_1,\\dots,i_s\\}$ with $s:=|S|\\le\\auglen{\\cM}{\\cF}$ such that\n$x+\\sum_{k=1}^s\\lambda_{i_k}m_{i_k}=y$. Since $p_{x,y}$ uses the edge\n$\\{u,v\\}$, there is $j\\in[s]$ such that $i_j=k$ and\n$\\lambda_{i_j}=\\mu$. Since $|\\lambda_{i_k}|\\le r_{i_k}$, there are at\nmost \n$$s!\\cdot (2r_{i_1}+1)\\cdots (2r_{i_{j-1}}+1)\\cdot(2r_{i_{j+1}}+1)\\cdots\n(2r_{i_s}+1)\\le\ns!\\cdot 3^{s-1}\\prod_{t\\in S\\setminus\\{k\\}}r_{t}$$\npaths in\n$\\Gamma_{\\{u,v\\}}$ that uses the edge $\\{u,v\\}$ and the moves\n$m_{i_1},\\dots,m_{i_{j-1}},m_{i_{j+1}}\\dots,m_{i_s}$. Since all the\npaths are minimal, they have length at most $\\auglen{\\cM}{\\cF}$ so\nindeed every path in $\\Gamma$ has that form.\n\\begin{equation*}\n\\frac{|\\Gamma_{u,v}|}{\\heatbath{\\pi}{f}{\\cF}{\\cM}(u,v)}\\le\n3^{\\auglen{\\cM}{\\cF}-1}\\frac{\\sum_{S\\in\\mathcal{S}}\\left(|S|!\\prod_{t\\in\nS\\setminus\\{k\\}}r_{t}\\right)}{f(m_{i_j})\\cdot\\frac{1}{|\\ray{u}{m_{i_j}}|}}\\le\\frac{3^{\\auglen{\\cM}{\\cF}-1}\\cdot\n\\auglen{\\cM}{\\cF}!\\cdot|\\mathcal{S}|\\cdot\nr_1r_2\\dots r_{\\auglen{\\cM}{\\cF}}}{f(m_{i_j})},\n\\end{equation*}\nwhere we have used the assumption $r_1\\ge r_2\\ge \\dots\\ge r_k$.\nBounding $|\\mathcal{S}|$ rigorously from above by $2^{|\\cM|}$, the claim follows\nfrom Lemma~\\ref{l:CanonicalPaths}.\n\\end{proof}\n\n\\begin{defn}\nLet $\\cF\\subset\\ZZ^d$ and $\\cM\\subset\\ZZ^d$ be finite sets. The longest\nray through $\\cF$ along vectors of $\\cM$ is\n$\\longestRay{\\cF}{\\cM}:=\\arg\\max\\{|\\polyray{u}{m}{\\cF}|: m\\in\\cM, u\\in\n\\cF\\}$.\n\\end{defn}\n\n\\begin{cor}\\label{c:AugmentingExpander}\nLet $(\\cF_i)_{i\\in\\NN}$ be a sequence of finite sets in $\\ZZ^d$ and let\n$\\pi_i$ be the uniform distribution on $\\cF_i$. Let $\\cM\\subset\\ZZ^d$ be\nan augmenting Markov basis for $\\cF_i$ with $\\auglen{\\cM}{\\cF_i}\\le\\dim(\\cF_i)$\nand suppose that\n$(|\\longestRay{\\cF_i}{\\cM}|)^{\\dim(\\cF_i)})_{i\\in\\NN}\\in\\mathcal{O}(|\\cF_i|)_{i\\in\\NN}$.\nThen for any positive mass function $f:\\cM\\to[0,1]$, there exists $\\epsilon>0$ such that\n$\\lambda(\\heatbath{\\pi_i}{f}{\\cF_i}{\\cM})\\le 1-\\epsilon$ for all\n$i\\in\\NN$.\n\\end{cor}\n\\begin{proof}\nThis is a straightforward application of\nTheorem~\\ref{t:MixingofAugmentingMarkovBases}. \n\\end{proof}\n\n\\begin{cor}\\label{c:ExpanderDilatation}\nLet $\\cP\\subset\\ZZ^d$ be a polytope, $\\cF_i:=(i\\cdot\\cP)\\cap\\ZZ^d$\nfor $i\\in\\NN$, and let $\\pi_i$ be the uniform distribution on $\\cF_i$.\nSuppose that $\\cM\\subset\\ZZ^d$ is an augmenting Markov basis\n$\\{\\cF_i:i\\in\\NN\\}$ such that $\\auglen{\\cM}{\\cF_i}\\le\\dim(\\cP)$ for\nall $i\\in\\NN$. \nThen for any positive mass function $f:\\cM\\to[0,1]$, there exists\n$\\epsilon>0$ such that $\\lambda(\\heatbath{\\pi_i}{f}{\\cF_i}{\\cM})\\le\n1-\\epsilon$ for all $i\\in\\NN$.\n\\end{cor}\n\\begin{proof}\nLet $r:=\\dim(\\cP)$. We first show that\n$(|\\longestRay{\\cF_i}{\\cM}|)_{i\\in\\NN}\\in\\mathcal{O}(i)_{i\\in\\NN}$.\nWrite $\\cM=\\{m_1,\\dots,m_k\\}$ and denote by\n$l_i:=\\max\\{|(u+m_i\\cdot\\ZZ)\\cap\\cP|: u\\in\\cP\\}$ be the length of the\nlongest ray through the polytope $\\cP$ along $m_i$. It suffices to\nprove that $i\\cdot(l_k+1)$ is an upper bound on the length of any ray\nalong $m_k$ through $\\cF_i$. For that, let $u\\in\\cF_i$ such that\n$u+\\lambda m_k\\in\\cF_i$ for some $\\lambda\\in\\NN$, then\n$\\frac{1}{i}u+\\frac{\\lambda}{i} m_k\\in\\cP$ and thus\n$\\lfloor\\frac{\\lambda}{i}\\rfloor\\le l_k$, which gives $\\lambda\\le\ni\\cdot(l_k+1)$. With $C:=\\max\\{l_1,\\dots,l_k\\}+1$ we have\n$|\\longestRay{\\cF_i}{\\cM}|\\le C\\cdot i$.\nEhrhart's theorem~\\cite[Theorem~3.23]{Beck2007} gives\n$(|\\cF_i|)_{i\\in\\NN}\\in\\Omega(i^r)_{i\\in\\NN}$ and\nsince $|\\longestRay{\\cF_i}{\\cM}|\\le C\\cdot i$, we have\n$(|\\longestRay{\\cF_i}{\\cM}|^r)_{i\\in\\NN}\\in\\mathcal{O}(|\\cF_i|)_{i\\in\\NN}$.\nAn application of Corollary~\\ref{c:AugmentingExpander} proves the claim.\n\\end{proof}\n\n\\begin{example}\\label{e:CrossPoly}\nFix $d,r\\in\\NN$ and let $\\cC_{d,r}:=\\{u\\in\\ZZ^d: \\|u\\|_1\\le r\\}$ be\nthe set of integers of the $d$-dimensional cross-polytope with radius\n$r$. The set $\\cM_d=\\{e_1,\\dots,e_d\\}$\nis a Markov basis for\n$\\cC_{d,r}$ for any $r\\in\\NN$. We show that $\\cM_d$ is an augmenting\nMarkov basis whose augmentation length is at most $d$. For that, let \n$u,v\\in\\cC_{d,r}$ distinct elements. We claim that there exists\n$i\\in[d]$ such that $x_i\\neq v_i$ and $u_i+(v_i-u_i)\\in\\cC_{d,r}$.\nLet $S\\subseteq[d]$ be the set of indices where $u$ and $v$ differ and let\n$s= r- ||u||_1$. If $|S|=1$, then the result is clear so\nsuppose $|S| \\geq 2$. If the result doesn't hold then for all\n$i\\in S$, $|v_i|-|u_i| > s$. It follows that \n\\begin{equation*} \n\\|v\\|_1 = \\sum_{i \\notin S} |u_i| + \\sum_{i \\in S} |v_i|\n > \\sum_{i \\notin S} |u_i| + \\sum_{i \\in S} s + |u_i| = |S_{uv}|\\cdot s + \\|u\\|_1\n = (|S|-1) \\cdot s + r.\n\\end{equation*}\nBut we assumed that $v \\in\\cC_{d,r}$. It follows that for any pair of\npoints $u,v$ in $\\cC_{d,r}$, there is a walk, using the unit vectors\nas moves, that uses each move at most once.\nCorollary~\\ref{c:AugmentingExpander} yield that for any $d\\in\\NN$, the\nsecond largest eigenvalue modulus of the heat-bath random walk on\n$\\cC_{d,r}$ with uniform as stationary distribution can be strictly\nbounded away from $1$ for $r\\to\\infty$.\n\\end{example}\n\nThe bound on the second largest eigenvalue in\nTheorem~\\ref{t:MixingofAugmentingMarkovBases} is quite general and can\nbe improved vastly, provided one has better control over the paths.\nFor example, this can be achieved for hyperrectangles intersected with\na halfspace.\n\n\\begin{prop}\\label{p:Hyperrectangles}\nLet $a\\in\\NN^d_{>0}$, $b\\in\\NN$,\n$\\cF=\\{u\\in\\NN^d: a^T\\cdot u\\le b\\}$, and\n$\\cM:=\\{e_1,\\dots,e_d\\}$. If $\\pi$ and $f$ are the uniform\ndistributions on $\\cF$ and $\\cM$ respectively, then\n\\begin{equation*}\n\\lambda(\\heatbath{\\pi}{f}{\\cF}{\\cM})\\le\n1-\\frac{|\\cF|}{d^2}\\prod_{i=1}^d\\frac{a_i}{b}.\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nObserve that $\\cM$ is a Markov basis for $\\cF$ since all nodes are connected\nwith $0\\in\\cF$. Let $u,v\\in\\cF$ be distinct. We first show\nthat there exists $k\\in[d]$ such that $u_k\\neq v_k$ and\n$u+(v_k-u_k)e_k\\in\\cF$. If $u\\le v$, the statement trivially holds.\nOtherwise, there exists $k\\in [d]$ such that $u_k>v_k$ and the vector\nobtained by replacing the $k$th coordinate of $u$ by $v_k$ remains in $\\cF$.\nNow, consider for the following path between $u$ and $v$: Choose the smallest index\n$k\\in[d]$ such that $u_k\\neq v_k$ and such that $u+(v_k-u_k)\\cdot\ne_k\\in\\cF$ and proceed recursively with $u+(v_k-u_k)$ and $v$. This\ngives a path $p_{u,v}$ between $u$ and $v$ of length at most $d$. Let\n$\\Gamma$ be the collection of all these paths.\nWe want to apply Lemma~\\ref{l:CanonicalPaths}. Thus, let $x\\in\\cF$ and\nconsider the edge $x\\rightarrow x+c\\cdot e_s$. Let us count the paths\n$p_{u,v}$ that use that edge. Let $u,v\\in\\cF$ and let $k_1,\\dots,k_r\\in\n[d]$ be distinct indices such that\n\\begin{equation*}\nu\\rightarrow u+(v_{k_1}-u_{k_1})e_{k_1}\\rightarrow\nu+(v_{k_1}-u_{k_1})e_{k_1}+(v_{k_2}-u_{k_2})e_{k_2}\\rightarrow\\cdots\\rightarrow\nv\n\\end{equation*}\nrepresents the path $p_{u,v}$ constructed by the upper rule. Assume\nthat $p_{u,v}$ uses the edge $\\{x,x+ce_s\\}$ and let $k_l=s$ and\n$(v_{k_l}-u_{k_l})=c$. In particular,\n\\begin{equation*}\n\\begin{split}\nu&+(v_{k_1}-u_{k_1})e_{k_1}+\\cdots +(v_{k_{l-1}}-u_{k_{l-1}})e_{k_{l-1}}=x\\\\\nx&+(v_{k_l}-u_{k_l})e_{k_l}+\\cdots +(v_{k_r}-u_{k_r})e_{k_r}=v.\n\\end{split}\n\\end{equation*}\nWe see that $v_{k_t}=x_{k_t}$ for all $t< l$ and that\n$u_{k_t}=x_{k_t}$ \nfor all $t\\ge l$. In particular, $v_{k_l}=u_{k_l}+c=x_{k_l}+c$ is also\nfixed. The coordinates $u_{k_t}$ and $v_{k_t}$ are bounded\nfrom above by $\\frac{b}{a_{k_t}}$ for\nall $t\\in[r]$, and hence there can be at most \n\\begin{equation*}\n\\left(\\prod_{t=1}^{l-1}\\frac{b}{a_{k_t}}\\right)\\cdot\n\\left(\\prod_{t=l+1}^{r}\\frac{b}{a_{k_t}}\\right).\n\\end{equation*}\nSince $k_1,\\dots,k_t$ are distinct coordinate indices, we have\n\\begin{equation*}\n\\frac{|\\Gamma_{x,x+c\\cdot e_s}|}{\\heatbath{\\pi}{f}{\\cF}{\\cM}(x,x+c\\cdot\ne_s)}\\le d\\cdot\\prod_{i=1}^d\\frac{b}{a_i}.\n\\end{equation*}\nLemma~\\ref{l:CanonicalPaths} finishes the proof.\n\\end{proof}\n\nIn fixed dimension, Proposition~\\ref{p:Hyperrectangles} leads to rapid\nmixing, but for $d\\to\\infty$, no statement can be made.\nIn~\\cite{Morris2004}, it was shown that the simple walk with an\nadditional halting probability on $\\{u\\in\\NN^d: a^tu\\le\nb\\}\\cap\\{0,1\\}^d$ has mixing time in $\\mathcal{O}(d^{4.5+\\epsilon})$.\nFor zero-one polytopes, simple and heat-bath walk coincide and we are\nconfident that a similar statement holds without the restriction on\nzero-one polytopes.\n\nThe heat-bath random walk mixes rapidly when an augmenting Markov\nbasis with a small augmentation length is used. We think that it is\ninteresting to question how might an augmenting Markov bases be obtained and\nhow their augmentation length can be improved.\n\n\\begin{question}\nLet $\\cM$ be an augmenting Markov basis of $A$. Can we find finitely\nmany moves $m_1,\\dots,m_k$ such that the augmentation length of\n$\\cM\\cup\\{m_1,\\dots,m_k\\}$ on $\\fiber{A}{b}$ is at most\n$\\dim(\\ker_\\ZZ(A))$ for all $b\\in\\cone{A}$?\n\\end{question}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n Since the lack of observational evidence of WIMP (Weakly interacting massive particle), primordial black hole (PBH) gradually becomes one of the promising dark matter (DM) candidate~\\cite{Hawking:1971ei,Chapline:1975ojl,Khlopov:2008qy,Carr:2016drx,Carr:2020gox,Carr:2020xqk,Green:2020jor} and provides convincing explanation~\\cite{Clesse:2016vqa,Bird:2016dcv,Sasaki:2016jop} for LIGO\/Virgo~\\cite{LIGOScientific:2016aoc,LIGOScientific:2016sjg,LIGOScientific:2017bnn} gravitational wave (GW) signals.\nSeveral production mechanisms have been proposed.\nAfter inflation, PBHs can be generated through the collapse\nof the overdensity regions,\nwhich are developed from the primordial fluctuation re-entering the horizon~\\cite{Carr:1974nx,Sasaki:2018dmp}.\nOther papers discussed PBHs directly form following the bubble collisions of first-order phase transition (FOPT)~\\cite{Hawking:1982ga,Moss:1994iq,Konoplich:1999qq,Kodama:1982sf,Gross:2021qgx,Baker:2021nyl}.\n\nIn this work, we consider an alternative PBHs production mechanism from FOPT in the early Universe.\nDuring FOPT,\nan intermediate state, dubbed as Fermi balls (FBs), were formed\nvia the expanding bubbles.\nThen, the attractive Yukawa interaction triggers a FB collapsing into a PBH~\\cite{Kawana:2021tde}.\nThe success of FB formation requires the fermionic particles $\\chi$'s\nbeing trapped and aggregated\ninside a false vacuum surrounding by the expanding bubbles of true vacuum.\n~\\cite{Hong:2020est,Marfatia:2021twj,Witten:1984rs,Bai:2018dxf}.\nSince the Standard Model (SM)\nelectroweak and QCD (quantum chromodynamics) phase transitions are smooth crossovers,\nwe therefore invoke the FOPT being generated in the dark sector.\nSpecifically, a commonly discussed\nquartic effective thermal potential of a scalar field $\\phi$ is adopted.\nIn addition, the scalar field couples to the $\\chi$ through Yukawa interaction,\nwhich serves two purposes.\nFirst, the non-zero VEV (vacuum expectation value) of the scalar increases the mass $\\chi$ in the true vacuum, due to the energy-momentum conservation at bubble wall, which keeps the $\\chi$'s in the false vacuum\nif the mass difference is larger than the critical temperature of the FOPT.\nThe other purpose, the Yukawa interaction induces attractive force between $\\chi$'s\nand triggers the collapse of a FB into a PBH.\nThis instability of FB happened when the range of interaction becomes comparable with the mean separation distance of $\\chi$'s\nin a FB~\\cite{Kawana:2021tde}.\n\n\n\nOn the astrophysical observations perspective,\nan excess amount of photons with energy 511 keV has been confirmed\nfrom the central region of Milky Way by SPI\/INTEGRAL~\\cite{Bouchet:2010dj}.\nIf someone identified this as a result of electron-positron annihilation,\nit requires the injection rate of $2\\times 10^{43}$ non-relativistic positrons per second\n~\\citep{Weidenspointner:2004my,Churazov:2004as,Weidenspointner:2007rs,Jean:2005af,Prantzos:2005pz}.\nVarious potential astrophysical sources have been proposed for the excess,\nbut encountered difficulties explaining the characteristic of the signal~\\cite{Prantzos:2010wi}. In addition, the MeV scale DM annihilation is hardly consistent with cosmic microwave background observation~\\cite{Wilkinson:2016gsy}.\nRecently, it has been suggested that the positrons responsible for the 511 keV line excess\nmight be produced through the Hawking evaporation\n~\\cite{Hawking:1975vcx,Gibbons:1977mu} of PBHs\n~\\cite{Keith:2021guq,Frampton:2005fk,Bambi:2008kx,Cai:2020fnq,Laha:2019ssq,DeRocco:2019fjq}.\nAssuming the PBHs distribution follows the NFW halo profile\nand concentrates in the inner Galaxy.\nThe PBH mass of $M_{\\rm PBH}\\sim \\mathcal{O}(5\\times 10^{16})~{\\rm g}$ and the $\\mathcal{O}(10^{-3})$ fraction of total DM abundance, has been shown,\nnot only produces the right amount of positron flux\nbut is also consistent with existing limits of\nCOMPTEL\/INTEGRAL gamma-ray observations~\\cite{Keith:2021guq}.\n\nIn this work, we aim to find the corresponding parameters space under the FOPT scenario, which generates the PBH mass and relic abundance to explain the 511 keV excess.\nAs a result, the correlated gamma-ray spectra form PBH evaporation and GW signals from FOPT are predicted.\nThis paper is organized as following: In section~\\ref{sec:PBH_511keV}, base on NFW and isothermal distributions,\nwe scrutinize the preferred PBH mass and abundance fraction for 511 keV excess.\nThe realization of PBH formation and correlated signals production\nare discussed in section~\\ref{sec:PBH_FOPT} and \\ref{sec:BP}, respectively.\nFinally, we summarize the results in section~\\ref{sec:conclusion}.\n\n\n\n\\bigskip\n\n\n\\section{PBH evaporation}\n\\label{sec:PBH_511keV}\n Hawking emission describes a PBH thermally produce primary particles with masses lighter than the PBH temperature $T_{\\rm PBH}=M^2_{\\rm Pl}\/M_{\\rm PBH}$, which numerically formulated as\n \\begin{eqnarray}\n T_{\\rm PBH}\\simeq 5.3~{\\rm MeV}\\times \\left( \\frac{10^{-18}M_\\odot}{M_{\\rm PBH}} \\right)\\,.\n \\end{eqnarray}\n The secondary particles productions come from the decays or fragmentation\n of primary particles (e.g $e^\\pm,\\mu^\\pm,\\pi^\\pm,\\pi^0$).\n A PBH dominantly emits photons and neutrinos,\n but for $M_{\\rm PBH}\/M_\\odot \\lesssim 10^{-16}$,\n PBH can significantly produces $e^\\pm$.\n %\n The emission rate of primary particle $i$ is given by~\\cite{Hawking:1974rv,Hawking:1975vcx}\n \\begin{eqnarray}\n \\frac{dN_i}{dE dt}=\n \\frac{n^{\\rm d.o.f}_i \\Gamma_i(E,M_{\\rm PBH})}{2\\pi (e^{E\/T_{\\rm PBH}}\\pm 1)}\\,,\n \\end{eqnarray}\n where $n^{\\rm d.o.f}_i$ indicates the degrees of freedom of $i$ particle,\n and the fermions (bosons) are distinguished by the $+(-)$ in the denominator.\n The graybody factor $\\Gamma_i(E,M)$ varies from particle to particle\n and is derived from considering a wave packet scattering\n in the PBH spacetime geometry from the PBH horizon to an observer\n at infinity.\n\n %\n Technically, we use the software package BlackHawk v2.1~\\cite{Arbey:2019mbc,Arbey:2021mbl}\n to compute the $\\gamma$ and $e^\\pm$ production rates\n including both primary and secondary components of PBH evaporation.\n The positron emission rate per single PBH can be numerically obtained after integrating out the evaporation spectrum\n \\begin{eqnarray}\n L_{e^+}= \\int dE\\, \\frac{dN_{e^+}}{dE dt}\\,.\n \\end{eqnarray}\n\n\\bigskip\n\n\\subsection{Galaxy center 511 keV line}\n\n\n The galaxy center 511 keV line observed by SPI\/INTEGRAL~\\cite{Bouchet:2010dj}\n exceeds the astrophysical contributions.\n In this paper, we focus on the explanation of PBHs in mass range from $10^{15}\\,$g\n to $2\\times 10^{17}\\,$g, which significantly produce particles by Hawking evaporation.\nPositrons are abundantly emitted from a PBH and more than 95\\% of them become non-relativistic via ionization. Among the non-relativistic positrons, fraction of $(1-f)$ directly annihilate with an electron to produce two 511 keV photons, while the others fraction of $f\\approx0.967$ form a positronium bound state with an electron~\\cite{Keith:2021guq}. Among the later case, 25\\% of positroniums annihilate to form pair of 511 keV photons, and the rest 75\\% yield three photons with energy less than 511 keV. As a Result, the number of 511 keV photons produced per positron is\n \\begin{eqnarray}\n 2(1-f)+\\frac{2f}{4}\\approx0.55\\,.\n \\end{eqnarray}\n The flux of 511 keV photons from PBHs near the galactic center,\n covers solid angle $\\Delta\\Omega$, is formulated as\n \\begin{eqnarray}\n \\Phi_{\\mathrm{PBH}}(\\Delta\\Omega)=\\frac{0.55L_{e^+}(M_{\\textrm{PBH}})f_{\\textrm{PBH}}}{4\\pi M_{\\textrm{PBH}}}\\int_{\\Delta\\Omega}{\\int_{\\textrm{l.o.s}}{\\rho(\\ell,\\Omega)d\\ell}d\\Omega} \\,,\n \\end{eqnarray}\n where $L_{e^+}(M_{\\textrm{PBH}})$ is the positron production rate from a PBH, $f_{\\textrm{PBH}}$ be the fraction of PBHs abundance to the DM relic abundance. The integrals are performed over solid angle and along the line-of-sight.\n We assume the PBHs distribution follows the Navarro-Frenk-White (NFW) DM halo profile:\n \\begin{eqnarray}\n \\rho(r)=\\frac{\\rho_0}{(r\/R_s)^\\gamma[1+(r\/R_s)]^{3-\\gamma}}\\,,\n \\end{eqnarray}\n where $r$ is the distance from the galactic center,\n and the normalization parameter $\\rho_0$ adjusts the DM density near our solar system,\n $\\rho=0.4$ $\\textrm{GeV}\/\\textrm{cm}^3$ at $r=8.25$ kpc.\n Here, we fix $\\gamma=1.6$ and scale radius $R_s=20$ kpc in this analysis.\n Combining with the galactic disk contribution,\n the angular distribution of 511 keV line predicted from PBHs emission\n is shown in Fig.~\\ref{fig:1}.\n It distributes as a function of galactic latitude $b$ after averaging over the longitude profile $-8^\\circ<\\ell<+8^\\circ$ and is compared with the INTEGRAL data of photon energy from 508.25 keV to 513.75 keV~\\cite{Bouchet:2010dj}.\n\n \\begin{figure}[t]\n \\centering\n \\includegraphics[width=13cm]{FIG\/1.jpg}\n \\caption{The orange curve is combined 511 keV gamma-ray flux\n from PBH evaporation with $M_{\\rm PBH}=10^{16}$ g, $f_{\\textrm{PBH}}=10^{-4}$ and astrophysical source~\\cite{Robin:2003uus}(blue curve).\n Comparing to the INTEGRAL data from 508.25 keV to 513.75 keV~\\cite{Bouchet:2010dj}}\\label{fig:1}\n \\end{figure}\n\n The best-fit normalization for the 511 keV INTEGRAL data is obtained by the $\\chi^2$ test, which is defined by\n \\begin{eqnarray}\n \\chi^2(f_\\mathrm{PBH})=\\sum_{i}{\\left[\\frac{\\Phi_{s}(b_i,f_\\mathrm{PBH})+\\Phi_{\\mathrm{disk}}(b_i)-\\Phi_{\\textrm{data}}(b_i)}{\\sigma_{\\Phi}(b_i)}\\right]^2}\\,,\n \\end{eqnarray}\n and $i$ runs over the INTEGRAL data points in Fig.\\ref{fig:1}.\n $\\Phi_{\\mathrm{disk}}$ stems from astrophysical positron emission in the disk. $\\Phi_{s}$ represents the PBHs contribution after taking into account the angular resolution of INTEGRAL, which is obtained by Gaussian smearing as\n \\begin{eqnarray}\n \\Phi_s(b_i,f_\\mathrm{PBH})=\\int^{b_i+3\\sigma_{b_i}}_{b_i-3\\sigma_{b_i}}{ F(b,b_i,\\sigma_{b_i})\\Phi_{\\mathrm{PBH}}(b,f_\\mathrm{PBH})db} \\,,\n \\end{eqnarray}\n where $b_i$ and $\\sigma_i$ respectively correspond to the central value and the horizontal error of each INTEGRAL data point, and then the smearing function is\n \\begin{eqnarray}\n F(b,b_i,\\sigma_{b_i})=\\frac{1}{\\sqrt{2\\pi \\sigma^2_{b_i}}}\\exp{\\left[-\\frac{(b_i-b)^2}{\\sigma^2_{b_i}}\\right]}\\,.\n \\end{eqnarray}\n The yellow region of Fig. \\ref{fig:2} shows\n the best-fit parameter region of $(M_{\\rm PBH},f_{\\rm PBH})$,\n which requires $\\chi^2-\\chi_{\\mathrm{min}}^2\\leq4$.\n For $M_{\\rm PBH}\\simeq 2\\times 10^{17}~{\\rm g}$,\n PBHs can serve as 100\\% DM abundance\n and explain the 511 keV line excess simultaneously.\n For $M_{\\rm PBH}\\lesssim 3\\times 10^{16}~{\\rm g}$,\n PBHs overproduce the extragalactic gamma-ray,\n and thus is excluded by current upper bound of COMPTEL.\n\n\n \\begin{figure}[t]\n \\centering\n \\includegraphics[width=13cm]{FIG\/2.jpg}\n \\caption{The yellow and cyan bands,\n which assume the NFW and Isothermal profiles respectively, can produce the 511 keV excess observed by INTEGRAL via the PBH evaporation.\n The constraints of COMPTEL+EGRET and sensitivities of eASTROGAM and AMEGO are shown by the solid curves, requiring the extragalactic photon from PBHs contribution does not exceed the observation data.\n The benchmark points ({\\bf BP}s) label the PBHs produced from the FOPT scenario, and their parameters are listed in Table~\\ref{table:1}.\n }\\label{fig:2}\n \\end{figure}\n\n\n \\bigskip\n\n\\subsection{Extragalactic gamma-ray}\n\n\n The PBH evaporation also produces extragalactic gamma-ray background, thus the current gamma-ray observational data can be converted into the upper limit of the PBHs abundance.\n To find the experiment limits in the parameter space, we calculate the PBH extragalactic photon contributions and compare with the current experimental data of COMPTEL\/EGRET\/FermiLAT and future sensitivities of AMEGO\/e-ASTROGAM as well.\n The extragalactic photon contributions due to PBH is given by~\\cite{Marfatia:2021hcp}\n \\begin{eqnarray}\n \\frac{d\\Phi}{dE}=\\int^{\\mathrm{min}(t_{\\mathrm{eva}},t_0)}_{t_\\mathrm{CMB}}{c[1+z(t)]\\frac{f_\\mathrm{PBH}\\rho_{\\mathrm{DM}}}{M_\\mathrm{PBH}}\\frac{d^2N_\\gamma}{d\\tilde{E}dt}\\bigg|_{\\tilde{E}=[1+z(t)]E}dt}\\,,\n \\end{eqnarray}\nwhere $\\rho_\\mathrm{DM}=1.27\\mkern5mu \\mathrm{GeV }\\mathrm{m}^{-3}$ is the average dark matter density in the extragalactic medium. From this expression,\nwe can see the extragalactic photon flux is independent from the uncertainties of DM density profile near the galactic center,\nthereby it provides more robust prediction than the galactic one.\nThe lower limit of temporal integral starts at the time of last scattering, $t_\\mathrm{CMB}=3.8\\times10^5\\mkern5mu\\mathrm{yr}$, after that photons decoupled from thermal plasma and can propagate freely. Meanwhile, the upper limit of integral is set by the shorter period of the lifetime of PBH ($t_\\mathrm{eva}$) or the age of Universe ($t_0=13.77\\times10^9\\mkern5mu\\mathrm{yr}$). Applying matter-dominated approximation to the evolution of Universe, the time-redshift relation is given by\n$1+z(t)=\\left(t_0\/t\\right)^{\\frac{2}{3}}$.\n For different masses of PBH,\n we show the upper limits of $f_\\mathrm{PBH}$ in Fig.~\\ref{fig:2},\n which are compatible with the present COMPTEL\/EGRET\/FermiLAT upper bounds (blue curve)\n and future AMGEGO\/e-ASTROGAM gamma-ray sensitivities (red\/black curves).\n\n\n\\bigskip\n\n\\section{First-order phase transition}\n\\label{sec:PBH_FOPT}\nThe PBH can be produced through a two-step processes during FOPT\nwhich is induced by the effective thermal potential of a scalar $\\phi$ in the hidden sector.\nThe FB stems from the aggregation of dark fermions $\\chi$'s and plays the role as intermediate state before PBH formation.\nBecause the $\\chi$'s inside a FB couple through $\\phi$, which provides an attractive Yukawa potential.\nThe length of Yukawa interaction increases as FB temperature decreases,\nand thus Yukawa energy eventually dominates the total FB energy\nwhich causing the FB collapses into a PBH~\\cite{Kawana:2021tde}.\n\n\n\n\n\n Specifically, the formation of PBHs can be realized by the Lagrangian~\\cite{Marfatia:2021hcp}\n \\begin{eqnarray}\n \\mathcal{L}\\supset \\bar{\\chi}(i\\slashed{\\partial}-M_i)\\chi -g_{\\chi}\\phi \\bar{\\chi}\\chi-V_{\\mathrm{eff}}(\\phi,T)\\,,\n \\end{eqnarray}\n where $M_i$ represents the bare mass of $\\chi$ in the false vacuum,\n $V_{\\mathrm{eff}}$ is the finite-temperature quartic effective potential\n commonly appears from theoretical models\n ~\\cite{Dine:1992wr,Adams:1993zs}\n \\begin{eqnarray}\n V_{\\mathrm{eff}}(\\phi,T)=D(T^2-T^2_0)\\phi^2-(AT+C)\\phi^3+\\frac{\\lambda}{4}\\phi^4\\,,\n \\end{eqnarray}\n which induces the cosmological FOPT.\n The FOPT happened when the temperature becomes lower than critical temperature $T_c$,\n which is determined by the condition $V_{\\rm eff}(0,T_c)=V_{\\rm eff}(v_\\phi(T_c),T_c)$,\n at this moment, the false vacuum ($\\langle \\phi \\rangle=0$)\n was tunnelling to the true vacuum ($\\langle \\phi \\rangle = v_\\phi$).\n We define the parameter $B$ to be the zero-temperature potential energy density difference\n between the false and true vacuum~\\cite{Marfatia:2021twj}.\n Finally, the quartic potential can be described by these input parameters~\\cite{Marfatia:2021twj}\n \\[ \\lambda, A, B, C, D\\,. \\]\n\n We use the analytical expression for the Euclidean action $S_3(T)\/T$\n of quartic potentials~\\cite{Adams:1993zs}\n to compute the bubble nucleation rate per unit volume $\\Gamma(T)$\n and then obtain the fraction of space in the false vacuum $F(t)$~\\cite{Marfatia:2021twj}.\n The phase transition temperature $T_\\star$ identified as percolation\n temperature can be obtained when a fraction $1\/e$ of the space\n remains in the false vacuum. Such that the corresponding time of phase transition $t_\\star$\n is given by $F(t_\\star)=1\/e\\simeq 0.37$.\n\n \nIn general, GW signals accompany the FOPT from bubble collisions, sound waves, and Magnetohydrodynamic turbulence.~\\cite{Huber:2008hg,Nakai:2020oit,Espinosa:2010hh,Caprini:2015zlo,Kehayias:2009tn}. In our case, since the bubble wall reaches a relativistic terminal velocity thus corresponds to a non-runaway bubble, the sound waves dominates the GW production~\\cite{Caprini:2015zlo}. We follow the semi-analytical approaches to evaluate the GW spectra~\\cite{Marfatia:2020bcs,Caprini:2015zlo}.\n GW characteristic frequency associates with\n the\n \n inverse time duration of the FOPT\n $\\beta\/H_\\star\\simeq T_\\star\\,\n \\left[d(S_3\/T)\/dT\\right]$~\\cite{Nakai:2020oit},\n which is normalized to Hubble time scale. In addition to this parameter,\n the GW spectrum also depends on $\\alpha$, the strength of FOPT,\n which is defined as the ratio between\n false vacuum energy to the total radiation energy.\n\n\n To find $n_{\\rm FB}$ the number density of FB, we need to know\n the critical volume $V_\\star$ which is defined from a volume in false vacuum satisfying the\n condition $\\Gamma(T_\\star)V_\\star R_\\star\\simeq v_w$~\\cite{Hong:2020est},\n and thus this volume would not further separate into smaller one during its shrinking.\n Here $R_\\star$ and $v_w$\n are the radius of critical volume and the bubble wall velocity, respectively.\n Under the assumption that each critical volume corresponds to one FB,\n the volume fraction of false vacuum implies the relation\n $n_{\\rm FB}|_{T_\\star}= F(t_\\star)\/V_\\star$ at $t_\\star$.\n\n Inside a FB, to avoid complete annihilation $\\bar{\\chi} \\chi \\to \\phi \\phi$,\n there must be a nonzero number density asymmetry\n $\\eta_{\\rm DM} \\equiv (n_\\chi-n_{\\bar{\\chi}})\/s$ (normalized to the entropy density).\n Then we define $Q_{\\rm FB}$ the total number of $\\chi$'s comprising a FB:\n $Q_{\\rm FB}\\equiv \\eta_{\\rm DM}(s\/n_{\\rm FB})|_{T_\\star}$~\\cite{Hong:2020est}.\n\n In general, the dark and SM sectors need not to be in thermal equilibrium,\n otherwise the dark radiation ($\\phi$ and $\\chi$) would contributes\n the effective number of extra neutrino species\n $\\Delta N_{\\rm eff}$\n and exceed the current observational upper bound.\n In fact, lowering the dark sector temperature than SM sector\n alleviates the tension between $\\Delta N_{\\rm eff}$ and observations~\\cite{Marfatia:2021twj}.\n Thus, we include the temperature ratio $r_T\\equiv T_\\star\/T_{\\rm SM_{\\star}}$\n at $t_\\star$ as one of the input parameter for FOPT.\n\n The dark fermion particles $\\chi$'s aggregated to form\n macroscopic FBs due to the FOPT.\n Subsequently, the attractive Yukawa force between $\\chi$'s mediated by\n $\\phi$ destabilizes the FBs to form PBHs.\n In the following section, we quantitatively discuss these processes.\n \n\n \n\n \n\n\n\n\n \\bigskip\n\n \\subsection{FB properties and PBH formations}\n The derivation for the FB mass and radius can be found in Ref.~\\cite{Marfatia:2021hcp}. Including the Fermi gas kinetic energy, Yukawa potential energy, and the temperature-dependent potential energy difference between the false and true vacua; the total energy of a FB is approximately written as\n\n \\begin{eqnarray}\n \\label{eq:FB_energy}\n E_{\\rm FB}&=&\n \\frac{3\\pi}{4} \\left( \\frac{3}{2\\pi} \\right)^{2\/3} \\frac{Q^{4\/3}_{\\rm FB}}{R}\n \\left[1+\\frac{4 \\pi}{9}\\left(\\frac{2\\pi}{3} \\right)^{1\/3} \\frac{R^2 T^2}{Q^{2\/3}_{\\rm FB}}\n \\left(1+\\frac{3}{2\\pi^2}\\frac{M^2_i}{T^2} \\right)\n \\right] \\nonumber \\\\\n &&\n -\\frac{3 g^2_\\chi}{8 \\pi} \\frac{Q^2_{\\rm FB} L^2_\\phi}{R^3}\n + \\frac{4 \\pi}{3} V_0 R^3 \\left(1+\\frac{T^2 M^2_i}{12 V_0} \\right) \\,,\n \\end{eqnarray}\n where $V_0(T)\\equiv V_{\\rm eff}(0,T) - V_{\\rm eff}(v_\\phi(T),T)$, which at zero temperature returns into $B$. Since the FB is a macroscopic object, we ignore the contribution from the surface tension.\n Inside a FB, each $\\chi$ has mass $M_i$ and couples to others via the attractive Yukawa interaction $g_{\\chi}\\phi \\bar{\\chi}\\chi$. The length of Yukawa interaction is given as\n \\begin{eqnarray}\n L_\\phi(T)\\equiv \\left(\\frac{d^2 V_\\mathrm{eff}}{d\\phi^2}\\bigg|_{\\phi=0}\\right)^{-1\/2}=(2D(T^2-T^2_0))^{-1\/2} \\,.\n \\end{eqnarray}\n\n To find $R_{\\rm FB}$, we require $dE_{\\rm FB}\/dR=0$,\n which yields a cubic polynomial equation with three solutions.\n The largest among the three roots\n gives the FB radius $R_{\\rm FB}$.\n Evaluating $E_{\\rm FB}$ at $R_{\\rm FB}$ gives rise the mass of the FB,\n we have\n \\begin{eqnarray}\n R_{\\rm FB} &=& \\left[ \\frac{1}{4}\\left( \\frac{3}{2\\pi} \\right)^{2\/3} \\frac{Q^{4\/3}_{\\rm FB}}{V_0} \\right]^{1\/4} X^{1\/2} \\\\\n M_{\\rm FB} &=& \\frac{3}{4} Q_{\\rm FB} \\left( 9 \\pi^2 V_0 \\right)^{1\/4} X^{-3\/2} \\nonumber \\\\\n &\\times& \\left\\lbrace X+ \\frac{4}{9}\\left(1+\\frac{T^2 M^2_i}{12V_0} \\right)X^3\n + \\left( \\frac{2\\pi T^2}{9 V^{1\/2}_0}+ \\frac{M^2_i}{3\\pi V^{1\/2}_0} \\right) X^2\n - \\frac{2g^2_\\chi L^2_\\phi V^{1\/2}_0}{3\\pi}\\right\\rbrace\\,,\n \\label{eq:FB_mass_radius}\n \\end{eqnarray}\n where denote\n \\begin{eqnarray}\n X&\\equiv& \\left(1+\\frac{T^2 M^2_i}{12V_0} \\right)^{-1} \\nonumber\\\\\n & \\times &\\Bigg[\\left( 1+ \\frac{13}{108}\\frac{T^2M^2_i}{V_0}+ \\frac{\\pi^2}{81} \\frac{T^4}{V_0} + \\frac{1}{36\\pi^2}\\frac{M^4_i}{V_0} \\right)^{1\/2} \\cos\\theta-\\frac{\\pi}{18} \\frac{T^2}{V^{1\/2}_0}\\left(1+\\frac{3}{2\\pi^2}\\frac{M^2_i}{T^2} \\right) \\Bigg]\\,. \\nonumber\n \\end{eqnarray}\nTo make a FB stable in the false vacuum, it further requires the conditions~\\cite{Kawana:2021tde}\n\\begin{eqnarray}\n\\label{eq:FB_stable}\n\\frac{dM_{\\rm FB}}{dQ_{\\rm FB}} \\leq g_\\chi v_\\phi+M_i\\,,~~~~\\frac{d^2M_{\\rm FB}}{dQ^2_{\\rm FB}}\\leq 0\\,,\n\\end{eqnarray}\nwhere the second one is automatically satisfied by the surface tension.\nHowever, in our region of interest, non-zero $M_i$ is always necessary to full fill the first condition with perturbative Yukawa coupling\n$|g_\\chi|\\leq \\sqrt{4 \\pi}$.\n\n\n If we neglect the Yukawa energy, $X$ reduces into\n \\begin{eqnarray}\n X\n &\\simeq& \\frac{\\sqrt{3}}{2}\\left(1-\\frac{1}{4\\sqrt{3}\\pi}\\frac{M^2_i}{V^{1\/2}_0}- \\frac{\\pi}{6\\sqrt{3}}\\frac{T^2}{V^{1\/2}_0}-\\frac{5}{216} \\frac{M^2_iT^2}{V_0} \\right)\\,.\n \\end{eqnarray}\n Under the assumption $V_0\\gg T$ and $M_i$, Eq.~(\\ref{eq:FB_mass_radius}) can be simplified into\n \\begin{eqnarray}\n \\label{eq:FB_mass_radius_simple}\n R_{\\rm FB}&=& \\left[ \\frac{3}{16} \\left( \\frac{3}{2\\pi} \\right)^{2\/3}\n \\frac{Q^{4\/3}_{\\rm FB}}{V_0} \\right]^{1\/4}\n \\left(1-\\frac{1}{8\\sqrt{3}\\pi}\\frac{M^2_i}{V^{1\/2}_0}\n - \\frac{\\pi}{12\\sqrt{3}}\\frac{T^2}{V^{1\/2}_0}-\\frac{5}{432} \\frac{M^2_iT^2}{V_0} \\right)\\,, \\\\\n M_{\\rm FB} &=& Q_{\\rm FB}\\left( 12 \\pi^2 V_0 \\right)^{1\/4}\n \\left(1+ \\frac{\\sqrt{3}}{8\\pi} \\frac{M^2_i}{V^{1\/2}_0}\n +\\frac{\\pi}{4\\sqrt{3}}\\frac{T^2}{V^{1\/2}_0}-\\frac{1}{16} \\frac{M^2_i T^2}{V_0} \\right)\\,.\n \\end{eqnarray}\n\n\n\nDue to the fact that the magnitude of Yukawa energy\n \\begin{eqnarray}\n |E_Y|\\simeq \\frac{3g^2_\\chi}{8 \\pi} \\frac{Q^2_{\\rm FB}}{R}\\, \\left(\\frac{L_\\phi}{R} \\right)^2\n \\end{eqnarray}\nincreases as the temperature decreases, at temperature $T_\\phi$ when $|E_Y|$ equals to the Fermi-gas kinetic energy, the FB becomes unstable and starts collapsing to a PBH. This roughly coincides when $L_\\phi$ equals to the mean separation distance of $\\chi$'s, i.e., $L_\\phi\\simeq R_{\\rm FB}\/Q^{1\/3}_{\\rm FB}$.\nTherefore, there are two scenarios of PBH formation:\ni) If $T_\\phi \\leq T_\\star$, the FB forms as an intermediate state before collapsing into PBH.\nii) If $T_\\phi > T_\\star$, the $\\chi$'s enclosed in a critical volume directly collapse into PBH without FB formation.\nIn the first scenario, we can adopt the above formulas of FB to compute the mass of PBH, i.e. $M_{\\rm PBH}(T_\\phi)=M_{\\rm FB}(T_\\phi)$, and the\nnumber density follows the adiabatic evolution of Universe $n_{\\rm PBH}|_{T_\\phi}=n_{\\rm FB}|_{T_\\star} s(T_\\phi)\/s(T_\\star)$~\\cite{Kawana:2021tde}.\nConsequently, the PBH relic abundance and fraction at present Universe is given by\n\\begin{eqnarray}\n\\Omega_{\\rm PBH}h^2 = \\frac{M_{\\rm PBH}|_0 n_{\\rm PBH}|_0}{3 M^2_{\\rm Pl}(H_0\/h)^2}\\,,~ f_{\\rm PBH}\\equiv \\frac{\\Omega_{\\rm PBH}h^2}{\\Omega_{\\rm DM}h^2}\\,,\n\\end{eqnarray}\nwhere the Hubble constant $H_0=2.13h\\times 10^{-42}~{\\rm GeV}$.\nIn the following section,\nwe will select the benchmark points belong to the first scenario.\n\n\n\n\n\n\\bigskip\n\n\\section{Correlated signals}\n\\label{sec:BP}\n\n \\begin{figure}[t]\n \\centering\n \\includegraphics[width=13cm]{FIG\/3.jpg}\n \\caption{The extragalactic photon contributions of {\\bf BP}s in Table~\\ref{table:1}. }\\label{fig:3}\n \\end{figure}\n\n \\begin{figure}[t]\n \\centering\n \\includegraphics[width=13cm]{FIG\/4.jpg}\n \\caption{\n Gravitational wave power spectra for {\\bf BP}s in Table~\\ref{table:1}.\n }\\label{fig:4}\n \\end{figure}\n\n Since the FOPT behaves as common origin for both PBHs and GW,\n it predicts the correlated signals among the 511 keV line,\n extragalactic photon spectrum,\n and GW spectrum.\n More specifically, the quartic potential dictates the FOPT\n and then the Yukawa interaction determines the period of PBHs formation.\n %\n Therefore, we select and scan the input parameters: including the coefficients of the effective potential,\n the asymmetry parameter $\\eta_\\mathrm{DM}$,\n the temperature ratio of the dark and SM sectors $r_T$,\n the Yukawa coupling $g_\\chi$, and the bare DM mass $M_i$,\n in the ranges\n\\begin{eqnarray}\n&& 0.1\\leq B^{1\/4}\/{\\rm MeV}\\leq 10^4, \\mkern15mu 0.1\\leq D\\leq 10, \\mkern15mu 0.05\\leq \\lambda \\leq 0.2 \\nonumber \\\\\n&& 0.01\\leq C\/{\\rm MeV} \\leq 10^4 , \\mkern15mu 0.3\\leq r_T\\leq 1, \\mkern15mu 0.01\\leq g_\\chi \\leq \\sqrt{4\\pi} \\nonumber \\\\\n&& 10^{-3} \\leq M_i\/B^{1\/4} \\leq 10\\,,\n\\end{eqnarray}\n where $A$ is fixed to be $0.1$.\n We pick six benchmark points (BPs) satisfying the 511 keV excess with the corresponding PBH mass and $f_\\mathrm{PBH}$ are listed in Table~\\ref{table:1} and Fig.~\\ref{fig:2}.\n \\begin{table}[h]\n \n \\centering\n \\resizebox{\\textwidth}{!}{\n \\begin{tabular}{c|c c| c c| c c }\n \\hline\n \\hline\n & \\bf{BP-1} & \\bf{BP-2} & \\bf{BP-3} & \\bf{BP-4}& \\bf{BP-5} & \\bf{BP-6} \\\\\n \\hline\n \\hline\n $B\/\\rm MeV^4$ & $5.94\\times10^{4}$ & $4.13\\times10^{5}$ & $5.69\\times10^{2}$ & $4.80\\times10^{5}$ & $3.99\\times10^{7}$ & $5.19\\times10^{6}$ \\\\\n $\\lambda$ & $0.1693$ & $0.0888$ & $0.0999$ & $0.0795$ & $0.0685$ & $0.1903$\\\\\n $D$ & $1.9103$ & $0.5360$ & $3.5249$ & $1.6074$ & $1.3413$ & $0.8706$ \\\\\n $\\eta_{\\mathrm{DM}}$ & $4.70\\times10^{-14}$ & $3.60\\times10^{-13}$ & $6.70\\times10^{-11}$ & $7.60\\times10^{-11}$ & $4.35\\times10^{-11}$ & $1.08\\times10^{-9}$ \\\\\n $r_T$ & $0.4377$ & $0.3512$ & $0.3149$ & $0.3845$ & $0.3891$ & $0.4707$\\\\\n $C\/\\rm MeV$ & $1.5257$ & $1.3976$ & $0.0920$ & $0.5830$ & $2.8082$ & $3.6441$\\\\\n $g_{\\chi}$ & $1.2728$ & $1.1556$ & $1.0275$ & $1.1958$ & $0.9141$ & $1.3278$\\\\\n $M_i\/B^{1\/4}$ & $0.5570$ & $0.0516$ & $0.4973$ & $0.0047$ & $0.3048$ & $1.4341$ \\\\\n \\hline\n $M_\\mathrm{PBH}\/M_{\\bigodot}$ & $9.32\\times 10^{-19}$ & $4.39\\times 10^{-18}$ & $1.52\\times 10^{-17}$ & $3.72\\times 10^{-17}$ & $4.90\\times 10^{-17}$ & $7.04\\times 10^{-17}$\\\\\n $f_\\mathrm{PBH}$ & $4.74\\times 10^{-6}$ & $8.20\\times 10^{-5}$ & $0.0013$ & $0.0100$ & $0.0235$ & $0.4350$\\\\\n \\hline\n \\end{tabular}\n }\n \\caption{\n The benchmark points form PBHs after FOPT with $A=0.1$ fixed.\n }\n \\label{table:1}\n \\end{table}\n\n\n According to the BPs in Table~\\ref{table:1}, in order to produce the desired PBH mass,\n one requires the energy scale $\\mathcal{O}(1)\\lesssim B^{1\/4}\/{\\rm MeV} \\lesssim \\mathcal{O}(100)$ of FOPT. By incorporating $\\eta_{\\rm DM}$, we can obtain the correct values of $f_{\\rm PBH}$ to generate 511 keV line.\n Under our scenario,\n FB was formed as an intermediate state and treated as a progenitor of PBH.\n In order to satisfy the stability condition of FB, i.e Eq.(\\ref{eq:FB_stable}), all BPs tend to have large values of $g_\\chi$, close to the perturbative limit,\n and nonzero values of $M_i$ are necessary.\n In addition, the observational upper bound of $\\Delta N_{\\rm eff}\\leq 0.5$\n was imposed to all the BPs,\n and thereby restricts the temperature ratio to be $0.3\\leq r_T\\leq 0.5$.\n Here, we assumed the dark sector and SM sector are thermally decoupled,\n such that it suppresses the dark sector contribution to the light degree of freedom.\n\n\n The extragalactic photon contributions of these BPs are shown\n in Fig.~\\ref{fig:3}. {\\bf{BP-4}}, {\\bf{BP-5}} and {\\bf{BP-6}} haven't\n been rule out by the present observations COMPTEL\/EGRET\/FermiLAT.\n This is because the associated PBH masses from {\\bf{BP-4}}, {\\bf{BP-5}} and {\\bf{BP-6}} are heavier\n comparing with {\\bf{BP-1}}, {\\bf{BP-2}}, and {\\bf{BP-3}}.\n Base on the property of Hawking radiation,\n the gamma-ray spectra of them peak\n at lower energy window between\n 0.1 MeV to 1 MeV, where the present observations do not have sufficient sensitivities.\n However, this window will be explored\n by future AMEGO and e-ASTROGAM observations.\n %\n On the other hand, the GW spectra of BPs, mainly arising from sound waves in the plasma after bubbles collision during the FOPT~\\cite{Caprini:2015zlo},\n are shown in Fig.~\\ref{fig:4}.\n They cover the frequency region from $10^{-5}$ Hz to $10^{-3}$ Hz,\n and their amplitudes can be substantially detected by future $\\mu$Ares telescope.\n\n\n\\bigskip\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nPBH with mass $3\\times 10^{-17} \\lesssim M_{\\rm PBH}\/m_\\odot \\lesssim 7\\times 10^{-17}$ and fractional abundance\n$0.01 \\lesssim f_{\\rm PBH} \\lesssim 0.5$ is favored to explain\nthe galaxy center 511 keV line excess.\nPBH evaporation emits not only positrons but also photons\ncontributing to the extragalactic gamma-ray flux.\nWe investigated the novel PBH production mechanism from the cosmological FOPT\naggregating $\\chi$'s into FB as intermediate state.\nThrough the attractive Yukawa interaction, FBs become unstable as temperature decreasing and eventually collapse to form PBHs.\nTo realise this scenario,\nwe selected the BPs satisfying\nseveral conditions: $\\Delta N_{\\rm eff}\\leq 0.5$,\nstability of FB in the false vacuum,\nand upper limits of gamma-ray flux from EGERT\/COMPTEL\/FermiLAT.\nIncorporate the $\\chi$ asymmetry $\\eta_{\\rm DM}\\simeq 10^{-10}$,\nthe phase transition with vacuum energy\n$\\mathcal{O}(1)\\lesssim B^{1\/4}\/{\\rm MeV} \\lesssim \\mathcal{O}(100)$\nproduces the desired PBH mass and abundance for 511 keV excess.\nConsequently, correlated signals of extragalactic $\\mathcal{O}({\\rm MeV})$ gamma-ray and GW spectrum with peak frequency\nfrom $10^{-5}$ Hz to $10^{-3}$ Hz are predicted.\nIn the future, these signals can be either confirmed or ruled out by AMEGO\/e-ASTROGAM\nand $\\mu$Ares.\n\n\n\n\\bigskip\n\n\\section*{Acknowledgment}\nP.Y.Tseng is supported in part by the Ministry of Sciences and Technology with\nGrant No. MoST-111-2112-M-007-012-MY3.\nY.M.Yeh is supported in part by Ministry of Education with Grant No. 111J0382I4.\n\n\n\n\n\\bigskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\t\n\nIn \\cite{Ni:Siena} considering a second order operator in ${\\mathbb R}^{1+2}$\n \\begin{equation}\n \\label{eq:model}\n P_{mod}=-D_0^2+2x_1D_0D_2+D_1^2+x_1^3D_2^2,\\quad x=(x_0, x')=(x_0, x_1, x_2)\n \\end{equation}\n %\n we have proved\n\\begin{them}\n\\label{thm:main}{\\rm(\\cite[Theorem 1.1]{Ni:Siena})}\nThe Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is not locally solvable at the origin in the Gevrey class of order $s$ for any $b_0, b_1, b_2\\in {\\mathbb C}$ if $s>6$. In particular the Cauchy problem for $P_{mod}$ is $C^{\\infty}$ ill-posed near the origin for any $b_0, b_1, b_2\\in {\\mathbb C}$.\n\\end{them}\nRecall that the Gevrey class of order $s$, denoted by $\\gamma^{(s)}({\\mathbb R}^n)$, is the set of all $f(x)\\in C^{\\infty}({\\mathbb R}^n)$ such that for any compact set $K\\subset {\\mathbb R}^n$, there exist $C>0$, $h>0$ such that\n\\begin{equation}\n\\label{eq:gevrey}\n|\\partial_x^{\\alpha}f(x)|\\leq Ch^{|\\alpha|}|\\alpha|!^s,\\quad x\\in K,\\;\\;\\alpha\\in {\\mathbb N}^{n}.\n\\end{equation}\nWe say that the Cauchy problem for $P=P_{mod}+\\sum_{j=0}^2b_jD_j$ is locally solvable in $\\gamma^{(s)}$ at the origin if for any $\\Phi=(u_0, u_1)\\in (\\gamma^{(s)}({\\mathbb R}^2))^2$ there exists a neighborhood $U_{\\Phi}$, may depend on $\\Phi$, of the origin such that the Cauchy problem \n\\begin{equation}\n\\label{eq:CP}\n\\begin{cases}\nPu=0\\quad \\text{in}\\;\\;U_{\\Phi},\\\\\nD_0^ju(0, x')=u_j(x'),\\quad x'\\in U_{\\Phi}\\cap\\{x_0=0\\},\\;\\;j=0, 1\n\\end{cases}\n\\end{equation}\nhas a solution $u(x)\\in C^2(U_{\\Phi})$.\nIn this note we remark that one can improve Theorem \\ref{thm:main} so that\n\\begin{them}\n\\label{thm:main:bis}\nThe Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is not locally solvable in $\\gamma^{(s)}$ at the origin for any $b_0, b_1, b_2\\in {\\mathbb C}$ if $s>5$. \n\\end{them}\n$P_{mod}$ plays a special role in the well-posedness of the Cauchy problem for differential operators with non-effectively hyperbolic characteristics. Before explaining it we give a short introduction to the general context. Let $P$ be a differential operator of order $m$ with principal symbol $p(x, \\xi)$. At a singular point $\\rho$ of $p=0$, the Hamilton map $F_p(\\rho)$ is defined as the linearization at $\\rho$ of the Hamilton vector field $H_p$. The first fundamental result is\n\\begin{them}\n\\label{thm:IPH}{\\rm(\\cite{IP}, \\cite{Ho1})} Let $\\rho$ be a singular point of $p=0$ and assume that $F_p(\\rho)$ has no non-zero real eigenvalues. If the Cauchy problem for $P$ is $C^{\\infty}$ well posed it is necessary that\n\\begin{equation}\n\\label{eq:IPH}\n{\\mathsf{Im}}P_{sub}(\\rho)=0,\\quad \\big|P_{sub}(\\rho)\\big|\\leq {\\rm Tr^+}F_p(\\rho)\/2\n\\end{equation}\nwhere ${\\rm Tr^+}F_p(\\rho)=\\sum|\\mu_j|$ and $\\mu_j$ are the eigenvalues of $F_p(\\rho)$, counted with multiplicity, and $P_{sub}$ is the subprincipal symbol of $P$. \n\\end{them}\nWe call \\eqref{eq:IPH} the Ivrii-Petkov-H\\\"ormander condition (IPH condition, for short). If the strict inequality holds in \\eqref{eq:IPH} we call it the strict Ivrii-Petkov-H\\\"ormander condition (strict IPH condition, for short). For the sufficiency of the IPH condition, we assume that the set of singular points $\\Sigma$ of $p=0$ is a $C^{\\infty}$ manifold and the following conditions are satisfied:\n\\begin{equation}\n\\label{eq:jyoken}\n\\begin{cases}\n\\text{$F_p$ has no non-zero real eigenvalues at each point of $\\Sigma$},\\\\\n\\text{Near each point of $\\Sigma$, $p$ vanishes exactly of order $2$} \\\\\n\\text{and the rank of $\\sum_{j=0}\n^nd\\xi_j\\wedge dx_j$ is constant. }\n\\end{cases}\n\\end{equation}\nAccording to the spectral type of $F_p(\\rho)$, two different possible cases may arise\n\\begin{align}\n\\label{eq:type:1}\n{\\rm Ker}\\,F_p^2(\\rho)\\cap {\\rm Im}\\,F_p^2(\\rho)= \\{0\\}, \\\\\n\\label{eq:type:2}\n{\\rm Ker}\\,F_p^2(\\rho)\\cap {\\rm Im}\\,F_p^2(\\rho)\\neq \\{0\\}.\n\\end{align}\nWe say that $p$ is of spectral type $1$ (resp. type $2$) near $\\rho\\in\\Sigma$ if there is a conic neighborhood $V$ of $\\rho$ such that \\eqref{eq:type:1} (resp. \\eqref{eq:type:2}) holds in $V\\cap \\Sigma$. We say that there is no transition (of spectral type) if for any $\\rho\\in \\Sigma$ one can find a conic neighborhood $V$ of $\\rho$ such that either \\eqref{eq:type:1} or \\eqref{eq:type:2} holds in $V\\cap\\Sigma$. \n\\begin{them}\n\\label{thm:jyubun} {\\rm(\\cite[Theorem 5.1]{Ni:book})} Assume \\eqref{eq:jyoken} and that there is no transition and no bicharacteristic tangent to $\\Sigma$. Then the Cauchy problem for $P$ is $C^{\\infty}$ well posed under the strict IPH condition.\n\\end{them}\nThe principal symbol $p=-\\xi_0^2+2x_1\\xi_0\\xi_2+\\xi_1^2+x_1^3\\xi_2^2$ of $P_{mod}$ is a typical example of spectral type $2$ with tangent bicharacteristics (note that there is no tangent bicharacteristic if $p$ is of spectral type $1$, see \\cite{Iv2}). The set of singular point of $p=0$ with $\\xi\\neq 0$ is $\\Sigma=\\{x_1=0, \\xi_0=\\xi_1=0\\}$ which is a $C^{\\infty}$ manifold on which $p$ vanishes exactly of order $2$ and a tangent bicharacteristic is given explicitly by\n\\begin{equation}\n\\label{eq:bitokusei}\n(x_1, x_2)=(-x_0^2\/4, x_0^5\/80),\\quad \n(\\xi_0, \\xi_1, \\xi_2)=(0, x_0^3\/8, 1)\n\\end{equation}\nwhich is parametrized by $x_0$. The operator $P_{mod}$ shows how the situation becomes to be complicated when a tangent bicharacteristic exists. We now give some such results: The Cauchy problem for $P_{mod}$ is not locally solvable in $\\gamma^{(s)}$ for $s>5$, in particular ill-posed in $C^{\\infty}$ while the Cauchy problem for general second order operator $P$ of spectral type $2$ satisfying $P_{sub}=0$ on $\\Sigma$ (note that the subprincipal symbol of $P_{mod}$ is identically zero) is well posed in the Gevrey class of order $13$ (Proposition \\ref{pro:main} below) while the Cauchy problem for general second order operator $P$ of spectral type $2$ with ${\\rm codim}\\,\\Sigma=3$ is well posed in the Gevrey class of order $15$.\n\\end{lemma}\nThus in order to prove Theorem \\ref{thm:main:bis} it suffices to prove the following result which also improves \\cite[Theorem 1.3]{BN:JHDE}.\n\\begin{pro}\n\\label{pro:main}The Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is not locally solvable in $\\gamma^{(s)}$ at the origin for any $b_0, b_1\\in {\\mathbb C}$ and $0\\neq b_2\\in{\\mathbb C}$ if $s>3$.\n\\end{pro}\nTo prove Proposition \\ref{pro:main} we repeat the proof of \\cite[Theorem 1.3]{BN:JHDE} with obvious minor changes. Look for a family of solutions to $(P_{mod}+\\sum_{j=0}^2b_jD_j)U_{\\lambda}=0$ in the form\n\\begin{equation}\n\\label{eq:teigi:U}\nU_{\\lambda}(x)=e^{ i\\xi_0\\lambda x_0}V_{\\lambda}(x'),\\quad V_{\\lambda}(x')=e^{\\pm \\lambda^5x_2-i(b_1\/2)x_1}u(\\lambda^2x_1),\\;\\;\\xi_0=\\xi_0(\\lambda)\n\\end{equation}\nthat is, we look for $u(x)$ satisfying\n\\begin{equation}\n\\label{eq:a8}\nu''(x)=\\big(x^3+ 2\\xi_0 x+ b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4\\big)u(x).\n\\end{equation}\nTo study solutions to \\eqref{eq:a8} we consider \n\\begin{equation}\n\\label{eq:aa1}\nu''(x)=(x^3+a_2x+a_3)u(x),\\quad x\\in{\\mathbb C},\\;\\;a_j\\in{\\mathbb C},\\;\\;j=2, 3.\n\\end{equation}\nLet ${\\mathcal Y}_0(x;a)$, $a=(a_2,a_3)$ be the solution given in \\cite[Chapter 2]{Si} to (\\ref{eq:aa1}) which has asymptotic representation\n\\begin{equation}\n\\label{eq:zenkin:a}\n{\\mathcal Y}_0(x; a)\\simeq x^{-3\/4}\\big[1+\\sum_{N=1}^{\\infty}B_Nx^{-N\/2}\\big]e^{-E(x,a)}\n\\end{equation}\nas $x$ tends to infinity in any closed subsector of the open sector $|\\arg x|<3\\pi\/5$ where\n\\[\nE(x,a)=\\frac{2}{5}x^{5\/2}+a_2x^{1\/2}\n\\]\nand $A_N$, $B_N$ are polynomials in $(a_2,a_3)$. Let $\\omega=e^{2\\pi i\/5}$ and set\n\\begin{equation}\n\\label{eq:setuzoku}\n{\\mathcal Y}_k(x;a)={\\mathcal Y}_0(\\omega^{-k}x; \\omega^{-2k}a_2,\\omega^{-3k}a_3),\\quad k=0, 1, 2, 3, 4\n\\end{equation}\nwhich are also solutions to (\\ref{eq:aa1}). Recall that \\cite[Chpater 17]{Si}\n\\[\n{\\mathcal Y}_k(x;a)=C_k(a){\\mathcal Y}_{k+1}(x;a)-\\omega {\\mathcal Y}_{k+2}(x;a)\n\\]\nwhere $C_k(a)$ are entire analytic in $a=(a_2,a_3)$ and $C_k(a_2,a_3)=C_0(\\omega^{-2k}a_2,\\omega^{-3k}a_3)$. Choose \n\\begin{equation}\n\\label{eq:aa2}\n\\begin{split}\n&u(x)={\\mathcal Y}_0(x; a)=C_0(a){\\mathcal Y_1}(x; a)-\\omega{\\mathcal Y_2}(x; a),\\\\\n &a_2=2\\xi_0,\\quad a_3= b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4\n\\end{split}\n\\end{equation}\nwhich solves \\eqref{eq:a8}. We require that $V_{\\lambda}(x')$ is bounded as $\\lambda\\to +\\infty$ when $|x'|$ remains in a bounded set. Note \\eqref{eq:setuzoku} and\n\\begin{equation}\n\\label{eq:heiho}\n\\begin{split}\n(\\omega^{-1}x)^{5\/2}=-i|x|^{5\/2},\\quad 2\\omega^{-2} \\xi_0(\\omega^{-1}x)^{1\/2}=-2i \\xi_0|x|^{1\/2},\\\\\n(\\omega^{-2}x)^{5\/2}=i|x|^{5\/2},\\quad 2\\omega \\xi_0(\\omega^{-2}x)^{1\/2}=2i \\xi_0|x|^{1\/2}\n\\end{split}\n\\end{equation}\nfor $x<0$. Then $|{\\mathcal Y}_1(\\lambda^2x; a_2, a_3)|$ and $|{\\mathcal Y}_2(\\lambda^2x;a_2, a_3)|$ behaves like $e^{2{\\mathsf{Re}}(i\\xi_0)\\lambda |x|^{1\/2}}$ and $e^{-2{\\mathsf{Re}}(i\\xi_0)\\lambda |x|^{1\/2}}$ respectively as $x\\to -\\infty$. Since $\\omega\\neq 0$, taking \\eqref{eq:aa2} into account, the requirement for boundedness implies that\n\\begin{align}\n\\label{eq:Stokes}\nC_0(2\\xi_0, a_3)=0,\\\\\n\\label{eq:expo}\n-{\\mathsf{Re}}(i\\xi_0)={\\mathsf{Im}}\\,\\xi_0<0.\n\\end{align}\n\nInstead of solving directly the ``Stokes equation\" (\\ref{eq:Stokes}) we go rather indirectly. Let us consider\n\\[\nH(\\beta)=p^2+x^2+i\\beta x^3\n\\]\nas an operator in $L^2({\\mathbb R})$ with the domain $D(H(\\beta))=D(p^2)\\cap D(x^3)$. Here $p^2$ denotes the self-adjoint realization of $-d^2\/dx^2$ defined in $H^2({\\mathbb R})$ and by $D(x^3)$ we mean the domain of the maximal multiplication operator by the function $x^3$.\n\n\\begin{pro}\n\\label{pro:Caliceti_1}{\\rm \\cite[Corollary 2.16, Lemma 3.1]{CGM})}\n Let $k\\in {\\mathbb N}_0$ and $\\epsilon>0$ be given. Then there is a $B>0$ such that for $|\\beta|0$, $H(\\beta)$ has exactly one eigenvalue $E_k(\\beta)$ near $2k+1$. Such eigenvalues are analytic functions of $\\beta$ for $|\\beta|0$, and admit an analytic continuation across the imaginary axis to the whole sector $|\\beta|0$ and $E(\\beta)$ is an eigenvalue of the problem\n\\begin{equation}\n\\label{eq:a1}\n-u''(x)+( x^2+i\\beta x^3)u(x)=E(\\beta)u(x)\n\\end{equation}\nthat is $(\\ref{eq:a1})$ has a solution $0\\neq u\\in D(H(\\beta))$. Then we have\n\\begin{equation}\n\\label{eq:a2}\nC_0\\Big(-\\frac{\\omega^2}{3}\\beta^{-\\frac{8}{5}},\\, \\omega^3\\big\\{\\frac{2}{27}\\beta^{-\\frac{12}{5}}+\\beta^{-\\frac{2}{5}}E(\\beta)\\bigl\\}\\Big)=0\n\\end{equation}\nwhere $\\beta^{-j\/5}=(\\beta^{-1\/5})^{j}$ and the branch $\\beta^{\\pm1\/5}$ is chosen such that $|\\arg \\beta^{\\pm1\/5}|<\\pi\/10$.\n\\end{lemma}\nLet $E(\\beta)$, $|\\beta|0$ be an eigenvalue which is analytically continued to the sector $|\\beta|0\n\\end{equation}\nwhere the second requirement comes from \\eqref{eq:expo}. Assume that \n\\begin{equation}\n\\label{eq:argA}\n0<\\arg b_2<\\pi \\quad \\text{or}\\quad -\\pi \\leq \\arg b_2<-\\pi\/2\n\\end{equation}\nand denote $27b_2 \/2$ by $A$ for notational simplicity and look for $\\zeta(\\lambda)$ in the form\n\\[\n\\left\\{\\begin{array}{l}\n\\zeta(\\lambda)=e^{-\\pi i\/5}A^{1\/6}\\big(1+\\lambda^{-5\/6}z\\big)\\lambda^{1\/6}\n\\quad (0<\\arg A<\\pi)\n,\\\\[5pt]\n\\zeta(\\lambda)=e^{2\\pi i\/15}A^{1\/6}\\big(1+\\lambda^{-5\/6}z\\big)\\lambda^{1\/6}\\quad \n(-\\pi\\leq \\arg A<-\\pi\/2).\\end{array}\\right.\n\\]\n It is clear that $\\zeta(\\lambda)$ verifies (\\ref{eq:cond}) provided that $z$ is bounded and $\\lambda$ is large. Note that $E(\\zeta^{-5\/2})$ is analytic in $|\\arg \\zeta|<\\pi\/4-\\epsilon$ for large $|\\zeta|$ and verifies\n\\[\n\\big|E(\\zeta^{\n-5\/2})-a_0\\big|\\leq C|\\zeta|^{-5}\n\\]\nwith some $a_0=2k+1$, $k\\in{\\mathbb N}$ uniformly in $|\\arg \\zeta|<\\pi\/4-\\epsilon$ when $|\\zeta|\\to \\infty$. We insert $\\zeta$ into (\\ref{eq:reduced}) to get\n\\begin{equation}\n\\label{eq:A:la}\n\\begin{split}\nA\\lambda(1+\\lambda^{-5\/6}z)^6+\\lambda^{1\/6}H(z)+d_1\\lambda^{-2\/3}(1+\\lambda^{-5\/6}z)^8\\\\\n+d_2\\lambda^{-7\/3}(1+\\lambda^{-5\/6}z)^4=A\\lambda+d_3\\lambda^{-4},\\quad d_i\\in{\\mathbb C}\n\\end{split}\n\\end{equation}\nwhere $H(z)$ is analytic in $|z|R$.\n\\end{pro}\nIf we write \n\\begin{equation}\n\\label{eq:a:b:teigi}\n2\\xi_0=\\lambda^{2\/3}a(\\lambda),\\quad b_2\\lambda-\\xi_0^2\\lambda^{-2}+ b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4=\\lambda b(\\lambda)\n\\end{equation}\n it is clear that $|a(\\lambda)|$, $|b(\\lambda)|0$ for $\\lambda\\geq R_1$.\n\n\n\n\\subsection{Asymptotics of ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda )$ for large $|x|$ and $\\lambda$ }\n\nRecalling \\eqref{eq:a:b:teigi} we study how ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda )$ behaves for large $|x|$ and large $\\lambda$ where $|a|, |b| \\leq M$ is assumed. \nIn what follows $f=o_{a}(1)$ means that there are positive constants $C_{a}>0$ and $\\delta_a>0$ such that\n\\[\n|f|\\leq C_a\\lambda^{-\\delta_a}, \\quad \\lambda\\to+\\infty.\n\\]\nWe make the asymptotic representation \\eqref{eq:zenkin:a} slightly precise. \n\\begin{pro}\n\\label{pro:Esti:Y}\nLet $\\rho>1\/3$ be given. Then one can write\n\\begin{gather*}\n{\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)\n\\simeq (1+p_{\\rho}(x,\\lambda))e^{R_{\\rho}(x,\\lambda)}x^{-3\/4}\\exp{\\{-E_{\\rho}(x; a, b, \\lambda)\\}},\\\\\n{\\mathcal Y}_0'(x;a\\lambda^{2\/3},b\\lambda)\n\\simeq (-1+p_{\\rho}(x,\\lambda))e^{R_{\\rho}(x,\\lambda)}x^{3\/4}\\exp{\\{-E_{\\rho}(x; a, b, \\lambda)\\}}\n\\end{gather*}\nas $x$ tends to infinity in any closed subsector of\n\\[\nS_{\\lambda}=\\{x;|\\arg x|<3\\pi\/5, |x|> \\lambda^{\\rho}\\}\n\\]\nwhere \n\\[\nE_{\\rho}(x; a, b, \\lambda)=\\frac{2}{5}x^{5\/2}+a\\lambda^{2\/3} x^{1\/2}-b\\lambda x^{-1\/2}+r_{\\rho}(x,\\lambda)\n\\]\nand $r_{\\rho}$ is a polynomial in $x^{-1\/2}$ such that\n\\[\n|r_{\\rho}(x,\\lambda)|=C\\lambda^{4\/3-3\\rho\/2}(1+o_{\\rho}(1)),\\quad x\\in S_{\\lambda}\n\\]\nand $p_{\\rho}(x,\\lambda)$, $R_{\\rho}(x,\\lambda)$ are holomorphic in $S_{\\lambda}$ and in any closed subsector of $S_{\\lambda}$\n\\begin{gather*}\n|p_{\\rho}(x,\\lambda)|\\leq C_{\\rho}\\lambda^{-2(\\rho-1\/3)},\\quad |R_{\\rho}(x,\\lambda)|\\leq C_{\\rho}\\lambda^{-1},\\quad x\\in S_{\\lambda}\n\\end{gather*}\nand $R_{\\rho}(x,\\lambda)\\to 0$, $p_{\\rho}(x,\\lambda)\\to 0$ as $|x|\\to\\infty$, $x\\in S_{\\lambda}$.\n\\end{pro}\n\\begin{proof} \nWe follow Sibuya \\cite[Chapter 2]{Si} and \\cite[Proposition 2.3]{BN:JHDE} with needed modifications. \n\\end{proof}\n\\if0\n\\begin{proof} \nWe follow Sibuya \\cite{Si} and \\cite{BN2} with needed modifications. \nLet $y(x,\\lambda)$ satisfy\n\\begin{equation}\n\\label{eqn:1}\ny''(x,\\lambda)=(x^3+a\\lambda^{2\/3} x+b\\lambda)y(x,\\lambda)\n\\end{equation}\nwhere $|a|,|b|\\leq M$. \nSet\n\\[\nx=\\xi^2,\\quad \\left[\\begin{array}{l}\ny\\\\\ny'\\end{array}\\right]=\\left[\\begin{array}{cc}\n1&0\\\\\n0&\\xi^3\\end{array}\\right]v\n\\]\nthen $v$ satisfies \n\\[\n\\frac{dv}{d\\xi}=\\xi^4\\left[\\begin{array}{cc}\n0&2\\\\\n2+2a\\lambda^{2\/3}\\xi^{-4}+2b\\lambda\\xi^{-6}&-3\\xi^{-5}\\end{array}\\right]v.\n\\]\nLet us set again\n\\[\nv=\\left[\\begin{array}{cc}\n1&1\\\\\n-1&1\\end{array}\\right]w\n\\]\nso that $w$ verifies\n\\[\n\\frac{dw}{d\\xi}=\\xi^4\\left[\\begin{array}{cc}\n\\alpha_1(\\xi)&\\beta_1(\\xi)\\\\\n\\beta_2(\\xi)&\\alpha_2(\\xi)\\end{array}\\right]w\n\\]\nwhere\n\\[\n\\left\\{\n\\begin{array}{ll}\n\\alpha_1(\\xi)=-2-a\\lambda^{2\/3}\\xi^{-4}-\\frac{3}{2}\\xi^{-5}-b\\lambda\\xi^{-6},&\\beta_1(\\xi)=-a\\lambda^{2\/3}\\xi^{-4}+\\frac{3}{2}\\xi^{-5}-b\\lambda\\xi^{-6}\n,\\\\\n\\alpha_2(\\xi)=2+a\\lambda^{2\/3}\\xi^{-4}-\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6},&\n\\beta_2(\\xi)=a\\lambda^{2\/3}\\xi^{-4}+\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6}.\n\\end{array}\\right.\n\\]\nLet us set \n\\[\nw=\\left[\\begin{array}{ll}\n1\\\\\np\\end{array}\\right]e^{\\int^{\\xi}\\eta^4\\gamma(\\eta)d\\eta}\n\\]\nwhere now $p$ and $\\gamma$ are unknowns. It is clear that\n\\[\n\\left\\{\\begin{array}{ll}\n\\gamma(\\xi)=\\alpha_1(\\xi)+\\beta_1(\\xi)p,\\\\\n\\displaystyle{\\frac{dp}{d\\xi}=\\xi^4(\\beta_2(\\xi)+(\\alpha_2(\\xi)-\\gamma(\\xi))p)}\\end{array}\\right.\n\\]\nwhich gives\n\\[\n\\frac{dp}{d\\xi}=\\xi^4(\\beta_2+(\\alpha_2-\\alpha_1)p-\\beta_1p^2).\n\\]\nDenoting\n\\[\n\\left\\{\n\\begin{array}{lll}\nf(\\xi)=\\beta_2(\\xi)=a\\lambda^{2\/3}\\xi^{-4}+\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6},\\\\[3pt]\nh(\\xi)=\\alpha_2(\\xi)-\\alpha_1(\\xi)=4+2a\\lambda^{2\/3}\\xi^{-4}+2b\\lambda \\xi^{-6},\\\\[3pt]\ng(\\xi)=-\\beta_1(\\xi)=a\\lambda^{2\/3}\\xi^{-4}-\\frac{3}{2}\\xi^{-5}+b\\lambda\\xi^{-6}\n\\end{array}\\right.\n\\]\none can write\n\\begin{equation}\n\\label{eqn:2}\n\\frac{dp}{d\\xi}=\\xi^4\\bigl[f(\\xi)+h(\\xi)p+g(\\xi)p^2\\bigr].\n\\end{equation}\nPutting \n\\[\n\\theta=\\xi^{-5},\\quad \\phi=a\\lambda^{2\/3}\\xi^{-4},\\quad \\psi=b\\lambda\\xi^{-6}\n\\]\nsuch that $f(\\xi)=\\phi+\\frac{3}{2}\\theta+\\psi$, $h(\\xi)=4+2\\phi+2\\psi$, $g(\\xi)=\\phi-\\frac{3}{2}\\theta+\\psi$ we look for a formal solution to (\\ref{eqn:2}) of the form\n\\[\n{\\hat p}=\\sum_{k+\\ell+m\\geq 1}C_{k,\\ell,m}\\theta^k\\phi^{\\ell}\\psi^m.\n\\]\nSince \n\\[\n\\xi^{-4}\\frac{d}{d\\xi}(\\theta^k\\phi^{\\ell}\\psi^m)=-(5k+4\\ell+6m)\\theta^{k+1}\\phi^{\\ell}\\psi^m\n\\]\nit is easy to see that $C_{k,\\ell,m}$ are uniquely determined by \n\\begin{eqnarray*}\nC_{100}=-3\/8,\\;\\;C_{010}=-1\/4,\\;\\;C_{001}=-1\/4,\\\\\n4C_{k,\\ell,m}=F_j(C_{k,\\ell,m}; k+\\ell+m \\leq j-1),\\;\n k+\\ell+m=j,\\;\\; j\\geq 2.\n\\end{eqnarray*}\nLet us take\n\\[\n{\\hat p}=\\sum_{1\\leq k+\\ell+m1\/6$ and note that \n\\[\n|\\psi|=|b\\lambda\\xi^{-6}|\\leq M|\\xi|^{-(6-1\/\\rho)},\\quad |\\phi|=|a\\lambda^{2\/3}\\xi^{-4}|\\leq M|\\xi|^{-2(6-1\/\\rho)\/3}\n\\]\nprovided $\n|\\xi|\\geq \\lambda^{\\rho}$\nand, taking $N$ large so that $2(N-1)(6-1\/\\rho)\/3\\geq 1$ one has\n\\begin{gather*}\n|\\mu(\\xi)|\\leq C|\\xi|^{-1-2(6-1\/\\rho)\/3},\\; |{\\tilde \\lambda}(\\xi)|\\leq C|\\xi|^{-2(6-1\/\\rho)\/3},\\\\\n |\\nu(\\xi)|\\leq C|\\xi|^{-2(6-1\/\\rho)\/3},\\;|{\\hat p}(\\xi)|\\leq C|\\xi|^{-2(6-1\/\\rho)\/3}.\n\\end{gather*}\nWe now repeat the same argument as in Section 10 of \\cite{Si} \nin $ {\\mathcal S}_{\\delta,M}$ (for the notation ${\\mathcal S}_{\\delta,M}$, \nsee \\cite{Si} and here $M$ depends on $\\lambda$) then we get a solution $q(\\xi)$ to (\\ref{eqn:4}) \nsuch that $|q(\\xi)|\\leq C|\\xi|^{-1}$ for $|\\xi|\\geq \\lambda^{\\rho}$. Since we can choose arbitrarily large $N=N(\\rho)$ in \\eqref{eq:teigi:mu} we conclude that\n\\begin{equation}\n\\label{eqn:6}\n|q(\\xi)|\\leq C|\\xi|^{-N},\\quad |\\xi|\\geq \\lambda^{\\rho}.\n\\end{equation}\n Thus we obatain a solution $p(\\xi)=q(\\xi)+{\\hat p}(\\xi)$. We turn back to $v$;\n\\[\nv=\\left[\\begin{array}{cc}\n1&1\\\\\n-1&1\\end{array}\\right]\\left[\\begin{array}{c}\n1\\\\\np(\\xi)\\end{array}\\right]e^{\\int^{\\xi}\\eta^4\\gamma(\\eta)d\\eta}.\n\\]\nConsider\n\\[\n\\eta^4\\gamma(\\eta)=-\\{2\\eta^4+a\\lambda^{2\/3}+\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2}+(a\\lambda^{2\/3}-\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2})p(\\eta)\\}\n\\]\nand hence\n\\begin{eqnarray*}\n\\int^{\\xi}\\eta^4\\gamma(\\eta)d\\eta=-\\{\\frac{2}{5}\\xi^5+a\\lambda^{2\/3}\\xi+\\frac{3}{2}\\log\\xi-b\\lambda\\xi^{-1}\\}\\\\\n-\\int^{\\xi}(a\\lambda^{2\/3}-\\frac{3}{2}\\eta^{-1}+b\\lambda\\eta^{-2})p(\\eta)d\\eta.\n\\end{eqnarray*}\nDenoting\n\\[\n{\\hat p}=\\sum_{1\\leq k+\\ell+m0$. Let $X>0$. There exist $\\ell$, $c>0$, $C>0$ such that for any $0\\leq \\mu<5\/6$ we have\n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x,\\; a\\lambda^{2\/3},b\\lambda)\\big|\\leq C(1+o_{\\mu}(1))\\lambda^{\\ell}\\\\\n\\times\\exp{\\big\\{-\\delta \\lambda^{5\/3}|x|^{1\/2}(1+o_{a}(1))+C_{\\mu}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu})\\big\\}}\n\\end{gather*}\nfor $k=0, 1$ and $|x|\\geq \\lambda^{-2\\mu}X$.\n\\end{lemma}\n\\begin{proof} In Proposition \\ref{pro:Esti:Y} choose $\\rho=2(1-\\mu)$ and estimate ${\\mathcal Y_0}$ in $x\\leq -\\lambda^{-2\\mu}X$ first. Recall that for $x<0$ we have\n\\begin{gather*}\n{\\mathcal Y}_0(x; a\\lambda^{2\/3},b\\lambda)=-\\omega{\\mathcal Y_2}(x; a\\lambda^{2\/3}, b\\lambda)\n=-\\omega {\\mathcal Y}_0(e^{\\pi i\/5}|x|;\\omega a\\lambda^{2\/3},\\omega^{-1}b\\lambda).\n\\end{gather*}\nDenote\n\\[\n\\phi^{-}(x,\\lambda)=E_{\\rho}(e^{\\pi i\/5}\\lambda^2|x|; \\omega a, \\omega^{-1}b, \\lambda),\\quad\\phi^{+}(x,\\lambda)=E_{\\rho}(\\lambda^2 x; \\omega a, \\omega^{-1}b, \\lambda)\n\\]\nthen we have\n\\begin{equation}\n\\label{eq:phi:x:fu}\n\\begin{split}\n\\phi^{-}(x,\\lambda)=\\frac{2}{5}i\\lambda^5 |x|^{5\/2}+i a\\lambda^{5\/3}|x|^{1\/2}\n+ib|x|^{-1\/2}+r_{\\rho}(e^{\\pi i\/5}\\lambda^{2}|x|,\\lambda).\n\\end{split}\n\\end{equation}\nSince $\\big|r_{\\rho}(e^{\\pi i\/5}\\lambda^{2}|x|,\\lambda)\\big|\\leq C_{\\rho}\\lambda^{-5\/3+3\\mu}$ for $|x|\\geq \\lambda^{-2\\mu}X$ and then\n\\begin{equation}\n\\label{eq:x:fu}\n-{\\mathsf{Re}}\\,\\phi^{-}(x,\\lambda)\n\\leq -\\delta\\lambda^{5\/3}|x|^{1\/2}(1+o_{a}(1))+C_{\\rho}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu}).\n\\end{equation}\n For $x\\geq \\lambda^{-2\\mu}X>0$ note that\n\\begin{equation}\n\\label{eq:x:sei}\n\\begin{split}\n-{\\mathsf{Re}}\\;\\phi^{+}(x,\\lambda)\\leq -c\\lambda^5 x^{5\/2}+C\\lambda^{5\/3}x^{1\/2}\n+C_{\\rho}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu})\\\\\n=-c\\lambda^{5-4\\mu} x^{1\/2}(1+o_{a}(1))+C_{\\rho}(\\lambda^{\\mu}+\\lambda^{-5\/3+3\\mu})\n\\end{split}\n\\end{equation}\nfor $\\mu<5\/6$. Then the assertion follows from \\eqref{eq:x:fu} and \\eqref{eq:x:sei}. \n\\end{proof}\n\\begin{lemma}\n\\label{lem:X:1:2}Assume that \\eqref{eq:rela} and \\eqref{eq:Im:a} hold with some $\\delta>0$ and that $\\mu< 5\/6$. Let $00$, $\\ell$ and $c>0$ such that \n\\[\n\\big|{\\mathcal Y}_0(\\lambda^{2} x; a\\lambda^{2\/3}, b\\lambda)\\big|\\geq C\\lambda^{\\ell} e^{-c\\lambda^{5\/3-\\mu}},\\quad \\lambda^{-2\\mu}X_1\\leq -x\\leq \\lambda^{-2\\mu}X_2.\n\\]\n\\end{lemma}\n\\begin{proof}It is clear from \\eqref{eq:phi:x:fu} that there exists $C_1>0$ such that\n\\[\n-\\lambda^{5\/3-\\mu}\/C_1\\leq -{\\mathsf{Re}}\\,\\phi^{-}(x, \\lambda)\n\\leq -C_1\\lambda^{5\/3-\\mu}\n\\]\nwhen $\\lambda^{-2\\mu}X_1\\leq -x\\leq \\lambda^{-2\\mu}X_2$. This proves the assertion.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:outside:many} Under the same assumptions as in Lemma \\ref{lem:1:ten} there exist $c>0$, $A>0$ such that \n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x,\\; a\\lambda^{2\/3},b\\lambda)\\big|\\leq C_{\\mu}A^{k+1}(1+k^{3}+\\lambda^{5\/2})^{k}e^{c\\lambda^{5\/6}},\\;\\;k\\in {\\mathbb N}\n\\end{gather*}\nfor $|x|\\geq \\lambda^{-2\\mu}X$. \n\\end{lemma}\n\\begin{proof} We first estimate $\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2}x;a\\lambda^{2\/3},b)\\big|$ in $x\\leq -\\lambda^{-2\\mu}X$. \nFrom Proposition \\ref{pro:Esti:Y} with $\\rho=2(1-\\mu)$ we have\n\\[\n{\\mathcal Y}_0(\\lambda^{2}x;a\\lambda^{2\/3},b\\lambda)=C(1+p_{\\mu}(x))\\lambda^{-3\/2}x^{-3\/4}e^{-\\phi^{-}(x, \\lambda)+R_{\\mu}(x)}\n\\]\nwhere $p_{\\mu}(x)$ and $R_{\\mu}(x)$ are holomorphic and bounded in $|x|>\\lambda^{-2\\mu}X$, $|{\\rm arg}\\, x|<3\\pi\/5$. Since $|x|^{-1}\\leq \\sqrt{X}\\lambda^{2\\mu}\\leq \\sqrt{X}\\lambda^{5\/3}$ we have\n\\begin{gather*}\n|d^k(-\\phi^{-}(x, \\lambda)+R_{\\mu}(x))\/dx^k|\\\\\n\\leq C_{\\mu}A^kk!(\\lambda^5 |x|^{3\/2}+\\lambda^{5\/3}|x|^{-1\/2}+C_{\\mu}|x|^{-3\/2})|x|^{1-k}\\\\\n\\leq C_{\\mu}A^kk!(1+\\lambda^5 |x|^{3\/2}+\\lambda^{5\/2})\\lambda^{5(k-1)\/3}\\\\\n\\leq C_{\\mu}A^kk!\\lambda^{5k\/3}(\\lambda^{10\/3}|x|^{3\/2}+\\lambda^{5\/6}),\\quad |x|\\geq \\lambda^{-2\\mu}X,\\;\\;k\\geq 1.\n\\end{gather*}\nTherefore it follows that for $x\\leq -\\lambda^{-2\\mu}X$ \n\\[\n\\big|d^ke^{-\\phi^{-}(x, \\lambda)+R_{\\mu}(x)}\/dx^k|\\leq CA^k\\lambda^{5k\/3}(\\lambda^{10\/3}|x|^{3\/2}+\\lambda^{5\/6}+k)^ke^{-{\\mathsf{Re}}\\,(-\\phi(x)+R_{\\mu}(x))}.\n\\]\nSince $-{\\mathsf{Re}}\\,(-\\phi^{-}(x, \\lambda)+R_{\\mu}(x))\\leq -c\\lambda^{5\/3}|x|^{1\/2}+C_{\\mu}\\lambda^{5\/6}$ for $ x\\leq -\\lambda^{-2\\mu }X$ \nwith $c>0$ independent of $\\mu$ and \n\\[\n|x|^{3k\/2}e^{-c\\lambda^{5\/3}|x|^{1\/2}}\\leq C^{k+1}\\lambda^{-5k}k^{3k}\n\\]\nwe conclude that\n\\[\n|d^ke^{-\\phi^{-}(x, \\lambda)+R_{\\mu}(x)}\/dx^k|\\leq CA^k(\\lambda^{5\/2}+k^3)^ke^{c\\lambda^{5\/6}}\n\\]\nwhich proves the assertion for $x\\leq -\\lambda^{-2\\mu}X$. For $x\\geq \\lambda^{-2\\mu}X$ it is enough to repeat the same arguments noting that \\eqref{eq:x:sei} and $5-4\\mu>5\/3$.\n\\end{proof}\n\\begin{cor}\n\\label{cor:outside} Assume that ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)$ verifies \\eqref{eq:rela} and \\eqref{eq:Im:a}. Then there exist $c>0$, $A>0$, $C>0$ such that for any $\\epsilon>0$ there is $\\lambda_{\\epsilon}$ such that\n\\[\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x,\\; a\\lambda^{2\/3},b\\lambda)\\big|\\leq C_{\\epsilon}A^{k+1}(1+k^{3}+\\lambda^{5\/2})^{k}e^{c\\lambda^{5\/6}},\\;\\;k\\in {\\mathbb N}\n\\]\nfor $\\lambda^{-5\/3+\\epsilon}\\leq |x|$, $\\lambda\\geq \\lambda_{\\epsilon}$. \n\\end{cor}\n\\begin{proof} Choose $\\mu=5\/6-\\epsilon\/2$ in Lemma \\ref{lem:outside:many}.\n\\end{proof}\n\n\n\n\nNext we estimate ${\\mathcal Y}_0(\\lambda^{2}x;a\\lambda^{2\/3},b\\lambda)$ for $|x|\\leq \\lambda^{-5\/3+\\epsilon}$. \n\n\n\\begin{lemma}\n\\label{lem:time:T}{\\rm (\\cite[Lemma 6.5, Lemma 6.7]{BN:JHDE})} Assume that $y(x,\\lambda)$ satisfies \n\\begin{equation}\n\\label{eqn:1}\ny''(x,\\lambda)=(x^3+a\\lambda^{2\/3} x+b\\lambda)y(x,\\lambda),\\quad |a|,\\;|b|\\leq M.\n\\end{equation}\nThen there are $c>0$, $C>0$ and $\\ell_i>0$ such that for any $T>0$ we have\n\\begin{eqnarray*}\n\\big|(d\/dx)^ky(x,\\lambda)\\big| \\leq C^{k+1}(k+\\lambda^{1\/3}+|x|)^{3k\/2}\\lambda^{\\ell_1}(1+T)^{\\ell_2\n}e^{c\\lambda^{5\/6}(1+\\lambda^{-1\/3}T)^{5\/2}}\\\\\n\\times \\big\\{|y(T,\\lambda)|+|y'(T,\\lambda)|\\big\\},\\quad |x|\\leq T,\\;\\;k\\in{\\mathbb N},\\;\\;\\lambda\\geq 1.\n\\end{eqnarray*}\n\\end{lemma}\n\n\n\\begin{pro}\n\\label{pro:inside:many} Assume that ${\\mathcal Y}_0(x;a\\lambda^{2\/3},b\\lambda)$ verifies \\eqref{eq:rela} and \\eqref{eq:Im:a}.\nThen there are $\\ell$, $c>0$, $A>0$, $C>0$ such that for any $\\epsilon>0$ there is $\\lambda_{\\epsilon}$ such that\n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2} x; a\\lambda^{2\/3},b\\lambda)\\big|\\leq CA^{k+1}\n \\lambda^{\\ell}\n(k^{3}+\\lambda^{4})^{k}e^{c\\lambda^{5\/6+\\epsilon}},\\;\\;k\\in{\\mathbb N}\n\\end{gather*}\nfor $\\lambda\\geq \\lambda_{\\epsilon}$.\n\\end{pro}\n %\n \\begin{proof}\n Applying Lemma \\ref{lem:1:ten} with $\\mu=5\/6-\\epsilon\/2$. we have\n %\n \\[\n \\big|(d\/dx)^k{\\mathcal Y}_0(\\pm\\lambda^{1\/3+\\epsilon})\\big|\\leq C\\lambda^{\\ell}e^{c_1\\lambda^{5\/6}},\\;\\;\\lambda\\geq \\lambda_{\\epsilon},\\;\\; k=0, 1.\n \\]\n %\nSince ${\\mathcal Y}_0(x)={\\mathcal Y}_0(x, a\\lambda^{2\/3}, b\\lambda)$ satisfies \\eqref{eqn:1}, choosing $T=\\lambda^{1\/3+\\epsilon}$ in Lemma \\ref{lem:time:T} we get\n\\[\n\\big|(d\/dx)^k{\\mathcal Y}_0(x)\\big|\\leq C^{k+1}\\lambda^{\\ell}(k+\\lambda^{1\/3+\\epsilon})^{3k\/2}e^{c\\lambda^{5\/6+3\\epsilon}},\\;\\;|x|\\leq \\lambda^{1\/3+\\epsilon},\\;\\; k\\in {\\mathbb N}\n\\]\nfor $\\lambda\\geq \\lambda_{\\epsilon}$. This proves that\n\\begin{gather*}\n\\big|(d\/dx)^k{\\mathcal Y}_0(\\lambda^{2}x)\\big|\\leq C^{k+1}\\lambda^{\\ell}(\\lambda^{2}k^{3\/2}+\\lambda^{5\/2+2\\epsilon})^{k}e^{c\\lambda^{5\/6+3\\epsilon}}\n\\end{gather*}\nfor $|x|\\leq \\lambda^{-5\/3+\\epsilon}$. Since $\\lambda^{k(5\/2+2\\epsilon)}e^{-\\lambda^{5\/6+3\\epsilon}}\\leq k^{3k}$ and $\n\\lambda^{2}k^{3\/2}\\leq C(\\lambda^{4}+k^3)$ \ncombining Corollary \\ref{cor:outside} with the above-obtained estimates we conclude the assertion.\n\\end{proof}\nRecalling\n\\[\nV_{\\lambda}(x')=e^{i\\lambda^5x_2-i(b_1\/2)x_1}{\\mathcal Y_0}(\\lambda^2 x_1; \\lambda^{2\/3}a(\\lambda), \\lambda b(\\lambda))\n\\]\nwith $\\lambda^{2\/3}a(\\lambda)=2\\xi_0$ and $b(\\lambda)=b_2+b_0\\xi_0\\lambda^{-4}-\\xi_0^2\\lambda^{-3}-b_1^2\\lambda^{-5}\/4$ one has\n\\begin{lemma}\n\\label{lem:V:bibun} There exist $c>0$, $A>0$, $C>0$ and $\\lambda_0>0$ such that \n\\begin{equation}\n\\label{eq:V:bibun}\n\\big|\\partial_{x_1}^{k}V_{\\lambda}(x')\\big|\\leq CA^{k}(k!)^3e^{c\\lambda^{4\/3}},\\quad k\\in{\\mathbb N},\\;\\lambda\\geq \\lambda_0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nNoting $\\lambda^{4k}\\leq C_1^{k}(k!)^3e^{3\\lambda^{4\/3}}$ the assertion follows from Proposition \\ref{pro:inside:many}.\n\\end{proof}\n \n\\section{Proof of Theorem \\ref{thm:main:bis}}\n\n\nAssume that $b_2\\neq 0$ satisfies \\eqref{eq:argA}. Following Section \\ref{sec:solv} we have a family of exact solutions $\\{U_{\\lambda}\\}$ satisfying $(P_{mod}+\\sum_{j=0}^2b_jD_j)U_{\\lambda}=0$. We show that $\\{U_{\\lambda}\\}$ \ndoes not satisfy apriori estimates derived from $\\gamma^{(s)}$ local solvability of the Cauchy problem if $s>3$. Let $h>0$ and a compact set $K$ be fixed and denote by $\\gamma_0^{(s),h}(K)$ the set of all $f(x')\\in \\gamma^{(s)}({\\mathbb R}^2)$ such that ${\\rm supp}\\,f\\subset K$ and \\eqref{eq:gevrey} holds with some $C>0$ for all $\\alpha\\in {\\mathbb N}^2$. Note that $\\gamma_0^{(s), h}(K)$ is a Banach space with the norm\n\\[\n\\sup_{\\alpha, x}\\frac{|\\partial_x^{\\alpha}f(x')|}{h^{|\\alpha|}|\\alpha|!^s}.\n\\]\n\\begin{pro}[Holmgren]\n\\label{pro:Holmgren}Denote $\nD_{\\epsilon}=\\{x\\in {\\mathbb R}^3\\mid |x'|^2+|x_0|<\\epsilon\\}$. \nThere exists $\\epsilon_0>0$ such that for $\\epsilon$ satisfying $0<\\epsilon<\\epsilon_0$ if $u(x)\\in C^2(D_{\\epsilon})$ satisfies\n\\[\n\\begin{cases}\n\\big(P_{mod}+\\sum_{j=0}^2b_jD_j\\big)u=0\\;\\;\\text{in}\\;\\; D_{\\epsilon},\\\\\\;D_0^ju(0, x')=0\\;\\;(j=0,1),\\;\\;x\\in D_{\\epsilon}\\cap\\{x_0=0\\}\n\\end{cases}\n\\]\nthen $u(x)\\equiv 0$ in $D_{\\epsilon}$.\n\\end{pro}\n\\begin{lemma}\n\\label{lem:futosiki}{\\rm(e.g.\\cite[Proposition 4.1, Theorem 4.2]{Miz}, \\cite{Lax})}\nAssume that the Cauchy problem for $P_{mod}+\\sum_{j=0}^2b_jD_j$ is locally solvable in $\\gamma^{(s)}$ at the origin. Then there exists $\\delta>0$ such that for any $0<\\epsilon_1<\\delta$ and any $\\Phi=(u_j(x'))\\in \\gamma_0^{(s),h}(\\{|x'|\\leq \\epsilon_1\\})$ there is a unique solution $u(x)\\in C^2(D_{\\delta})$ to the Cauchy problem \\eqref{eq:CP} with $U_{\\Phi}=D_{\\delta}$ and for any compact set $L\\subset D_{\\delta}$ there exists $C>0$ such that \n\\begin{equation}\n\\label{eq:G:hyoka}\n|u(x)|_{C^2(L)}\\leq C\\sum_{j=0}^1\\sup_{\\alpha, x'}\\frac{|\\partial_{x'}^{\\alpha}u_j(x')|}{h^{|\\alpha|}|\\alpha|!^s}\n\\end{equation}\nholds.\n\\end{lemma}\nSince $\\lambda^{5k}\\leq k!^se^{s\\lambda^{5\/s}}$ and $U_{\\lambda}(0, x')=V_{\\lambda}(x')$, $D_0 U_{\\lambda}(0, x')= \\xi_0(\\lambda)\\lambda V_{\\lambda}(x')$ it is clear from Lemma \\ref{lem:V:bibun} that one can find $c_1>0$, $C>0$ such that\n\\begin{equation}\n\\label{eq:U:futosiki}\n\\sum_{j=0}^1\\sup_{\\alpha, x'}\\frac{|\\partial_{x'}^{\\alpha}D_0^jU_{\\lambda}(0, x')|}{h^{|\\alpha|}|\\alpha|^{s|\\alpha|}}\\leq Ce^{c_1\\lambda^{\\max{\\{5\/s, 4\/3\\}}}},\\quad s\\geq 3.\n\\end{equation}\nOn the other hand thanks to Lemma \\ref{lem:X:1:2} \nand Proposition \\ref{pro:sol} there is $c_0>0$ such that\n\\begin{equation}\n\\label{eq:U:x:0}\n\\big|U_{\\lambda}( x_0, -\\lambda^{-2\\mu}, 0)\\big|\\geq C\\lambda^{\\ell}e^{c_0\\lambda^{5\/3} x_0-c\\lambda^{5\/3-2\\mu}},\\quad x_0>0\n\\end{equation}\nwhere $\\mu$ is chosen such that $0<\\mu<5\/6$. Let $\\chi(x')\\in \\gamma^{(s)}({\\mathbb R}^2)$ be such that $\\chi(x')=0$ for $|x'|\\geq \\sqrt{\\epsilon_1}$ and $\\chi(x')=1$ for $|x'|\\leq \\sqrt{\\epsilon_2}<\\sqrt{\\epsilon_1}$. Since $\\Phi_{\\lambda}=\\chi(x')(U_{\\lambda}(0, x'), D_0U_{\\lambda}(0, x'))\\in \\gamma_0^{(s), h}(\\{|x'|\\leq \\epsilon_1\\})$, thanks to Lemma \\ref{lem:futosiki}, there is a unique solution $u_{\\lambda}(x)\\in C^{2}(D_{\\delta})$ to the Cauchy problem with Cauchy data $\\Phi_{\\lambda}(x')$ which satisfies \\eqref{eq:U:futosiki}. \nThanks to Proposition \\ref{pro:Holmgren} we see that $u_{\\lambda}=U_{\\lambda}$ in $D_{\\epsilon_2}$. Take a compact set $L\\subset D_{\\epsilon_2}$ that contains $( x_0, -\\lambda^{-2\\mu}, 0)$ with small $x_0>0$ and large $\\lambda$. If $s>3$ hence $\\max{\\{5\/s, 4\/3\\}}<5\/3$ the inequalities \\eqref{eq:U:futosiki} and \\eqref{eq:U:x:0} are not compatible which proves Theorem \\ref{thm:main:bis}.\n\nWhen $b_2\\neq 0$ does not satisfy \\eqref{eq:argA} we make a change of local coordinates $(x_0, x_1, x_2)\\mapsto (-x_0, x_1, -x_2)$ such that $P_{mod}+\\sum_{j=0}^2b_jD_j$ will be\n\\begin{equation}\n\\label{eq:sin:eq}\nP_{mod}-b_0D_0+b_1D_1-b_2D_2.\n\\end{equation}\nin the new local coordinates. Since the local solvability in $\\gamma^{(s)}$ at the origin is invariant under (analytic) change of local coordinates and $-b_2$ obviously satisfies \\eqref{eq:argA}, we conclude the same assertion also in this case.\n\n\n\n\\begin{comment}\n\n\\subsection{Remaining case}\n\nIn this section we treat the case that $0\\neq b_2$ does not satisfy \\eqref{eq:iti}, that is\n\\[\n-4\\pi\/5\\leq \\arg b_2\\leq -\\pi\/5.\n\\]\n Look for a family of exact solutions $U_{\\lambda}$ to $P_{mod}+\\sum_{j=0}^2b_jD_j$ in the form\n\\begin{equation}\n\\label{eq:teigi:U:bis}\nU_{\\lambda}(x)=e^{i\\xi_0\\lambda(\\tau-x_0)-i\\lambda^5x_2}W_{\\lambda}(x_1),\\quad W_{\\lambda}(x_1)=e^{-i(b_1\/2)x_1}u(\\lambda^2x_1)\n\\end{equation}\nwhere $\\tau>0$ is a constant, that is we look for $u(x)$ satisfying\n\\begin{equation}\n\\label{eq:a8:bis}\nu''(x)=\\big(x^3+ 2\\xi_0 x-b_2\\lambda-b_0\\xi_0\\lambda^{-3}-\\xi_0^2\\lambda^{-2}-b_1^2\\lambda^{-4}\/4\\big)u(x).\n\\end{equation}\nSince $-b_2$ clearly satisfies \\eqref{eq:argA} one can apply Proposition \\ref{pro:sol} to find $\\xi_0(\\lambda)$ satisfying\n\\begin{equation}\n\\label{eq:case:b}\n\\begin{split}\n2\\xi_0=c\\,\\lambda^{2\/3}(1+\\lambda^{-5\/6}z(\\lambda))=\\lambda^{2\/3}a(\\lambda),\\quad {\\mathsf{Im}}\\quad c<0,\\\\\nC_0(2\\xi_0,\\, -b_2\\lambda-\\xi_0^2\\lambda^{-2}-b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4)=0.\n\\end{split}\n\\end{equation}\nChoose $u(x)={\\mathcal Y_0}(x; 2\\xi_0, a_2)$ with $a_2=-b_2\\lambda-\\xi_0^2\\lambda^{-2}-b_0\\xi_0\\lambda^{-3}-b_1^2\\lambda^{-4}\/4$. Suppose that the Cauchy problem \\eqref{eq:CP} is locally solvable in $\\gamma^{(s)}$ at the origin with some $s>3$. \nTake $\\chi(t)\\in \\gamma^{(s_1)}({\\mathbb R})$ with $0\\leq \\chi(t)\\leq 1$ such that $\\chi=1$ on $|t|\\leq1\/4$ and $\\chi=0$ in $|t|\\geq 1\/2$ and $\\int \\chi(t)ds=1$ and denote\n\\[\n\\chi_{\\lambda}(x_1)=\\chi(\\lambda^{2\\mu}(x_1+\\lambda^{-2\\mu})),\\quad \\varphi_{\\lambda}(x')=\\psi(x_2)\\chi_{\\lambda}(x_1)\n\\]\nwhere $\\psi\\in \\gamma^{(s)}({\\mathbb R})$ is supported in a small neighborhood of $x_2=0$. Choosing $s_1>1$ so that $s-s_1>5\/4$ and hence $k!^{s_1}\\lambda^{2\\mu k}\\leq k!^{s}e^{c\\lambda^{4\/3}}$ for $\\mu<5\/6$ it is clear that\n\\[\n|\\partial_{x'}^{\\alpha}\\varphi_{\\lambda}(x')|\\leq CA^{|\\alpha|}|\\alpha|!^{s}e^{c_1\\lambda^{4\/3}},\\quad \\alpha\\in {\\mathbb N}^2.\n\\]\nWe define $v_{\\lambda}(x')$ by\n\\[\nv_{\\lambda}(x')=\\varphi_{\\lambda}(x')W_{\\lambda}(x_1)\n\\]\nthen it follows easily from Lemma \\ref{lem:V:bibun} that\n\\begin{equation}\n\\label{eq:shokiti}\n\\sum_{j=0}^1\\sup_{\\alpha, x'}\\frac{|\\partial_{x'}^{\\alpha}v_{\\lambda}(x')|}{h^{|\\alpha|}|\\alpha|!^s}\\leq Ce^{c_2\\lambda^{4\/3}},\\quad s\\geq 3.\n\\end{equation}\nLet $u_{\\lambda}(x)$ be a solution to the Cauchy problem \\eqref{eq:CP} with the Cauchy data $(u_0, u_1)=(0, v_{\\lambda}(x'))$. \nThanks to Lemma \\ref{lem:futosiki} and \\eqref{eq:shokiti} we have\n\\begin{equation}\n\\label{eq:u:lam}\n\\big|u_{\\lambda}(x)\\big|_{C^1(L)}\\leq Ce^{c_2\\lambda^{4\/3}}.\n\\end{equation}\nFrom the Holmgren's uniqueness theorem (e.g. \\cite[Theorem 4.2]{Miz}), taking $\\tau>0$ and the support of $\\psi$ enough small, one can assume that the support of $u_{\\lambda}(x_0, x')$ with respect to $x'$ is small uniformly in large $\\lambda$ and $0\\leq t\\leq \\tau$. Noting that $P=P_{mod}+\\sum_{j=0}^2b_jD_j$ verifies $P^*=P$ we have\n\\begin{gather*}\n\\int_0^{\\tau}(U_{\\lambda}, Pu_{\\lambda})dx_0\n=\\int_0^{\\tau}(PU_{\\lambda}, u_{\\lambda})dx_0-i(D_0U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))\\\\\n-i(U_{\\lambda}(\\tau), D_0u_{\\lambda}(\\tau))-2i\\lambda^5(x_1 U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))+i(U_{\\lambda}(0), D_0u_{\\lambda}(0)).\n\\end{gather*}\nFrom this, it follows that\n\\begin{gather*}\n(U_{\\lambda}(0), D_0u_{\\lambda}(0))=e^{i\\xi_0(\\lambda)\\lambda\\tau}\\int e^{-i\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'\\\\\n=(D_0U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))\n+(U_{\\lambda}(\\tau), D_0u_{\\lambda}(\\tau))-2\\lambda^5(x_1U_{\\lambda}(\\tau), u_{\\lambda}(\\tau)).\n\\end{gather*}\nFrom Lemma \\ref{lem:V:bibun} and \\eqref{eq:u:lam} it is clear that \n\\begin{gather*}\n|(D_0U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))|\n+|(U_{\\lambda}(\\tau), D_0u_{\\lambda}(\\tau))|\n+|2\\lambda^5(x_1U_{\\lambda}(\\tau), u_{\\lambda}(\\tau))|\n\\end{gather*}\nis bounded by $Ce^{c_3\\lambda^{4\/3}}$ with some $c_3>0$, $C>0$ which proves\n\\begin{equation}\n\\label{eq:u:0:U}\n\\Big|e^{i\\xi_0(\\lambda)\\lambda\\tau}\\int e^{-i\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'\\Big|\\leq Ce^{c_3\\lambda^{4\/3}}.\n\\end{equation}\nHere note that\n\\begin{gather*}\n\\int e^{-\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'={\\hat \\psi}(\\lambda^5)\\int\\chi_{\\lambda}(x_1)|W_{\\lambda}(x_1)|^2dx_1\\\\\n={\\hat \\psi}(\\lambda^5)\\int\\chi_{\\lambda}(x_1)e^{({\\mathsf{Im}}\\,b_2)x_1}\\big|{\\mathcal Y_0}(\\lambda^2x_1; 2\\xi_0, \\lambda b(\\lambda))\\big|^2dx_1\n\\end{gather*}\nwhere ${\\hat \\psi}$ is the Fourier transform of $\\psi$. Thanks to Lemma \\ref{lem:X:1:2} it follows that $\\big|{\\mathcal Y}_0(\\lambda^{2} x_1; 2\\xi_0, \\lambda b(\\lambda))\\big|\\geq C\\lambda^{\\ell}\\exp{(-c\\lambda^{5\/3-2\\mu})}$ on the support of $\\chi_{\\lambda}$. Therefore we have\n\\[\n\\int\\chi_{\\lambda}(x_1)e^{({\\mathsf{Im}}\\,b_2)x_1}\\big|{\\mathcal Y_0}(\\lambda^2x_1; 2\\xi_0, \\lambda b(\\lambda))\\big|^2dx_1\\geq C\\lambda^{2\\ell-2\\mu}e^{-2c\\lambda^{5\/3-2\\mu}-3|{\\mathsf{Im}}\\,b_2|\\lambda^{-2\\mu}}.\n\\]\nSince $\\xi_0(\\lambda)$ verifies \\eqref{eq:case:b} we have from \\eqref{eq:u:0:U} that\n\\begin{equation}\n\\label{eq:psi:gevrey}\n\\begin{split}\nCe^{c_3\\lambda^{4\/3}}\\geq \\Big|e^{i\\xi_0(\\lambda)\\lambda\\tau}\\int e^{-i\\lambda^5x_2}\\varphi_{\\lambda}(x')|W_{\\lambda}(x_1)|^2dx'\\Big|\\\\\n\\geq C\\lambda^{2\\ell'}e^{-{\\mathsf{Im}}\\xi_0(\\lambda)\\lambda\\tau-c'\\lambda^{5\/3-2\\mu}}|{\\hat \\psi}(\\lambda^5)|\\geq C\\lambda^{2\\ell'}e^{{\\hat c}\\,\\lambda^{5\/3}\\tau}|{\\hat \\psi}(\\lambda^5)|\n\\end{split}\n\\end{equation}\nwith some ${\\hat c}>0$. If we choose $\\psi$ to be real and even we conclude from \\eqref{eq:psi:gevrey} that \n\\[\n|{\\hat \\psi}(\\xi)|\\leq Ce^{-c|\\xi|^{1\/3}}\n\\]\nwith some $c>0$ which implies $\\psi\\in \\gamma^{(3)}$. This gives a contradiction since $s>3$. This completes the proof.\n\n\\end{comment}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}