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+{"text":"\\section{Introduction}\n\t\n\t\\noindent In this paper, we address the universality of the determinant of a class of random Hermitian matrices.\n\tBefore discussing results specific to this symmetry assumption, we give a brief history of results in the non-Hermitian setting. In both settings, a priori bounds preceded estimates on moments of determinants, and the distribution of\n\tdeterminants for integrable models of random matrices. The universality of such determinants has then been the \n\tsubject of recent active research.\t\n\t\n\t\n\t\\subsection{Non-Hermitian matrices.}\\ \n\tEarly papers on this topic treat non-Hermitian matrices with independent and identically distributed entries.\n\t\tMore specifically, Szekeres and Tur\\'an first studied an extremal problem on  the determinant of $\\pm 1$  matrices \\cite{SzeTur1937}. \n\tIn the 1950s, a series of papers \\cite{For1951,ForTuk1952, NyqRicRio1954, Tur1955, Pre1967} calculated \n\tmoments of the determinant of random matrices of fixed size (see also \\cite{Gir1980}). \n\tIn general, explicit formulae are unavailable for high order moments of the determinant except\n\twhen the entries of the matrix have particular distribution (see, for example, \\cite{Dem1989} and the references therein).\n\tEstimates for the moments and the Chebyshev inequality give upper bounds\n\ton the magnitude of the determinant. \\\\\n\t\n\t\\noindent Along a different line of research,\n\tfor an $N\\times N$ non-Hermitian random matrix $A$,\n\tErd\\H{o}s asked whether $\\det A$ is non-zero with probability\n\ttending to one as $N$ tends to infinity.  In \\cite{Kom1967, Kom1968}, Kolm\\'{o}s proved\n\tthat for random matrices with Bernoulli entries,\n\tindeed $ \\det A \\neq 0$ with probability converging to 1 with $N$. \n\tIn fact, this method works for more general models, and following \\cite{Kom1967},\n\t\\cite{KahKomSze1995, TaoVu2006, TaoVu2007, BouVuWoo2010} give improved, exponentially small bounds on the probability that $\\det A = 0$. \\\\\n\t\n\n\t\\noindent \n\tIn \\cite{TaoVu2006}, the authors made the first steps towards quantifying the typical size of $\\left| \\det A \\right|$,\n\tproving that for Bernoulli random matrices, with probability\n\ttending to 1 as $N$ tends to infinity,\n\t\t\\begin{equation}\n\t\t\\label{tao eq 1}\n\t\t\\sqrt{N!}\\exp\\left(-c \\sqrt{N\\log N}\\right) \\leq \\left| \\det A\\right| \\leq \\omega(N)\\sqrt{N!}, \n\t\t\\end{equation}\n\tfor any function $\\omega(N)$ tending to infinity with $N$. In particular, with overwhelming probability\n\t\t\\[ \\log \\left| \\det A \\right| = \\left( \\frac{1}{2} + \\oo(1) \\right) N\\log N. \\]\n\t\n\t\\noindent In \\cite{Goo1963}, Goodman considered $A$ with independent standard real Gaussian entries.\n\tIn this case, he was able to express $\\left| \\det A \\right|^2$ as the product of independent\n\tchi-square variables. This enables one to identify the asymptotic distribution\n\tof $\\log\\left| \\det A \\right|$. Indeed, one can prove that\n\t\t\\begin{equation}\n\t\t\\label{GOE result}\n\t\t \\frac{\\log \\left| \\det A \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{2}\\log N}{\\sqrt{\\frac{1}{2}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1),\n\t\t\\end{equation}\n\t(see \\cite{RemWes2005}). In the case of $A$ with independent complex Gaussian entries, a similar analysis yields \n\t\t\\[ \\frac{\\log \\left| \\det A \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{4}\\log N}{\\sqrt{\\frac{1}{4}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1). \\]\n\t\n\t\\noindent \n\tIn \\cite{NguVu2014}, the authors proved (\\ref{GOE result}) holds under just\n\tan exponential decay hypothesis on the entries. Their method yields an explicit  rate of convergence\n\tand extends to handle the complex case. Then in \\cite{BaoPanZho2015}, the authors extended \n\t(\\ref{GOE result}) to the case where the matrix entries only require bounded fourth moment.\\\\\n\t\n\t\\noindent The analysis of determinants of non-Hermitian random matrices relies crucially on the\n\tassumption that the rows of the random matrix are independent. The fact that this independence\n\tno longer holds for Hermitian random matrices forces one to look for new methods to prove similar\n\tresults to those of the non-Hermitian case. \n\tNevertheless, the history of this problem mirrors the history of the non-Hermitian case. \n\t\n\t\\subsection{Hermitian matrices.}\\ In the 1980s, Weiss posed the Hermitian analogs of \\cite{Kom1967, Kom1968} as an open problem.\n\tThis problem was solved, many years later in \\cite{CosTaoVu2006}, and \n\tthen in \\cite[Theorem 34]{TaoVu2011} the authors proved the Hermitian analog of \n\t(\\ref{tao eq 1}). This left open the question of describing the limiting\n\tdistribution of the determinant. \\\\\t\n\t\n\t\\noindent In \\cite{DelLeC2000}, Delannay and Le Ca\\\"{e}r used\n\tthe explicit formula for the joint distribution of the eigenvalues to prove\n\tthat for $H$ an $N \\times N$ matrix drawn from the GUE,\n\t\t\\begin{equation}\n\t\t\\label{GUE CLT}\n\t\t \\frac{\\log \\left| \\det H \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{4}\\log N}{\\sqrt{\\frac{1}{2}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1). \n\t\t\\end{equation}\n\tAnalogously, one has\n\t\t\\begin{equation}\n\t\t\\label{GOE CLT}\n\t\t\\frac{\\log \\left| \\det H \\right|  - \\frac{1}{2}\\log N! + \\frac{1}{4}\\log N}{\\sqrt{\\log N}}\n\t\t\t\\to \\mathscr{N}(0, 1)\n\t\t\\end{equation}\n\twhen $H$ is drawn from the GOE.\n\tProofs of these central limit theorems also appear in \\cite{TaoVu2012, CaiLiaZho2015, BorLaC2015,\n\tEdeLaC2015}. For related results concerning other models of random matrices, see \\cite{Rou2007} and the references therein.\\\\\n\n\t\n\t\\noindent While the authors of \\cite{TaoVu2012} give their own proof of (\\ref{GUE CLT}) and (\\ref{GOE CLT}), their main interest is\n\tto establish such a result in the more general setting of Wigner matrices. \n\tIndeed, they show that in (\\ref{GOE CLT}), we may replace $H$ by $W$, a Wigner matrix whose entries' first\n\tfour moments match those of $\\mathscr{N}(0,1)$.\n\tThey also prove the analogous result in the complex case.\n\tIn this paper, we will relax this four moment matching assumption to a \n\ttwo moment matching assumption (see Theorem \\ref{main theorem}). \\\\\n\t\n\t\\noindent Finally, we mention that new interest in averages of determinants of random (Hermitian) matrices \n\thas emerged from the study of complexity of high-dimensional landscapes \\cite{FyoWil2007,AufBenCer2013}.\n\t\n\t\n\t\\subsection{Statement of results: The determinant. }\nThis subsection gives our main result and suggests extensions in connection with the general class of log-correlated random fields. Our theorems apply to Wigner matrices as defined below. \n\t\n\n\t\n\t\\begin{definition}\n\t\\label{def wigner}\n        \tA complex Wigner matrix, $W = \\left(w_{ij}\\right)$, is an $N\\times N$ \n\tHermitian matrix with entries  \n\t\t\\[ W_{ii} = \\sqrt{\\frac{1}{N}}\\,x_{ii},\\, i = 1, \\dots, N, \\quad W_{ij} = \\frac{1}{\\sqrt{2N}} \\left(x_{ij} + {\\rm i} y_{ij}\\right), \\, \n\t\t1 \\leq i < j \\leq N.\\]\n\tHere \n\t$\\{x_{ii}\\}_{1\\leq i \\leq N}$, $\\{x_{ij}\\}_{1 \\leq i < j \\leq N}$, $\\{y_{ij}\\}_{1 \\leq i <  j \\leq N}$ are \n\tindependent identically distributed random variables satisfying\n\t\t$\n\t\t\\mathbb{E} \\left(x_{ij}\\right) = 0, \\mathbb{E}\\left(x_{ij}^2\\right) = \\mathbb{E}\\left(y_{ij}^2\\right) = 1. \n\t\t$\n\tWe assume further \n        \tthat the common distribution $\\nu$ of $\\{x_{ii}\\}_{1\\leq i \\leq N}$, $\\{x_{ij}\\}_{1 \\leq i < j \\leq N}$, $\\{y_{ij}\\}_{1 \\leq i <  j \\leq N}$, has subgaussian decay, i.e.\\ there exists \n        \t$\\delta_0 > 0$ such that\n        \t\t\\begin{equation}\n\t\t\\label{subgaussian} \n\t\t\\int_\\mathbb{R} e^{\\delta_0 x^2}{\\rm d}\\nu(x) < \\infty. \n\t\t\\end{equation}\n\tIn particular, this means that all the moments of the entries of the matrix are bounded.\n\tIn the special case $\\nu = \\mathscr{N}(0,1)$, \n\t$W$ is said to be drawn from the Gaussian Unitary Ensemble (GUE). \n\t\n\tSimilarly, we define a real Wigner matrix to have entries of the form\n\t\t$ W_{ii} = \\sqrt{\\frac{2}{N}} x_{ii}$, $W_{ij} = \\sqrt{\\frac{1}{N}} x_{ij}$, \n\twhere $\\left\\{x_{ij}\\right\\}_{1\\leq i, j \\leq N}$ are independent identically distributed random variables satisfying\n\t\t$ \\mathbb{E}\\left(x_{ij}\\right) = 0, \\mathbb{E}\\left(x^2_{ij}\\right) = 1.$\n\tAs in the complex case, we assume\n\tthe common distribution $\\nu$ satisfies (\\ref{subgaussian}).\n\tIn the special case $\\nu = \\mathscr{N}(0,1)$,\n\t$W$ is said to be drawn from the Gaussian Orthogonal Ensemble (GOE).\n\t\\end{definition}\n\t\t\t\n\t\\noindent Our main result extends (\\ref{GUE CLT}) and  (\\ref{GOE CLT}) to the above class of Wigner matrices.  In particular,\n\tthis answers a  conjecture from \\cite[Section 8]{TaoVu2014}, which asserts that the central limit theorem \n\t(\\ref{GOE CLT})\n\tholds for Bernoulli ($\\pm 1$) matrices. Note that in the following statement, our centering\n\tdiffers from (\\ref{GUE CLT}) and (\\ref{GOE CLT}) because we normalize our matrix entries\n\tto have variance of size $N^{-1}$. \n\t\t\t\n\n\t\\begin{theorem}\\label{main theorem}\n\t\tLet $W$ be a real Wigner matrix satisfying (\\ref{subgaussian}). Then\n\t\t\\begin{equation}\\label{main}\n\t\t\\frac{\\log\\left| \\det W\\right| + \\frac{N}{2} }{\\sqrt{\\log N}} \\to \\mathscr{N}(0, 1).  \n\t\t\\end{equation}\n\t\tIf $W$ is a complex Wigner matrix satisfying (\\ref{subgaussian}), then\n\t\t\\begin{equation}\n\t\t\t\\frac{\\log\\left| \\det W \\right| + \\frac{N}{2}}{\\sqrt{\\frac{1}{2}\\log N}} \n\t\t\t\\to \\mathscr{N}(0, 1). \n\t\t\\end{equation}\n\t\\end{theorem}\n\t\n\t \\noindent Assumption (\\ref{subgaussian}) may probably  be relaxed to a finite moment assumption, but we will not pursue this direction here. Similarly, it is likely that the matrix entries do not need to be identically distributed;\n\t only the first two moments need to match. However we consider the case of a \n\t unique $\\nu$ in this paper. \n\t\n\t\n\t\\begin{remark}\\label{rem:multidim}\n\tLet $H$ be drawn from the GUE normalized so that in the limit as $N\\to\\infty$, the distribution\n\tof its eigenvalues is supported on $[-1,1]$, and let\n\t\t\\[D_N(x) = -\\log\\left| \\det\\left( H - x\\right) \\right|. \\]\n\tIn \\cite{Kra2007}, Krasovsky proved that  for $x_k \\in (-1,1)$, $k =1, \\hdots, m$, \n\t$x_j \\neq x_k$, uniformly in $\\Re \\left(\\alpha_k\\right) > - \\frac{1}{2}$, $\\mathbb{E}\\left( e^{-\\sum_{k=1}^m \\alpha_k D_N\\left(x_k\\right)} \\right)$ is asymptotic to\n\t\t\\begin{align}\\label{krasovsky}\n\t\t\t\\prod_{k=1}^m \\left( C\\left(\\frac{\\alpha_k}{2}\\right) \\left(1-x_k^2\\right)^{\\frac{\\alpha_k^2}{8}} \n\t\t\t N^{\\frac{\\alpha_k^2}{4}} e^{\\frac{\\alpha_k N}{2}\\left( 2x_k^2 -1 - 2\\log 2 \\right)}\\right)\n\t\t\t &\\prod_{1\\leq \\nu < \\mu \\leq m} \\left(2\\left| x_\\nu - x_\\mu \\right| \\right)^{-\\frac{\\alpha_\\nu \\alpha_\\mu}{2}}\n\t\t\t \\left(1 + {\\rm O}\\left( \\frac{\\log N}{N} \\right) \\right),\n\t\t\\end{align}\n\tas $N\\to\\infty$. Here $C(\\cdot)$ is the Barnes function.\n\tSince the above estimate holds uniformly for $\\Re \\left(\\alpha_k\\right) > -\\frac{1}{2}$,  \n\t(\\ref{krasovsky}) shows that\n\tletting\n\t\t\\[\t\n\t\t\t\\widetilde D_N(x) = \\frac{D_N(x) - N\\left( x^2 - \\frac{1}{2} - \\log 2 \\right)}{\\sqrt{\\frac{1}{2} \\log N}},\n\t\t\\]\n\tthe vector $\\left( \\widetilde D_N\\left(x_1\\right), \\hdots, \\widetilde D_N\\left(x_m\\right) \\right)$ converges in distribution \n\tto a collection of $m$ independent standard Gaussians. Our proof of Theorem\n\t\\ref{main theorem} automatically extends this result to Hermitian Wigner matrices as defined above.\n\tIf one were to prove an analogous convergence for the GOE, our proof of Theorem\n\t\\ref{main theorem} would extend the result to real symmetric Wigner matrices as well. \n\t\\end{remark}\n\t\n\t\n\\begin{remark} \\label{rem:multidim2}\n\tWe note that (\\ref{krasovsky}) was proved for fixed, distinct $x_k$'s. If  (\\ref{krasovsky}) holds for collapsing\n\t$x_k$'s, this means that fluctuations of the log-characteristic polynomial of the GUE  become log-correlated for large dimension, \n\tas in the case of the Circular Unitary Ensemble \\cite{BourgadeZeta}. More specifically, let\n\t$\\widetilde D_N(\\cdot)$ be as above, and let $\\Delta$ denote the distance between two points $x, y$ in\n\t$(-1, 1)$. For $\\Delta \\geq 1\/N$, we expect the covariance between $\\widetilde D_N(x)$ and $\\widetilde D_N(y)$\n\tto behave like $\\frac{\\log\\left(1\/\\Delta\\right)}{\\log N}$, and for $\\Delta \\leq 1\/N$, we expect it to\n\tconverge to 1.\n\\end{remark}\n\n\n\\noindent \n\tOur method  automatically\n\t establishes the content of Remark \\ref{rem:multidim2} for Wigner matrices,\n\t conditional on the knowledge of GOE and GUE cases. \n\t The exact statement is as follows, and we omit the proof, strictly similar to Theorem \\ref{main theorem}. Denote\n\t \t$$\n\t \tL_N(z)=\\log|\\det(W-z)|-N\\int_{-2}^2\\log|x-z|\\,{\\rm d} \\rho_{\\rm sc}(x).\n\t \t$$\n\t \t\n\t \t\n\\begin{theorem}\\label{multldim}\n\t\tLet $W$ be a real Wigner matrix satisfying (\\ref{subgaussian}). Let \n\t\t$\\ell\\geq 1$, $\\kappa>0$ and let $E^{(1)}_N)_{N\\geq 1},\\dots$, $(E^{(\\ell)}_N)_{N\\geq 1}$\n\t\tbe energy levels in $[-2+\\kappa,2-\\kappa]$.\n\t\tAssume that for all $i\\neq j$, for some constants $c_{ij}$ we have\n$$\n\\frac{\\log |E_N^{(i)}-E_N^{(j)}|}{-\\log N}\\to c_{ij}\\in[0,\\infty]\n$$\t\t\nas $N\\to\\infty$. Then\n\\begin{equation}\\label{eqn:logcor}\n\\frac{1}{\\sqrt{\\frac{1}{2}\\log N}}\\left(\nL_N\\left( \\left(E^{(1)}_N\\right) \\right),\\dots,\nL_N\\left(\\left(E^{(\\ell)}_N\\right)\\right)\n\\right)\n\\end{equation}\nconverges in distribution to a Gaussian vector with covariance $ (\\min(1,c_{ij}))_{1\\leq i,j\\leq N}$ (with diagonal $1$ by convention), provided the same result holds for GOE. \\\\\n\n\\noindent The same result holds for Hermitian Wigner matrices, assuming it is true in the GUE case, up to a change in the normalization from $\\sqrt{\\frac{1}{2}\\log N}$ to $\\sqrt{\\log N}$ in (\\ref{eqn:logcor}).\n\\end{theorem}\t \t\n\t \t\n\t \t\\noindent Theorem \\ref{multldim} says $L_N$ converges to a log-correlated field, provided this result holds for the Gaussian ensembles. It therefore suggests that the universal limiting behavior of extrema and convergence to Gaussian multiplicative chaos conjectured for unitary matrices in \\cite{FyoHiaKea12} extends to the class of Wigner matrices. \nTowards these conjectures, \\cite{ArgBelBou15,ChaMadNaj16,PaqZei17,FyoSim2015,LamPaq16} proved asymptotics on the maximum of characteristic polynomials of circular unitary and invariant ensembles, and \\cite{BerWebWon2017,NikSakWeb2018,Web2015} established convergence to the Gaussian multiplicative chaos, for the same models.\nWe refer to  \\cite{Arg16} for a survey on log-correlated fields and their connections with random matrices, branching processes, the Gaussian free field, and analytic number theory. \n\n\n\n\n\\subsection{Statement of results: Fluctuations of Individual Eigenvalues. }\t\nWith minor modifications, the proof of Theorem \\ref{main theorem} also extends the results of \\cite{Gus2005} and \\cite{ORo2010} \nwhich describe the fluctuations of individual eigenvalues in the GUE and GOE cases, respectively. \nBy adapting the method of \\cite{TaoVu2011}, \\cite{ORo2010} proves the following theorem under\nthe assumption that the first four moments of the matrix entries match those of a standard Gaussian.\nIn Appendix B, we show that the individual eigenvalue fluctuations of the GOE (GUE) also hold for real (complex) Wigner matrices in the sense of Definition \\ref{def wigner}.\nIn particular, the fluctuations of eigenvalues of Bernoulli matrices are Gaussian in the large dimension limit, which answers a question from \\cite{TaoVu2014}. \\\\\n\n\n\\noindent To state the following theorem, we follow the notation of Gustavsson \\cite{Gus2005} and write \n$k(N) \\sim N^\\theta$ to mean that $k(N) = h(N)N^\\theta$ where $h$ is a function such that for \nall ${\\varepsilon} > 0$, for large enough $N$,\n\t\\begin{equation}\n\t\\label{gustavsson sim}\n\tN^{-{\\varepsilon}}\\leq h(N)\\leq N^{\\varepsilon}.\n\t\\end{equation}\n\tIn the following, $\\gamma_k$ denotes the $k$\\textsuperscript{th} quantile of the semicircle law,\n\t\t\t\\begin{equation}\\label{quantiles}\n\t\t\t \\frac{1}{2\\pi}\\int_{-2}^{\\gamma_k} \\sqrt{\\left(4 - x^2 \\right)_+}{\\rm d}x = \\frac{k}{N}.\n\t\t\t\\end{equation}\n\n\\begin{theorem}\n\\label{fluctuations of individual eigenvalues}\n\tLet $W$ be a Wigner matrix satisfying (\\ref{subgaussian}) with eigenvalues $\\lambda_1 < \\lambda_2 <\n\t\\dots < \\lambda_N$. Consider $\\left\\{ \\lambda_{k_i} \\right\\}_{i=1}^m$ such that  $0 < k_{i+1} - k_{i} \\sim N^{\\theta_i},\n\t0 < \\theta_i \\leq 1$, and $k_i \/N \\to a_i \\in (0,1)$ as $N \\to \\infty$.\n\tLet\n\t\\begin{equation}\n\t\\label{def: X_i}\n\t\tX_i = \\frac{\\lambda_{k_i} - \\gamma_{k_i}}{\\sqrt{ \\frac{4\\log N}{\\beta \\left(4 - \\gamma_{k_i}^2 \\right) N^2}  }},\n\t\t\\quad i = 1, \\dots, m,\n\t\\end{equation}\n\twith $\\beta =1$ for real Wigner matrices, and $\\beta = 2$ for complex Wigner matrices.\n\tThen as $N \\to \\infty$,\n\t\\[\n\t\t\\P\\left\\{ X_1 \\leq \\xi_1, \\dots, X_m \\leq \\xi_m \\right\\} \\to \\Phi_{\\Lambda}\\left(\\xi_1, \\dots, \\xi_m \\right),\n\t\\]\n\twhere $\\Phi_\\Lambda$ is the cumulative distribution function for the $m$-dimensional normal distribution with\n\tcovariance matrix $\\Lambda_{i,j} = 1 - \\max\\left\\{ \\theta_k: i \\leq k < j < m \\right\\}$ if \n\t$i < j$, and $\\Lambda_{i,i} = 1$.\n\\end{theorem}\n\n\t\n\\noindent The above theorem  has been known to follow from the  homogenization result in \\cite{BouErdYauYin2016} (this technique gives a simple expression for the relative {\\it individual} positions of coupled eigenvalues from GOE and Wigner matrices) and fluctuations of mesoscopic linear statistics; see \\cite{LanSos2018} for a proof of eigenvalue fluctuations for Wigner and invariant ensembles.  However, the technique from  \\cite{BouErdYauYin2016} is not enough for Theorem \\ref{main theorem}, as the determinant depends on the positions of {\\it all} eigenvalues.\n\t\n\t\n\t\\subsection{Outline of the proof. }\\  In this section, we give the main steps of\n\tthe proof of Theorem  \\ref{main theorem}. Our outline discusses the real case, but\n\tthe complex case follows the same scheme. \\\\\n\t\n\t\\noindent The main conceptual idea of the proof follows the three step strategy of \n\t\\cite{ErdPecRmSchYau2010,ErdSchYau2011}. With a priori localization of eigenvalues (step one, \\cite{ErdYauYin2012Rig, CacMalSch2015}), one can prove that the determinant has universal fluctuations after a adding a small Gaussian noise (this second step relies on a stochastic advection equation from \\cite{BourgadeExtreme}). The third step proves by a density argument that the Gaussian noise does not change the distribution of the determinant, thanks to a perturbative moment matching argument as in \\cite{TaoVu2011,ErdYauYin2012Bulk}. We include Figure \\ref{diagram of implications} below to \n\thelp summarize\n\tthe argument.\\\\\n\t\n\t\n\n\t\\noindent {\\it First step: small regularization.}\n\tIn Section \\ref{smoothing section}, \n\twith Theorems \\ref{schlein} and \\ref{fixed energy thm}, we reduce the proof of Theorem \\ref{main theorem} to showing the convergence\n\\begin{equation}\\label{eqn:probaconv}\n \\frac{\\log\\left| \\det (W+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_0)\\right| + c_N}{\\sqrt{\\log N}} \\to \\mathscr{N}(0, 1)\n\t\t\t\t\\end{equation}\n\twith  some explicit deterministic $c_N$, and the small regularization parameter\n\t\t\\begin{equation}\n\t\t\\label{def:eta0}\n\t\t\t\\eta_0 = \\frac{e^{ \\left(\\log N\\right)^{ \\frac{1}{4}  } }}{N}.\n\t\t\\end{equation}\n\t\n\n\t\n\t\n\t\n\t\n\t\n\t\n\\noindent {\\it Second step: universality after coupling.} \nLet $M$ be a symmetric matrix which serves as the initial condition for the\n\tmatrix Dyson's Brownian Motion (DBM) given by \n\t\t\\begin{equation}\n\t\t\t\\label{eq:DBM evolution} \n\t\t{\\rm d} M_t = \\frac{1}{\\sqrt{N}}{\\rm d}B^{(t)} - \\frac{1}{2}M_t{\\rm d}t.\n\t\t\\end{equation}\nHere $B^{(t)}$ is a symmetric $N \\times N$ matrix such that $B^{(t)}_{ij}\\left(i < j\\right)$ and $B_{ii}^{(t)}\/\\sqrt{2}$ are\nindependent standard Brownian motions.\n\tThe above matrix DBM induces \n\ta collection of independent standard Brownian motions (see \\cite{AndGuiZei2010}), $\\tilde{B}^{(k)}_t\/\\sqrt{2}$, $k = 1, \\dots, N$ such that the eigenvalues of $M$ satisfy  the system of stochastic differential equations\n\t\\begin{align}\n\t\t\\label{eqn:DBM}\n\t\t{\\rm d}x_k(t) &= \\frac{{\\rm d}\\tilde{B}_{t}^{(k)}}{\\sqrt{N}} + \\left( \\frac{1}{N} \\sum_{l \\neq k}\n\t\t\t\\frac{1}{x_k(t) - x_l(t)} - \\frac{1}{2} x_k(t)\\right){\\rm d}t \t\n\t\\end{align}\n\twith initial condition given by the eigenvalues of $M$.\n\tIt has been known since \\cite{McK1969} that the system \n\t(\\ref{eqn:DBM}) has a unique strong solution.\n\tWith this in mind, we follow \\cite{BouErdYauYin2016} and introduce the following\n\tcoupling scheme. First, run the matrix DBM taking $\\tilde{W}_0$, a Wigner matrix, as the initial condition. Using\n\tthe induced Brownian motions, run the dynamics given by (\\ref{eqn:DBM}) using the eigenvalues \n\t$y_1 < y_2 < \\dots < y_N$\n\tof $\\tilde{W}_0$ as the initial condition. Call the solution to this system ${\\bm y}(\\tau)$.\n\tUsing the very same (induced) Brownian motions, run\n\tthe dynamics given by (\\ref{eqn:DBM}) again, this time using the eigenvalues of a GOE matrix, ${\\bm x}(0)$,\n\tas the initial condition. Call the solution to this system ${\\bm x}(\\tau)$. \\\\\n\t\n\t\n\t\\noindent Now fix $\\epsilon > 0$ and let \n\t\\begin{equation}\n\t\t\\label{def:tau}\n\t\t\t\\tau = N^{-\\epsilon}.\n\t\t\\end{equation}\n\tUsing Lemma \\ref{extreme}, we show that \n\t\t\\begin{equation}\n\t\t\\label{coupling at tau}\n\t\t \\frac{\\sum_{k = 1}^N \\log \\left|x_k(\\tau) + {\\rm i}\\eta_0\\right| - \n\t\t\t\t\\sum_{k = 1}^N \\log \\left|y_k(\\tau) + {\\rm i}\\eta_0\\right| }{\\sqrt{\\log N}}\n\t\t\\end{equation}\n\tand\n\t\t\\begin{equation} \\label{smoothing t} \\frac{\\sum_{k = 1}^N \\log \\left|x_k(0) + z_\\tau\\right| - \n\t\t\t\t\\sum_{k = 1}^N \\log \\left|y_k(0) + z_\\tau\\right| }{\\sqrt{\\log N}}\n\t\t\\end{equation}\n\tare very close. Here $z_\\tau$ is as in (\\ref{def:ztau}) with $z = {\\rm i} \\eta_0$. \n\tThe  significance of this is that since\n\t$z_\\tau \\sim \\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\tau$, we can use\n\tLemma \\ref{expectation} and well-known central limit theorems which apply to nearly macroscopic scales \n\tto show that (\\ref{smoothing t}) has variance of order ${\\varepsilon}$. Consequently,\n\t(\\ref{coupling at tau}) is also small, and since ${\\bm x}(\\tau)$ is distributed as the eigenvalues of a GOE matrix,\n\twe have proved universality of the regularized determinant after coupling.\\\\\n\t\n\n\n\t\n\t\\noindent {\\it Third step: moment matching}. In Section \\ref{conclusion}, we conclude the proof \n\tof Theorem \\ref{main theorem}. First, we choose\n\t$\\tilde{W}_0$ so that $\\tilde{W}_\\tau$ and $W$ have entries whose first four moments are\n\tclose, as in \\cite{ErdYauYin2012Bulk}. With this approximate moment matching, we use a perturbative argument, as in \\cite{TaoVu2012},\n\tto prove that (\\ref{eqn:probaconv})\n\tholds for $W$ if and only if it holds for $\\tilde{W}_\\tau$. \t\n\tBut as (\\ref{coupling at tau}) is small, this means (\\ref{eqn:probaconv})\n\tholds for $W$ if and only if it holds for a GOE matrix. By (\\ref{GOE CLT}), this concludes the proof.\n\t\n\t\n\n\n\t\n\t\n\t\\begin{wrapfigure}[22]{l}{8cm}\n\t\\vspace{-1.2cm}\n\t\\begin{center}\n\t\\begin{tikzcd}[row sep = 3cm, column sep = 3cm] \n\t\t& W \\\\\n\t\t\\tilde{W}_0 \\arrow{r} \\arrow[rightarrow]{r} [anchor=center,yshift=2ex] \n\t\t\t{\\text{Matrix DBM } {\\rm d} B_{ij}} & \\tilde{W}_\\tau  \\arrow[leftrightarrow]{u} [anchor=center,rotate=-90,yshift=2ex]{\\text{Moment Matching (3)}\n\t\t\\\\ [-85pt]\n\t\t{\\bm y(0)} \\arrow{r} [anchor=center,yshift=-2ex]{\\text{Eigenvalues DBM } {\\rm d} \\tilde{B}_k}& {\\bm y(\\tau)} \\\\ \n\t\t[-30pt]\n\t\t{{\\bm x}(0)} \\arrow[rightarrow]{r}[anchor=center,yshift=2ex]{ \\text{Eigenvalues DBM }{\\rm d} \\tilde{B}_k}& {{\\bm x}(\\tau)}\n\t\t\\arrow{u} [anchor=center,rotate=-90,yshift=2ex]{\\text{Coupling (2)}} \t\n\t\\end{tikzcd} \n\t\\caption{\n       \t\tWe show (\\ref{main}) holds for $\\tilde{W}_\\tau$ if and only\n\t\tif it holds for $W$, and\n\t\twe prove that (\\ref{main}) holds for\n\t\t${\\bm x}(\\tau)$ if and only if (\\ref{main}) holds for $\\tilde{W}_\\tau$. \n\t\tSince ${\\bm x}(\\tau)$ is distributed as the eigenvalues of a GOE matrix, it satisfies\n\t\t(\\ref{GOE CLT}) and we conclude the proof.\n\t\tNote that $\\log| \\det \\tilde{W}_\\tau | = \\sum \\log \\left|y_k(\\tau) \\right|$ pathwise because\n\t\t$B$ induces $\\tilde{B}$. \n    } \\label{diagram of implications}\n\t\\end{center}\n\\end{wrapfigure}\n\n\t\n\t\\subsection{Notation. }\n\tWe shall make frequent use of the notations $s_W$ and $m_{sc}$ in the remainder of this paper. We \n\tstate their definitions here for easy reference.\n\tLet $W$ be a Wigner matrix with eigenvalues $\\lambda_1 < \\lambda_2 < \\dots < \\lambda_N$.\n\tFor $\\Im (z) > 0$, define\n\t\t\\begin{equation}\n\t\t\\label{def:stransform}\n\t\ts_W(z) = \\frac{1}{N} \\sum_{k=1}^N \\frac{1}{\\lambda_k - z},\n\t\t\\end{equation}\n\tthe Stieltjes transform of $W$. \n\tNext, let\n\t\t\\begin{equation}\n\t\t\\label{def:msc}\n\t\t\tm_{sc}(z) =  \\frac{-z + \\sqrt{z^2 -4}}{2},\n\t\t\\end{equation}\n\twhere the square root $\\sqrt{z^2 -4}$ is chosen with the branch cut in $[-2, 2]$ so that $\\sqrt{z^2-4} \\sim z$ \n\tas $z \\to \\infty$. Note that\n\t\t\\begin{equation}\n\t\t\t\\label{self consistent m}\n\t\t\tm_{sc}(z) + \\frac{1}{m_{sc}(z)} + z = 0.\n\t\t\\end{equation} \n\n\\noindent Finally, throughout this paper, unless indicated otherwise,\n\t$C$ ($c$) denotes a large (small) constant independent of all other parameters of the problem. \n\tIt may vary from line to line.\t\n\t\n\t\t\t\t\t\n\\section{Initial Regularization}\\label{smoothing section}\n\n\t\\noindent Let $y_1 < y_2 < \\dots <y_N$ denote \n\tthe eigenvalues of $W$, a real Wigner matrix\n\tsatisfying (\\ref{subgaussian}). \n\tWe first prove \n\twe only need to show Theorem \\ref{main theorem} for a slight regularization of the logarithm.\n\t\n\t\n\t\\begin{proposition}\\label{smoothing}\n\t\tSet\n\t\t\\begin{equation*}\n\t\tg(\\eta) = \\sum_{k} \\left(\\log \\left| y_k+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta\\right| - \\log \\left| y_k\\right|\\right)-\\int_0^{\\eta}N\\Im \n\t\t\t\\left( m_{sc}(\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} s)\\right) {\\rm d} s\n\t\t\\end{equation*}\n\t\tand recall\n\t\t$ \\eta_0 =\\frac{e^{ \\left(\\log N\\right)^{ \\frac{1}{4}  } }}{N} $\n\t\tas in (\\ref{def:eta0}).\n\t\tThen we have the convergence in probability\n\t$$\\frac{g(\\eta_0)}{\\sqrt{\\log N}}\\to 0.$$\n\t\\end{proposition}\n\t\\noindent To prove Proposition \\ref{smoothing}, we will use Theorems \\ref{schlein} and \\ref{fixed energy thm}\n\tas input. In \\cite{CacMalSch2015}, Theorem \\ref{schlein} is stated for complex Wigner matrices, however,\n\tthe argument there proves the same statement for real Wigner matrices.\n\t\\begin{theorem}[Theorem 1 in \\cite{CacMalSch2015}]\\label{schlein}\n\t\tLet $W$ be a Wigner matrix and\n\t\tfix $\\tilde{\\eta} > 0$. \n\t\tFor any $\\tilde{E} > 0$, there exist constants $M_0, N_0, C, c, c_0 >0$ such that \n\t\t\t\t\t\\[ \\mathbb{P}\\left( \\left| \\Im \\left(s_W\\left(E + {\\rm i}\\eta\\right)\\right) - \n\t\t\t\t\t\\Im \\left(m_{sc}\\left(E + {\\rm i}\\eta\\right)\\right) \\right| \n\t\t\t\t\t\t\\geq \\frac{K}{N\\eta} \\right) \n\t\t\t\t\t\t\\leq \\frac{\\left(Cq\\right)^{cq^2}}{K^q} \\]\n\t\tfor all $\\eta \\leq \\tilde{\\eta}$, $|E| \\leq \\tilde{E}$, $K > 0$, $N > N_0$ such that $N\\eta > M_0$,\n\t\tand\n\t\t$q \\in \\mathbb{N}$ with $q \\leq c_0\\left(N\\eta\\right)^{\\frac{1}{8}}$.\n\t\\end{theorem}\t\n\t\\begin{remark}\n\t\tIn  \\cite{ErdYauYin2012Rig}, the authors proved that for some positive constant $C_0$, and $N$ large enough,\n\t\t\t\\[ \\left| s_W\\left(E + {\\rm i}\\eta\\right) - m_{sc}\\left(E + {\\rm i}\\eta\\right) \\right| \\leq\n\t\t\t\t\\frac{e^{C_0(\\log\\log N)^2}}{N\\eta} \\] \n\t\tholds with high probability. Though this estimate is weaker than the estimate\n\t\tof Theorem \\ref{schlein}, it holds for a more general model of Wigner matrix in which\n\t\tthe entries of the matrix need not have identical variances. On the other hand, we require the  \n\t\tstronger estimate in Theorem \\ref{schlein} in our proof of Proposition\n\t\t\\ref{smoothing}, and so we restrict ourselves to Wigner matrices as defined\n\t\tin Definition \\ref{def wigner}.\n\t\tThe proof of Lemma \\ref{expectation} also relies on Definition \\ref{def wigner}. \n\t\\end{remark}\n\t\\begin{theorem}[Theorem 2.2 in \\cite{BouErdYauYin2016}]\\label{fixed energy thm}\n\t\tLet $\\rho_1$ denote the first correlation function for the eigenvalues of an $N\\times N$ Wigner matrix,\n\t\tand let $\\rho(x) = \\frac{1}{2\\pi} \\sqrt{\\left( 4 - v^2 \\right)_+}$. \n\t\tThen for any $F: \\mathbb{R} \\to \\mathbb{R}$ continuous and compactly supported, and for any $\\kappa > 0$,\n\t\twe have, \n\t\t\t\\begin{equation}\\label{FEU} \\lim_{N\\to\\infty} \\sup_{E \\in [-2 + \\kappa, 2 - \\kappa]} \\left |\\frac{1}{\\rho(E)} \n\t\t\t\\int F(v)\\rho_1\\left( E + \\frac{v}{N\\rho(E)}\\right){\\rm d}v -\n\t\t\t\\int F(v) \\rho(v)\\, {\\rm d}v \\right| = 0.\\end{equation}\n\t\\end{theorem}\n\t\\begin{remark}\n\t\tIn fact Theorem 2.2 in \\cite{BouErdYauYin2016} makes a much stronger statement, namely it states the\n\t\tanalogous convergence for all correlation functions in the case of generalized Wigner matrices.\n\t\\end{remark}\n\t\n\t\\begin{corollary}\\label{repulsion}\n\t\tFor any small fixed $\\kappa,\\gamma > 0$ there exists $C,N_0>0$ such that for any $N\\geq N_0$ and any interval $I \\subset [-2 + \\kappa, 2 - \\kappa]$ we have\n\t\t\t\\[ \\mathbb{E}\\left( \\left| \\left\\{ y_k: y_k \\in I \\right\\} \\right|\\right)\\leq CN |I| + \\gamma. \\]\n\t\t\t\t\\end{corollary}\n\t\\begin{proof}\n\t\tIn Theorem \\ref{fixed energy thm}, choosing $F$ to be an indicator of an interval of length\n\t\t$1$ gives an expected value ${\\rm O}(1)$.\n\t\tSince the statement of Theorem \\ref{fixed energy thm} holds uniformly in $E$, \n\t\twe may divide the interval $I$ into sub-intervals of length order $1\/N$\n\t\tto conclude. \n\t\\end{proof}\n\n\t\\begin{corollary}\\label{micro fixed energy}\n\t\tLet $E\\in[-2+\\kappa,2-\\kappa]$ be fixed and $I_\\beta = (E -  \\beta\/2, E + \\beta\/2)$ with $\\beta=\\oo(N^{-1})$. Then\n\t\t\t\\[ \\lim_{N\\to\\infty} \\mathbb{P}\\left( \\left| \\left\\{ y_k \\in I_\\beta \\right\\}\\right| = 0 \\right) = 1. \\]\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tLet ${\\varepsilon}$ be any fixed small constant.\n\t\tLet $f$ be fixed, smooth, positive, equal to $1$ on $[-1,1]$ and $0$ on $[-2,2]^c$.\n\t\tThen\n\t\t$$\n\t\t\t\t\\mathbb{P}\\left( \\left| \\left\\{ y_k \\in I_\\beta \\right\\}\\right| \\geq 1 \\right) \\leq\n\t\t\t\t\\mathbb{E} \\left( \\left| \\left\\{ y_k \\in I_\\beta \\right\\}\\right| \\right) \\leq\n\t\t\t\t\\mathbb{E} \\left( \\sum_k f\\left(N(y_k-E)\/{\\varepsilon}\\right) \\right) \\leq 10{\\varepsilon},\n\t\t$$\n\t\twhere the last bound holds for large enough $N$ by Theorem \\ref{fixed energy thm}. \n\t\\end{proof}\n\t\\begin{proof}[Proof of Proposition \\ref{smoothing}]\n\tWe first choose $\\tilde{\\eta}<\\eta_0$ so that we can use Theorem \\ref{schlein} to estimate\n\t\t\\[\\mathbb{E}\\left( \\left| g\\left(\\eta_0\\right) - g\\left(\\tilde{\\eta}\\right) \\right| \\right),\\]\n\tand then take care of the remaining error using Corollaries \\ref{repulsion} and \\ref{micro fixed energy}.\n\tLet\n\t\t\\[ \\tilde{\\eta} = \\frac{d_N}{N}, \\ \\ {\\rm with} \\ \\ d_N = \\left( \\log N \\right)^{\\frac{1}{4}},\\]\n\tand observe that\n\t\t\\begin{equation}\\label{eqn:inter1} \\mathbb{E}\\left( \\left| g\\left(\\eta_0\\right) - g\\left(\\tilde{\\eta}\\right) \\right| \\right)= \n\t\t  \\mathbb{E}\\left( \\left|\\int_{\\tilde{\\eta}}^{\\eta_0} N \\Im \\left( s_W(\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} t) -m_{sc}(\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} t)\\right)\n\t\t  \t{\\rm d}t \\right| \\right)\n\t\t  \\leq \n\t\t  \\int_{\\tilde{\\eta}}^{\\eta_0} \\mathbb{E} (N\\left|\\Im \\left( s_{W_1}\\left({\\rm i}t\\right) - \n\t\t  \tm_{sc} ({\\rm i}t) \\right|\\right))\\, {\\rm d}t. \\end{equation}\n\tIn estimating the right hand side above, we will use the notation\n\t\t\\[ \\Delta(t) = \\left|\\Im \\left( s_{W_1}({\\rm i}t) - \n\t\t  \tm_{sc}({\\rm i}t \\right) \\right|. \\]\n\tFor $N$ sufficiently large, by Theorem \\ref{schlein} with $q =2$, we can write the right hand side of (\\ref{eqn:inter1}) as\n\t\t\\begin{multline}\n\t\t\t\\int_{\\tilde{\\eta}}^{\\eta_0} \\int_0^\\infty \\mathbb{P}\\left(N  \\Delta\\left( t \\right) > u\\right){\\rm d}u{\\rm d}t \n\t\t\t= \\int_{\\tilde{\\eta}}^{\\eta_0}\\left( \n\t\t\t\t\\int_0^1 \\mathbb{P}\\left( \\Delta\\left( t \\right) > \\frac{K}{Nt}\\right)\\,\\frac{{\\rm d}K}{t} +\n\t\t\t\t\\int_{1}^\\infty \\mathbb{P}\\left( \\Delta\\left( t \\right) > \\frac{K}{Nt}\\right)\\frac{{\\rm d}K}{t} \\right){\\rm d}t \\\\\n\t\t\t\\leq \\int_{\\tilde{\\eta}}^{\\eta_0} \\left( \\frac{1}{t} + \\int_1^\\infty \\frac{C}{K^2} \\frac{dK}{t} \\right){\\rm d}t\n\t\t\t\\leq\\left(1 + C\\right)\\log\\left( \\frac{\\eta_0}{\\tilde{\\eta}}\\right) \n\t\t\t={\\rm o}\\left( \\sqrt{\\log N} \\right).\\label{est1}\n\t\t\\end{multline}\n\tNext we estimate\n\t $\\sum_{k} \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right)$, and \n\tthis will give us a bound for $\\mathbb{E}\\left(\\left| g(\\tilde{\\eta}) \\right|\\right)$. \n\tTaylor expansion yields\n\t\t\\[ \\sum_{|y_k| > \\tilde{\\eta}}  \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right) \\leq\n\t\t\\sum_{|y_k| > \\tilde{\\eta}} \\frac{\\tilde{\\eta}^2}{y_k^2}.\n \\]\n\tDefine\n\t\t$N_1(u) = \\left| \\left\\{ y_k : \\tilde{\\eta} \\leq |y_k| \\leq u \\right\\}\\right|$.\n\tUsing integration by parts and Corollary \\ref{repulsion}, we have\n\t\t\\begin{equation} \\mathbb{E} \\left( \\sum_{|y_k| > \\tilde{\\eta}} \\frac{\\tilde{\\eta}^2}{y_k^2} \\right) =\n\t\t\t\\mathbb{E}\\left( \\int_{\\tilde{\\eta}}^\\infty \\frac{\\tilde{\\eta}^2}{y^2}\\,{\\rm d}N_1(y) \\right)\n\t\t\t= 2\\tilde{\\eta}^2 \\int_{\\tilde{\\eta}}^\\infty \\frac{\\mathbb{E}\\left(N_1(y)\\right)}{y^3}\\,{\\rm d}y = {\\rm O}\\left (d_N\\right).\\label{eqn:est2}\n\t\t\t\\end{equation}\n\tWe now estimate $\\sum_{|y_k| \\leq \\tilde{\\eta}}  \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right)$.\n\tWe consider two cases.\n\tFirst, let $A_N = b_N\/N$ for some very small $b_N$, for example \n\t\t\\[ b_N = e^{-\\left(\\log N\\right)^{\\frac{1}{4}}}.\\]\n\tFor $u > 0$ we denote\n\t\t$ N_2(u) = \\left| \\left\\{ y_k \\,:\\, A_N < | y_k |\\leq u \\right\\} \\right|$. Then again using integration by parts and Corollary \\ref{repulsion}  we obtain\n\t\t\\begin{multline*}\n\t\t\t\\mathbb{E}\\left(\n\t\t\t\\sum_{A_N <| y_k| < \\tilde{\\eta}}  \\left(\\log \\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left| y_k\\right|\\right)\n\t\t\t\\right)\n\t\t\t= \\mathbb{E}\\left( \\int_{A_N}^{\\tilde{\\eta}} \n\t\t\t\t\\left( \\log \\left| y + {\\rm i}\\tilde{\\eta}\\right| - \\log \\left|y\\right|\\right){\\rm d}N_2(y)  \\right)\\\\\n\t\t\t\t\\leq \\log\\left(\\sqrt{2}\\right) \\mathbb{E}\\left( N_2\\left( \\tilde{\\eta} \\right) \\right) + \n\t\t\t\t\\int_{A_N}^{\\tilde{\\eta}} \\frac{\\mathbb{E}\\left(N_2(y)\\right)}{y}{\\rm d}y\n\t\t\t\t= {\\rm O}\\left(d_N + d_N\\log\\left( \\frac{d_N}{b_N} \\right)\\right) = \\oo\\left( \\sqrt{\\log N}\\right).\n\t\t\\end{multline*}\nIt remains\n\tto estimate $\\sum_{|y_k| < A_N}\\left( \\log\\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log\\left|y_k\\right|\\right)$. \n\tBy Corollary \\ref{micro fixed energy}, we have\n\t\t\\begin{equation}\\label{est3}\n\t\t \\mathbb{P}\\left(\\sum_{|y_k| < A_N}\\left( \\log\\left| y_k + {\\rm i}\\tilde{\\eta}\\right| - \\log\\left|y_k\\right| \\right) = 0\\right) \n\t\t \t\\geq \\mathbb{P}\\left( \\left|\\left\\{ y_k \\in [-A_N,A_N]\\right\\} \\right|= 0 \\right) \\to 1.\n\t\t\\end{equation}\n\t\\noindent \n\tThe estimates (\\ref{est1}) and (\\ref{eqn:est2}) along with Markov's inequality, and the bound (\\ref{est3}),\n\tconclude the proof.\n\t\\end{proof}\n\\section{Coupling of Determinants}\\label{DBM}\n\t\n\t\\noindent In this section, we use the coupled Dyson Brownian Motion introduced in \\cite{BouErdYauYin2016} to\n\tcompare (\\ref{smoothing t}) and (\\ref{coupling at tau}). \n\tDefine $\\tilde{W}_\\tau$ by running the matrix Dyson Brownian Motion\n\t(\\ref{eq:DBM evolution}) with initial condition $\\tilde{W}_0$\n\twhere $\\tilde{W}_0$ is a Wigner matrix with eigenvalues ${\\bm y}$.\n\tRecall that this induces a collection of Brownian motions\n\t$\\tilde{B}^{(k)}_t$ \n\tso that the system (\\ref{eqn:DBM}) with initial condition ${\\bm y}$\n\thas a (unique strong) solution \n\t${\\bm y}(\\cdot)$, and ${\\bm y}(\\tau)$ are the eigenvalues of $\\tilde{W}_\\tau$. Using the same (induced) Brownian motions as we used to define ${\\bm y}(\\tau)$, define\n\t${\\bm x}(\\tau)$ by running the dynamics (\\ref{eqn:DBM}) with initial condition given by the eigenvalues of a GOE matrix.\n\tUsing the result of Section \\ref{smoothing section} as an input to Lemma \\ref{extreme}, we now prove Proposition\n\t\\ref{prop:advection} which says that (\\ref{coupling at tau}) and (\\ref{smoothing t})  are asymptotically equal in law. \\\\\n\n\n\t\n\t\\noindent To study the coupled dynamics of ${\\bm x}(t)$ and ${\\bm y}(t)$, we\n\tfollow \\cite{LanSosYau2016, BourgadeExtreme}. For $\\nu \\in [0,1]$, let\n\t\t\\begin{equation}\n\t\t\\label{def:initialcondition}\n\t\t\t\\lambda^\\nu_k(0) = \\nu x_k + \\left(1 - \\nu\\right)y_k\n\t\t\\end{equation}\n\twhere ${\\bm x}$ is the spectrum of a GOE matrix, and ${\\bm y}$ is the spectrum of $\\tilde{W}_0$.\n\tWith this initial condition, we denote the (unique strong) solution to (\\ref{eqn:DBM}) by ${\\bm \\lambda}^{(\\nu)}(t)$.\n\tNote that\n$\n\t\t\t{\\bm \\lambda}^{(0)}(\\tau) = {\\bm y}(\\tau)$ and $\n\t\t\t{\\bm \\lambda}^{(1)}(\\tau) = {\\bm x}(\\tau).\n$ \nLet\n\t\t\\begin{equation}\n\t\t\\label{eqn:ft}\n\t\tf^{(\\nu)}_t(z) =e^{-\\frac{t}{2}} \\sum_{k=1}^N \\frac{u_k(t)}{\\lambda^{(\\nu)}_k(t)-z}, \n\t\t\\quad  u_k(t) = \\frac{{\\rm d}}{{\\rm d}\\nu}\\lambda^{(\\nu)}_k(t),\n\t\t\\end{equation}\n\t(see \\cite{LanSosYau2016} for existence of this derivative) and observe that\n\t\t\\begin{equation}\n\t\t\\label{main obs}\n\t\t\\frac{{\\rm d}}{{\\rm d}\\nu} \\sum_k \\log\\left| \\lambda_k^{(\\nu)}(t) - z \\right| = e^{\\frac{t}{2}}\\Re  \\left(f_t(z)\\right).\n\t\t\\end{equation}\n\tLemma \\ref{extreme} below from \\cite[Proposition 3.3]{BourgadeExtreme}, tells us that we may estimate\n\t$f_\\tau(z)$ by $f_0\\left(z_\\tau\\right)$, with $z_\\tau$ as in (\\ref{def:ztau}) and $\\tau$ as in (\\ref{def:tau}). \n\t\n\t\\begin{lemma}\n\t\t\\label{extreme}\n\t\tThere exists $C_0>0$ such that with $\\varphi=e^{C_0(\\log\\log N)^2}$, \n\t\tfor any $\\nu\\in[0,1]$, $\\kappa>0$ (small) and $D>0$ (large),  \n\t\t there exists $N_0(\\kappa,D)$ so that for any $N\\geq N_0$ we have\n\t\t\t\\[ \\mathbb{P}\\left(\\left| f^{(\\nu)}_t(z) - f^{(\\nu)}_0\\left(z_t\\right) \\right| <  \\frac{\\varphi}{N\\eta}\\ {\\rm for\\ all}\\ 0<t<1\\ {\\rm and}\\ z=E+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i} \\eta, \\frac{\\varphi}{N}<\\eta<1,|E|<2-\\kappa\\right)\n\t\t\t\t\\geq 1-N^{-D}.\\]\n\t\tIn the above, $z_t$ is given by\n\t\t\t\\begin{equation}\n\t\t\t\\label{def:ztau} z_t = \\frac{1}{2}\\left( e^{\\frac{t}{2}}\\left(z + \\sqrt{z^2 - 4}\\right) + \n\t\t\t\te^{-\\frac{t}{2}}\\left(z - \\sqrt{z^2 - 4}\\right) \\right).\n\t\t\t\\end{equation}\n\t\\end{lemma}\t\n\t\\noindent For $z = {\\rm i}\\eta_0$, we have\n\t\t\\begin{equation}\n\t\t\\label{zt expansion}\n\t\t z_t = {\\rm i} \\left( \\eta_0 + \\frac{t \\sqrt{\\eta_0^2 + 4}}{2} \\right) + {\\rm O}\\left (t^2\\right),\n\t\t\\end{equation}\n\tand $\\eta_0$ is large enough to make use of Lemma \\ref{extreme}.\n\tTherefore,\n\tintegrating both sides of (\\ref{main obs}), we have by Lemma \\ref{extreme} that with overwhelming probability,\n\t\t\\begin{multline*}\n\t\t\t\\sum_k \\left( \\log \\left|x_k(\\tau) + {\\rm i}\\eta_0\\right| - \\log \\left|y_k(\\tau) + {\\rm i}\\eta_0\\right|  \\right) = \n\t\t\t\\int_0^1 \\frac{{\\rm d}}{{\\rm d}\\nu} \\sum_k \\log\\left| \\lambda_k^{(\\nu)}(\\tau) - z \\right| {\\rm d}\\nu \\\\\n\t\t\t= e^{\\frac{t}{2}} \\Re  \\int_0^1 f^{(\\nu)}_\\tau(z) {\\rm d}\\nu \n\t\t\t= e^{\\frac{t}{2}}\\Re  \\int_0^1 \n\t\t\t\\left( f_0^{(\\nu)}\\left(z_\\tau\\right) + {\\rm O}\\left ( \\frac{\\varphi}{N\\eta_0}\\right)\\right) {\\rm d}\\nu \n\t\t\t= e^{\\frac{t}{2}}\\Re  \\int_0^1 f_0^{(\\nu)}\\left(z_\\tau\\right){\\rm d}\\nu + {\\rm o}(1).\n\t\t\\end{multline*}\n\t\tMore precisely, the above estimates hold with probability $1-N^{-D}$ for large enough $N$, with rigorous justification by Markov's inequality based on the large moments \n\t\t$\\mathbb{E}((\\int_0^1 \\left( f_\\tau^{(\\nu)}\\left(z_0\\right)- f_0^{(\\nu)}\\left(z_\\tau\\right)  \\right) {\\rm d}\\nu)^{2p})$, which are bounded by Lemma \\ref{extreme}.\n\tAs a consequence, we have proved the following proposition.\n\t\\begin{proposition}\n\t\\label{prop:advection}\n\tLet $\\epsilon >0$, $\\tau = N^{-\\epsilon}$ and let $z_\\tau$ be as in (\\ref{def:ztau}) with $z = {\\rm i}\\eta_0$.\n\tThen for any $\\delta > 0$, \n\t\t\\[ \\lim_{N\\to\\infty} \\mathbb{P}\\left(\\left|\n\t\t\\sum_k \\left( \\log \\left|x_k(\\tau) + {\\rm i}\\eta_0\\right| - \\log \\left|y_k(\\tau) + {\\rm i}\\eta_0\\right|  \\right) \n\t\t - \\sum_k \\left( \\log \\left|x_k(0) + z_\\tau\\right| - \\log \\left|y_k(0) + z_\\tau\\right|  \\right)\\right| > \\delta  \\right) = 0.  \\]\n\t\\end{proposition}\n\n\\section{Conclusion of the Proof}\\label{conclusion}\n\n\t\\noindent We will conclude the proof of Theorem \\ref{main theorem} in the real symmetric case\n\tin two steps. The first step is to prove a Green's function comparison\n\ttheorem, and the second is to establish Theorem \\ref{main theorem} assuming Lemma\n\t\\ref{expectation}, proved  in the Appendix. \n\n\t\\subsection{Green's Function Comparison Theorem. }\\label{moment matching section}\n\tIn this section, we first use Lemma \\ref{moment matching} to \n\tchoose a $\\tilde{W}_0$ so that $\\tilde{W}_{\\tau}$ given by\n\t(\\ref{eq:DBM evolution})\n\tand initial condition $\\tilde{W}_0$,\n\tmatches $W$ closely up to fourth moment. We will then prove Theorem \\ref{4 moment matching theorem},\n\twhich by the result of Section \\ref{smoothing section}, says that\n\t$ \\log | \\det \\tilde{W}_\\tau |$ and \n\t$ \\log \\left| \\det W \\right|$ have the same law as $N\\to\\infty$.  \n\t\n\t\\begin{lemma}[Lemma 6.5 in \\cite{ErdYauYin2012Bulk}]\n\t\t\\label{moment matching}\n\t\tLet $m_3$ and $m_4$ be two real numbers such that\n\t\t\t\\begin{equation}\\label{moment condition}\n\t\t\t m_4 - m_3^2 - 1 \\geq 0, \\quad m_4 \\leq C_2 \n\t\t\t\\end{equation}\n\t\tfor some positive constant $C_2$. Let $\\xi^G$ be a Gaussian random variable with mean \n\t\t$0$ and variance $1$. Then for any sufficiently small $\\gamma > 0$ (depending on $C_2$),\n\t\tthere exists a real random variable $\\xi$, with subgaussian decay\n\t\tand independent of $\\xi^G$ such that the first four moments of\n\t\t\t\\[ \\xi' = \\left( 1 - \\gamma\\right)^{\\frac{1}{2}} \\xi_\\gamma + \\gamma^{\\frac{1}{2}}\\xi^G\\]\n\t\tare $m_1\\left(\\xi'\\right) = 0$, $m_2\\left(\\xi'\\right) = 1$, $m_3\\left(\\xi'\\right) = m_3$,\n\t\tand \n\t\t\t\\[ \\left|m_4\\left(\\xi'\\right) - m_4\\right| \\leq C\\gamma \\]\n\t\tfor some $C$ depending on $C_2$. \n\t\\end{lemma}\n\t\\noindent Now since $\\tilde{W}_\\tau$ is defined by independent Ornstein-Uhlenbeck processes in each entry, \n\tit has the same distribution as\n\t\t\\[ e^{-\\tau\/2} \\tilde{W}_0 + \\sqrt{1-e^{-\\tau}}W   \\]\n\twhere $W$ is a GOE matrix independent of $\\tilde{W}_0$.\n\tSo choosing $\\gamma = 1-e^{-\\tau}$,\n\tLemma \\ref{moment matching} says we can\n\tfind $\\tilde{W}_0$ so that the first three moments of the entries of $\\tilde{W}_\\tau$ \n\tmatch the first three moments of the entries of $W$, and the fourth moments of the entries\n\tof each differ by ${\\rm O}(\\tau)$.  \n\tOur next goal is to prove Theorem \\ref{4 moment matching theorem} which says that\n\twith $\\tilde{W}_\\tau$ constructed this way, if Theorem \\ref{main theorem} \n\tholds for $\\tilde{W}_\\tau$, then it holds for $W$. We first introduce stochastic domination and state\n\tTheorem \\ref{local law} which we will use in the proof.\n\t\n\t\\begin{definition}\n\t\tLet $X = \\left(X^N(u): N \\in \\mathbb{N}, u \\in U^N\\right), Y = \\left(Y^N(u): N \\in \\mathbb{N}, u \\in U^N\\right)$\n\t\tbe two families of nonnegative random variables, where $U^N$ is a possibly $N$-dependent parameter set.\n\t\tWe say that $X$ is stochastically dominated by $Y$, uniformly in $u$, if for every $\\epsilon > 0$ and $D > 0$, there exists $N_0(\\epsilon, D)$ such that\n\t\t\t\\[ \\sup_{u \\in U^N} \\mathbb{P}\\left[ X^N(u) > N^\\epsilon Y^N(u)\\right] \\leq N^{-D} \\]\n\t\tfor $N\\geq N_0$. Stochastic domination is always uniform in all parameters,\n\t\tsuch as matrix indices and spectral parameters, that are not explicitly fixed.\n\t\tWe will use the notation $X = O_\\prec(Y)$ or $X \\prec Y$ for the above property.\n\t\\end{definition}\n\t\n\t\\begin{theorem}[Theorem 2.1 in \\cite{ErdYauYin2012Rig}]\\label{local law}\n\tLet $W$ be a Wigner matrix satisfying (\\ref{subgaussian}). Fix $\\zeta > 0$ and define the domain\n\t\t\\[ S = S_N(\\zeta) := \\left\\{ E + {\\rm i}\\eta \\,:\\, |E| \\leq \\zeta^{-1}, \\,N^{-1+\\zeta} \\leq \\eta \\leq \\zeta^{-1} \\right\\}. \\]\n\tThen uniformly for $i,j = 1,\\hdots, N$ and $z \\in S$, we have\n\t\\begin{align*}\n\t\t s(z) &= m(z) +  {\\rm O}_\\prec\\left( \\frac{1}{N\\eta} \\right),\\\\\n\tG_{ij}(z)& = \\left(W -z\\right)^{-1}_{ij} = \n\t\tm(z)\\delta_{ij} + {\\rm O}_\\prec\\left( \\sqrt{\\frac{\\Im  \\left(m(z)\\right)}{N\\eta}} + \\frac{1}{N\\eta} \\right).\n\t\t\\end{align*}\n\t\\end{theorem}\n\n\t\\begin{theorem}\\label{4 moment matching theorem}\n\t\tLet $F: \\mathbb{R} \\to \\mathbb{R}$ be smooth with compact support, and\n\t\tlet $W$ and $V$ be two Wigner matrices \n\t\tsatisfying (\\ref{subgaussian}) such that for $1 \\leq i, j \\leq N$,\n\t\t\t\\begin{numcases}{\\mathbb{E}\\left(w_{ij}^a\\right) = }\n\t\t\t\t\\label{moment matching assumption 3}\n\t\t\t\t\\mathbb{E}\\left(v_{ij}^a\\right)  & $a \\leq 3$ \\\\\n\t\t\t\t\\label{moment matching assumption 4}\n\t\t\t\t\\mathbb{E}\\left(v_{ij}^a\\right)+ \\OO(\\tau) & $a = 4$,\n\t\t\t\\end{numcases}\n\t\twhere $\\tau$ is as in (\\ref{def:tau}). Further, let $c_N$ be any deterministic sequence and define\n\t\t\t\\[\n\t\t\t\tu_N(W) = \\frac{\\log | \\det \\left(W + \\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_0\\right)| +c_N}{\\sqrt{\\log N}}.\n\t\t\t\\]\n\t\twhere $\\eta_0$ is as in (\\ref{def:eta0}). Then\n\t\t\t\\begin{equation}\n\t\t\t\t\\lim_{N\\to \\infty} \\mathbb{E} \\left(F\\left( u_N(W)\\right) - F\\left( u_N(V) \\right)\\right) = 0.\\label{eqn:enough}\n\t\t\t\\end{equation} \n\t\\end{theorem}\n\t\\begin{proof}\t\t\n\t\tAs in \\cite{TaoVu2012}, where the authors also used the following technique to analyze\n\t\tfluctuations of determinants, we show that the effect of substituting $W_{ij}$ in place of $V_{ij}$ in $V$ is\n\t\tnegligible enough that making $N^2$ replacements, we conclude the theorem. \\\\\n\n\t\t\n\t\t\\noindent Fix $(i,j)$ and let\n\t\t$E^{(ij)}$ be the matrix whose elements are $E^{(ij)}_{kl} = \\delta_{ik}\\delta_{jl}$.\n\t\tLet $W_1$ and $W_2$ be two adjacent matrices in the swapping process described above. \n\t\tSince $W_1, W_2$ differ in just the $(i,j)$ and $(j,i)$ coordinates, we may write\n\t\t\t$$\n\t\t\t\tW_1 = Q + \\frac{1}{\\sqrt{N}} U, \\ \\ \\  \\ \\\n\t\t\t\tW_2 = Q + \\frac{1}{\\sqrt{N}} \\tilde{U}\n\t\t\t$$\n\t\twhere $Q$ is a matrix with $Q_{ij} = Q_{ji} = 0$, and\n\t\t\t$$\n\t\t\t\tU = u_{ij}E^{(ij)} + u_{ji}E^{(ji)}\\ \\ \\ \\ \\ \\ \n\t\t\t\t\\tilde{U} = \\tilde{u}_{ij}E^{(ij)} + \\tilde{u}_{ji}E^{(ji)}.\n\t\t$$\n\t\tImportantly $U,\\tilde{U}$ satisfy the same moment matching conditions\n\t\twe have imposed on $\\tilde{W}_\\tau$ and $W$. \n\t\tNow by the fundamental theorem of calculus, we have for any symmetric matrix $W$,\n\t\t\t\\begin{equation}\n\t\t\t\\label{log via fundamental theorem}\n\t\t\t\\log\\left|\\det(W + {\\rm i}\\eta_0)\\right| = \n\t\t\t\t\\sum_{k=1}^N \\log\\left| x_k + {\\rm i}\\eta_0\\right| = \\log\\left|\\det(W + {\\rm i})\\right| - N\\,\n\t\t\t\t \\Im  \\int_{\\eta_0}^1\n\t\t\t\ts_W\\left({\\rm i}\\eta \\right){\\rm d}\\eta.  \n\t\t\t\\end{equation}\n\t\t\tFrom the central limit theorems for linear statistics of Wigner matrices on macroscopic scales \\cite{LytPas2009}, \n\t\t\t $(\\log\\left|\\det(W + {\\rm i})\\right|-\\mathbb{E}(\\log\\left|\\det(W + {\\rm i})\\right|))\/\\sqrt{\\log N}$ \n\t\t\tconverges to $0$ in probability (the same result holds with $W$ replaced with $V$), and from Lemma \\ref{expectation} (which clearly holds with $1$ in place of $\\tau$), \n\t\t\t$(\\mathbb{E}(\\log\\left|\\det(W + {\\rm i})\\right|)-\\mathbb{E}(\\log\\left|\\det(V + {\\rm i})\\right|))\/\\sqrt{\\log N}\\to 0$.\n\t\t\tTherefore (\\ref{eqn:enough}) is equivalent to\n\t\t\t\t\\begin{equation}\n\t\t\\label{key expression}\n\t\t\t\\lim_{N\\to \\infty} \\mathbb{E} \\left(\\widetilde F\\left(N\\,\n\t\t\t\t \\Im  \\int_{\\eta_0}^1\n\t\t\t\ts_W\\left({\\rm i}\\eta \\right){\\rm d}\\eta\\right) - \\widetilde F\\left(N\\,\n\t\t\t\t \\Im  \\int_{\\eta_0}^1\n\t\t\t\ts_V\\left({\\rm i}\\eta \\right){\\rm d}\\eta\\right)\\right)= 0,\n\t\t\t\\end{equation}\n\t\t\twhere\n\t\t\t$$\n\t\t\t\\widetilde F(x)=F\\left(\\frac{\\mathbb{E}(\\log\\left|\\det(W + {\\rm i})\\right|)+c_N-x}{\\sqrt{\\log N}}\\right).\n\t\t\t$$\n\n\t\tWe now expand $s_{W_1}$ and $s_{W_2}$ around $s_{Q}$, and then\n\t\tto Taylor expand $\\widetilde F$.\n\t\tSo let \n\t\t\t\\[ R =R(z)= \\left(Q-z\\right)^{-1} \\text{ and } S=S(z)=\\left(W_1 - z\\right)^{-1}. \\]\n\t\tBy the resolvent expansion\n\t\t\t\\[ \n\t\t\tS = R - N^{-1\/2}RUR + \\hdots + N^{-2}(RU)^4R - N^{-5\/2}(RU)^5 S,\n\t\t\t\\]\n\t\twe can write\n\t\t\t\\[N \\int_{\\eta_0}^1 s_{W_1}({\\rm i}\\eta){\\rm d}\\eta = \\int_{\\eta_0}^1\\text{Tr}\\left(S({\\rm i}\\eta)\\right){\\rm d}\\eta = \n\t\t\t\\int_{\\eta_0}^1 \\text{Tr}\\left(R({\\rm i}\\eta)\\right){\\rm d}\\eta +\n\t\t\t\\left( \\sum_{m=1}^4 N^{-m\/2}\\hat{R}^{(m)}({\\rm i}\\eta) - N^{-5\/2}\\Omega\\right)\n\t\t\t\t:= \\hat{R} + \\xi  \\]\n\t\twhere \n\t\t\t\\[ \\hat{R}^{(m)} = (-1)^m \\int_{\\eta_0}^1 \\text{Tr}\\left((R({\\rm i}\\eta)U)^mR({\\rm i}\\eta)\\right){\\rm d}\\eta \n\t\t\t\\quad \\text{and} \\quad\n\t\t\t\\Omega = \\int_{\\eta_0}^1 \\text{Tr} \\left( (R({\\rm i}\\eta)U)^5S({\\rm i}\\eta) \\right){\\rm d}\\eta.\\]\n\t\tThis gives us an expansion of $s_{W_1}$ around $s_{Q}$. \n\t\tNow Taylor expand $\\widetilde F(\\hat{R} + \\xi)$ as\n\t\t\t\\begin{equation} \\label{taylor expansion} \\widetilde F\\left(\\hat{R}+\\xi\\right) = \\widetilde F\\left(\\hat{R}\\right) + \n\t\t\t\t\\widetilde F'\\left(\\hat{R}\\right)\\xi + \\hdots + \\widetilde F^{(5)}\\left(\\hat{R}+\\xi'\\right)\\xi^5 = \\sum_{m=0}^5 N^{-m\/2}A^{(m)}\n\t\t\t\\end{equation}\n\t\twhere $0 < \\xi' < \\xi$, and we have introduced the notation $A^{(m)}$ in order to arrange terms according to\n\t\tpowers of $N$. For example\n\t\t\t$$\n\t\t\t\tA^{(0)} = \\widetilde F\\left(\\hat{R}\\right),\\ \\ \n\t\t\t\tA^{(1)} = \\widetilde F'\\left(\\hat{R}\\right)\\hat{R}^{(1)}, \\ \\ \n\t\t\t\tA^{(2)} = \\widetilde F'\\left(\\hat{R}\\right)\\hat{R}^{(2)} + \\widetilde F''\\left(\\hat{R}\\right)\\left(\\hat{R}^{(1)}\\right)^2.\n\t\t\t$$\n\t\tMaking the same expansion for $W_2$, we record our two expansions as\n\t\t\t\\[ \\widetilde F\\left(\\hat{R} + \\xi_i\\right) = \\sum_{m=0}^5 N^{-m\/2} A^{(m)}_i, \\quad i = 1, 2, \\]\n\t\twith $\\xi_i$ corresponding to $W_i$. With this notation, we have\n\t\t\t\\begin{align*} \n\t\t\t\t\\mathbb{E}\\left(\\widetilde F\\left(\\hat{R} + \\xi_1\\right)\\right) - \\mathbb{E} \n\t\t\t\t\\left(\\widetilde F\\left(\\hat{R} + \\xi_2\\right)\\right)\n\t\t\t\t&=  \\mathbb{E} \\left(\\sum_{m=0}^5 N^{-m\/2}\\left( A_1^{(m)} - A_2^{(m)} \\right)\\right) .\n\t\t\t\\end{align*}\n\t\tNow only the first three moments of $U,\\tilde{U}$ appear in the terms corresponding to $m=1, 2, 3$, so\n\t\tby the moment\n\t\tmatching assumption (\\ref{moment matching assumption 3}), all of these terms are all identically zero. \n\t\tNext, consider $m = 4$. Every term with first, second, and third moments of\n\t\t$U$ and $\\tilde{U}$ is again zero, and what remains is\n\t\t\t\\[ \\mathbb{E} \\left(\\widetilde F'(\\hat{R}) \\left( \\hat{R}_1^{(4)} - \\hat{R}_2^{(4)}\\right)\\right). \\]  \n\t\t\\noindent So we can discard $A^{(4)}$ if \n\t\t\t\\begin{equation}\\label{4th moment}\n\t\t\t \\int_{\\eta_0}^1 \\left|\\mathbb{E} \\left( \n\t\t\t \t{\\rm Tr}\\left( (RU)^4R \\right) - {\\rm Tr}\\left( (R\\tilde{U})^4R\\right) \\right)   \\right| {\\rm d}\\eta \n\t\t\t \\end{equation}\n\t\tis small. To see that this is in fact the case, we expand the traces, and apply Theorem \\ref{local law}\n\t\talong with our fourth moment matching assumption (\\ref{moment matching assumption 4}). \n\t\tSpecifically,\n\t\t\t\\[ \\text{Tr}\\left( (RU)^4R\\right) = \\sum_j \\left(\\sum_{i_1, \\hdots, i_8} \n\t\t\t\tR_{ji_1}U_{i_1i_2}R_{i_2i_3}\\hdots U_{i_7i_8}R_{i_8j}\\right). \\]\n\t\tWriting the corresponding Tr for $W_2$ and applying the moment matching assumption, we see that we can\n\t\tbound (\\ref{4th moment}) by\n\t\t\t\\[ {\\rm O}(\\tau) \\int_{\\eta_0}^1\\sum_j \\sum_{i_1, \\hdots, i_8} \\mathbb{E} \\left(\n\t\t\t\t\\left| R_{ji_1}R_{i_2i_3}R_{i_4i_5}R_{i_6i_7}R_{i_8j} \\right| \\right){\\rm d}\\eta. \\]\n\t\tTo bound the terms in the sum, we need to count the number of diagonal and off-diagonal terms in each product.\n\t\tTo do this, let us say $U_{pq}, \\tilde{U}_{pq}$ and $U_{qp}, \\tilde{U}_{qp}$ are the only non-zero entries of \n\t\t$U, \\tilde{U}$. Then\n\t\teach of the sums over $i_1, \\hdots, i_8$ are just sums over $p, q$, and when $j \\notin \\{p,q\\}$, \n\t\t$R_{ji_1}$ and $R_{i_8j}$ are certainly off-diagonal entries of $R$. This means we can apply Cauchy-Schwartz\n\t\tto write that for any $\\gamma > 0$,\n\t\t\t\\begin{align*}\n\t\t\t\t{\\rm O}(\\tau) \\int_{\\eta_0}^1\\sum_{j \\notin \\{p, q\\}} \\sum_{i_1, \\hdots, i_8} \\mathbb{E} \\left(\n\t\t\t\t\t\\left| R_{ji_1}R_{i_2i_3}R_{i_4i_5}R_{i_6i_7}R_{i_8j} \\right| \\right){\\rm d}\\eta \n\t\t\t\t= {\\rm O}\\left (\\tau N^{1 + 2\\gamma}  \\int_{\\eta_0}^1 \\frac{1}{N\\eta}{\\rm d}\\eta\\right) = \n\t\t\t\t\t{\\rm O}\\left (N^{2\\gamma - \\epsilon} \\log(N)\\right).\n\t\t\t\\end{align*}\n\t\tSimilarly, \n\t\t\t\\begin{align*}\n\t\t\t\t{\\rm O}(\\tau) \\int_{\\eta_0}^1\\sum_{j \\in \\{p, q\\}} \\sum_{i_1, \\hdots, i_8} \\mathbb{E} \\left(\n\t\t\t\t\t\\left| R_{ji_1}R_{i_2i_3}R_{i_4i_5}R_{i_6i_7}R_{i_8j} \\right| \\right){\\rm d}\\eta \n\t\t\t\t= {\\rm O}\\left (\\tau N^{\\epsilon \/2} \\right) = \n\t\t\t\t\t{\\rm O}\\left (N^{- \\epsilon\/2} \\right).\n\t\t\t\\end{align*}\n\t\tSince $A^{(4)}$ has a pre-factor of $N^{-2}$ in (\\ref{taylor expansion}), and the above holds\n\t\tfor every choice of $\\gamma > 0$, in our entire \n\t\tentry swapping scheme starting from $V$ and ending with $W$, the corresponding error\n\t\tis ${\\rm o}(1)$.\\\\\n\t\t\n\t\t\\noindent Lastly we comment on the error term $A^{(5)}$. All terms in $A^{(5)}$ not involving \n\t\t$\\Omega$ can be dealt\n\t\twith as above. The only term involving $\\Omega$ is $\\widetilde F'(\\hat{R})\\Omega$, and to deal with this,\n\t\twe can expand the expression for $\\Omega$ as above. We do not have any\n\t\tmoment matching condition for the fifth moments of $U, \\tilde{U}$, but (\\ref{subgaussian}) means\n\t\tthat their fifth\n\t\tmoments are bounded which is enough for our purpose since $A^{(5)}$ has a pre-factor of $N^{-5\/2}$ above. \n\t\t\\end{proof}\t\n\t\t\n\t\\subsection{Proof of Theorem \\ref{main theorem}. }\\label{sub:end}\n\tIn this section we first prove Proposition \\ref{variance} and, using Lemma \n\t\\ref{expectation}, we\n\tconclude the proof of Theorem \\ref{main theorem}.\n\t\n\t\n\t\n\t\n\t\\begin{proposition}\\label{prop:variance} Recall $\\tau=N^{-{\\varepsilon}}$.\n\t\\label{variance} There exist ${\\varepsilon}_0,C$ such that for any fixed $0<{\\varepsilon}<{\\varepsilon}_0$, for large enough $N$, we have\n\t\\[ {\\rm Var}\\left( \\sum_k\\log|x_k(0)+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\tau|\\right) \\leq C (1+\\epsilon \\log N).\\]\n\\end{proposition}\t\n\\begin{proof}\nWe outline two proofs, which are trivial extensions of existing linear statistics asymptotics on global scales, to the case of almost macroscopic scales. The tool for this extension is the rigidity estimate from \\cite{ErdYauYin2012Rig}: for any $c,D>0$, there exists $N_0$ such that for any $N\\geq N_0$ and $k\\in\\llbracket 1,N\\rrbracket$ we have\n\\begin{equation}\\label{eqn:rigidity}\n\\mathbb{P}\\left(|x_k-\\gamma_k|>N^{-\\frac{2}{3}+c}\\min(k,N+1-k)^{-\\frac{1}{3}}\\right)\\leq N^{-D}.\n\\end{equation}\n\n\n\\noindent For the first proof, we use (\\ref{eqn:rigidity}) to bound all the error terms in the proof of \\cite[Theorem 3.6]{LytPas2009} (these error terms all depend on \\cite[Theorem 3.5]{LytPas2009}, which can be improved via \n(\\ref{eqn:rigidity}) to ${\\rm Var}(u_N(t))\\leq N^c(1+|t|)$ and ${\\rm Var}\\left(\\mathcal{N}_N(\\varphi)\\right)\\leq N^c\\|\\varphi\\|_{\\rm Lip}^2$). What we obtain is that if $\\varphi$ (possibly depending on $N$) satisfies $\\int|t|^{100}\\hat \\varphi(t)<N^{1\/100}$, then $\\sum\\varphi(x_k)-\\mathbb{E}(\\sum\\varphi(x_k))$ has limiting variance asymptotically equivalent to \n\t\t\\begin{equation}\n\\label{vwig} \n\t\tV_{\\rm Wig}[\\varphi] =  \\frac{1}{2\\pi^2} \\int_{(-2,2)^2}\n\t\t\t\\left( \\frac{\\Delta \\varphi}{\\Delta \\lambda}\\right)^2 \n\t\t\t\\frac{4 - \\lambda_1\\lambda_2}{\\sqrt{4 - \\lambda_1^2}\n\t\t\t\\sqrt{1- \\lambda_2^2}} \\,\n\t\t\t{\\rm d}\\lambda_1{\\rm d}\\lambda_2+ \\frac{\\kappa_4}{2\\pi^2}\n\t\t\t\\left( \\int_{-2}^{2} \\varphi(\\mu) \\frac{2 - \\mu^2}{\\sqrt{4 - \\mu^2}}\\,\n\t\t\t{\\rm d}\\mu\\right)^2,\n\t\t\\end{equation}\nwhere $\\Delta \\varphi = \\varphi\\left(\\lambda_1\\right) - \\varphi\\left(\\lambda_2\\right)$, $\\Delta \\lambda = \\lambda_1 - \\lambda_2$, $\\mu_4 = \\mathbb{E} \\left(W_{jk}^4\\right)$,\n\t$\\kappa_4 = \\mu_4 - 3$ is the fourth cumulant of the off-diagonal entries of $W$.\nWe choose $\\varphi(x)=\\varphi_N(x)=\\frac{1}{2}\\log(x^2+\\tau^2)\\chi(x)$ with $\\chi$ fixed, smooth, compactly supported, equal to 1 on $(-3,3)$. Note that for ${\\varepsilon}_0$ small enough, we have $\\int|t|^{100}\\hat \\varphi(t)<N^{1\/100}$. Then by (\\ref{eqn:rigidity}) and (\\ref{vwig}), \n$$\nV_{\\rm Wig}[\\log|\\cdot-\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\tau|]\\sim V_{\\rm Wig}[\\varphi]\\leq C \\iint \\left(\\frac{\\Delta\\varphi}{\\Delta\\lambda}\\right)^2\n\t\\, {\\rm d} \\lambda_1 {\\rm d} \\lambda_2\n\t= C \\int |\\xi|\\, \\left| \\hat{\\varphi}(\\xi) \\right|^2{\\rm d}\\xi,\n$$\nand the above right hand side can be bounded as follows.\n\tWe have\n\t\t\\begin{align*}\n\t\t \\left| \\hat{\\varphi}_N(\\xi) \\right| = \\left| \\frac{1}{2\\pi} \\int_{\\mathbb{R}}\n\t\t \t \\varphi_N(x)e^{-i\\xi x}\\, {\\rm d} x\\right|\n\t\t &\\leq \n\t\t C \\left| \\int_{-5}^{5} \\frac{x}{x^2 + \\tau^2} \\frac{e^{-i\\xi x}}{i\\xi}\\, {\\rm d} x\\right| \n\t\t =C\\left| \\frac{1}{\\xi} \\int_0^{5\/\\tau} \\frac{x}{x^2 + 1} \\sin(x \\xi \\tau) \\, {\\rm d} x \\right|.\n\t\t\\end{align*}\n\tFor $0 < \\xi < 5$, the inequality $|\\sin x| < x$ shows $\\left| \\hat{\\varphi}_N(\\xi) \\right| = {\\rm O}(1)$, and when\n\t$\\xi > 5\/\\tau$, integration by parts shows $\\left|\\hat{\\varphi}_N(\\xi)\\right| = {\\rm O}\\left (\\frac{1}{\\xi^2\\tau}\\right)$. \n\tWhen $5 < \\xi < 5\/\\tau$, first note\n\t\t\\begin{align*}\n\t\t\\int_0^{\\frac{5}{\\tau}} \\sin\\left(\\xi \\tau x\\right) \\frac{x}{x^2 + 1} \\, {\\rm d}x &= C + \n\t\t\t\\int_1^{\\frac{5}{\\tau}} \\frac{\\sin\\left(\\xi \\tau x\\right)}{x} \\,{\\rm d}x \n\t\t\t=  C + \\int_{\\xi \\tau}^{1} \\frac{\\sin y}{y} \\,{\\rm d}y + \\int_1^{5\\xi} \\frac{\\sin y}{y}\\,{\\rm d}y.\n\t\t\\end{align*}\n\tUsing $|\\sin y| < |y|$, we see that the first term is ${\\rm O}(1)$, and integrating by parts, we see\n\tthat the second term is ${\\rm O}(1)$ as well. This means \n\t\t\\[ \\int |\\xi| \\left| \\hat{\\varphi}_N(\\xi) \\right|^2{\\rm d}\\xi \\leq C +C \n\t\t\\int_{5}^{\\frac{5}{\\tau}} \\frac{1}{\\xi} \\,{\\rm d}\\xi = {\\rm O}\\left (1+|\\log \\tau|\\right), \n\t\t\\]\n\twhich concludes the proof.\\\\\n\t\n\\noindent The  second proof is similar but more direct. Theorem 3 in \\cite{KhoKhoPas} implies that for $z_1=\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_1,z_2=\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_2$ at macroscopic distance from the real axis, and $\\eta_1=\\im z_1>0,\\eta_2=\\im z_2<0$, we have\n\t$$\n\t\\left|{\\rm Cov}\\left(\\sum_k\\frac{1}{z_1-x_k},\\sum_k\\frac{1}{z_2-x_k}\\right)\\right|\\leq\\frac{C}{(\\eta_1-\\eta_2)^2}+f(z_1,z_2)+\\OO(N^{-1\/2}),\n\t$$\n\twhere $f$ is a function uniformly bounded on any compact subset of $\\mathbb{C}^2$.\n\tUsing (\\ref{eqn:rigidity}), one easily obtains that the formula above holds uniformly with $|\\im z_1|,|\\im z_2|>N^{-1\/10}$, and the deteriorated error term $\\OO(N^{-1\/10})$, for example.\t\nNote that\n$$\n\\log\\left|\\det(W + {\\rm i}\\eta)\\right| = \\log\\left|\\det(W + {\\rm i})\\right| - N\\,\n\t\t\t\t \\Im  \\int_{\\eta}^1\n\t\t\t\ts_W\\left({\\rm i}x \\right){\\rm d}x.  \n$$\nand $\\log\\left|\\det(W + {\\rm i})\\right|$ has fluctuations of order 1 due to the above macroscopic central limit theorems. \nFor for $\\eta>N^{-1\/10}$, the variance of the above integral can be bounded by \n$\n\\iint_{[\\eta,1]^2}\\frac{1}{|\\eta_1+\\eta_2|^2}\\,{\\rm d} \\eta_1{\\rm d} \\eta_2\\leq C|\\log \\eta|,\n$\nwhich concludes the proof.\n\t\\end{proof}\n\t\n\t\n\t\n\n\n\n\\noindent From (\\ref{GOE CLT}) and Proposition \\ref{smoothing}, for some explicit deterministic  $c_N$ we have\n\\begin{equation}\\label{eqn:1}\n\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|x_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}\\to\\mathscr{N}(0,1),\n\\end{equation}\nand Proposition \\ref{prop:advection} implies that\n$$\n\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|y_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}\n+\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|x_k(0) + z_\\tau\\right|-\\sum_{k =1}^N \\log \\left|y_k(0) + z_\\tau\\right|}{\\sqrt{\\log N}}\n\\to\\mathscr{N}(0,1).\n$$\n\n\n\t\\noindent Lemma \\ref{expectation} and Proposition \\ref{variance} show that the second term above, call it $X$, satisfies $\\mathbb{E}(X^2)<C{\\varepsilon}$, for some universal $C$. Thus for any fixed smooth and compactly supported function $F$,\n\t\\begin{align*}\n\t\\mathbb{E} \\left(F\\left(\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|y_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}\\right)\\right)&=\\mathbb{E} \\left(F\\left(\\frac{ \t\t\t\\sum_{k =1}^N \\log \\left|x_k(\\tau) + {\\rm i} \\eta_0\\right|+c_N}{\\sqrt{\\log N}}+X\\right)\\right)\n\t+\\OO\\left(\\|F\\|_{\\rm Lip}(\\mathbb{E}\\left(X^2\\right))^{1\/2}\\right)\\\\\n\t&=\\mathbb{E} \\left(F(\\mathscr{N}(0,1))\\right)+\\oo(1)+\\OO\\left({\\varepsilon}^{1\/2}\\right).\n\t\\end{align*}\nWith Theorem \\ref{4 moment matching theorem}, the above equation implies\n$$\n\\mathbb{E}\\left( F\\left(\\frac{\\log|\\det(W+\\mathrm{i}} %\\newcommand{\\mi}{\\mathrm{i}\\eta_0)|+c_N}{\\sqrt{\\log N}}\\right)\\right)=\\mathbb{E} \\left(F(\\mathscr{N}(0,1))\\right)+\\oo(1)+\\OO\\left({\\varepsilon}^{1\/2}\\right),\n$$\t\t\nand by Proposition \\ref{smoothing}, we obtain\n\\begin{equation}\\label{eqn:2}\n\\mathbb{E} \\left(F\\left(\\frac{\\log|\\det W|+\\frac{N}{2}}{\\sqrt{\\log N}}\\right)\\right)=\\mathbb{E} \\left(F(\\mathscr{N}(0,1))\\right)+\\oo(1)+\n\\OO\\left({\\varepsilon}^{1\/2}\\right).\n\\end{equation}\nSince ${\\varepsilon}$ is arbitrarily small, this concludes the proof.\n\\normalcolor\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\renewcommand{\\thetheorem}{A.\\arabic{theorem}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{IntroSec}\n\nThe \\emph{ping-pong argument} was first used in\n\\cite[\\S III,16]{Klein:Riemann} and \\cite[\\S II,3.8]{Fricke+Klein} to\nanalyze the actions of certain groups of linear fractional\ntransformations on the Riemann sphere.\nLater distillations and generalizations of the arguments\n(e.g., \\cite[Theorem 1]{MR0148731})\nwere used to establish that a given group is a free product.\nIn the current paper we will adapt the \\emph{ping-pong argument}\nto the setting of subgroups of \\({\\operatorname{Homeo}_+(I)}\\), the\ngroup of the orientation preserving homeomorphisms of the unit interval.\nOur main motivation is to develop a better understanding of the finitely generated\nsubgroups of the group \\({\\operatorname{PL}_+(I)}\\) of piecewise linear order-preserving homeomorphisms of\nthe unit interval.\nThe analysis in the current paper resembles the\noriginal \\emph{ping-pong argument} in that it establishes a tree structure on\ncertain orbits of a group action.\nHowever, the arguments in the current paper differ from the usual analysis\nas the generators of the group action have large sets of fixed points.\n\nThe focus of our attention in this article will be subgroups of \\({\\operatorname{Homeo}_+(I)}\\)\nwhich are specified by what we will term \\emph{geometrically fast} generating sets.\nOn one hand, our main result shows that the isomorphism types of the groups specified by\ngeometrically fast generating sets are determined by their \\emph{dynamical diagram} \nwhich encodes their qualitative dynamics;\nsee Theorem \\ref{CombToIso} below.\nThis allows us to show, for instance, that such sets generate groups which\nare always embeddable into Richard Thompson's group \\(F\\).\nOn the other hand, we will see that a broad class of subgroups\nof \\({\\operatorname{PL}_+(I)}\\) can be generated using such sets.\nThis is substantiated in part by Theorem \\ref{GeoProperGen} below.\n\nAt this point it is informative to consider an example.\nFirst recall the classical \\emph{Ping-Pong Lemma}\n(see \\cite[Prop. 1.1]{free_subgrp_linear}):\n\n\\begin{ping_pong_lem}\nLet \\(S\\) be a set and \\(A\\) be a set of permutations of \\(S\\)\nsuch that \\(a^{-1} \\not \\in A\\) for all \\(a \\in A\\).\nSuppose there is an assignment \\(a \\mapsto D_a \\subseteq S\\) of pairwise disjoint sets\nto each \\(a \\in A^\\pm := A \\cup A^{-1}\\) and an\n\\(x \\in S \\setminus \\bigcup_{a \\in A^\\pm} D_a\\) such that\nif \\(a \\ne b^{-1}\\) are in \\(A^\\pm\\), then\n\\[\n\\big(D_b \\cup \\{x\\}\\big)a \\subseteq D_a.\n\\] \nThen \\(A\\) freely generates \\(\\gen{A}\\).\n\\end{ping_pong_lem}\n\n\\noindent\n(We adopt the convention of writing permutations to the right of their arguments;\nother notational conventions and terminology will be reviewed in Sections \\ref{DefSec} and \\ref{FastBumpsSec}.)\nIn the current paper, we relax the hypothesis so that the\ncontainment \\(D_b a \\subseteq D_a\\)\nis required only when \\(D_b\\) is contained in the \\emph{support}\nof \\(a\\); similarly \\(x a \\in D_a\\) is only required when\n\\(x a \\ne x\\).\n\nConsider the three functions \\((b_i \\mid i < 3)\\) in\n\\({\\operatorname{Homeo}_+(I)}\\) whose graphs are shown in Figure \\ref{FastF3}.\n\\begin{figure}\n\\[\n\\xy\n(0,36); (0,0)**@{-}; (36,0)**@{-}; (0,0); (36,36)**@{-};\n(10,9.6)*{\\dvarbump{2}{14}{16}}; (5,15)*{b_0}; \n(26,25.6)*{\\dvarnbump{2}{14}{16}}; (31,21)*{b_2};\n(18,17.6)*{\\dvarbump{2}{14}{16}}; (15,24)*{b_1};\n\\endxy\n\\]\n\\caption{Three homeomorphisms}\\label{FastF3}\n\\end{figure}\n\\newcommand{\\varbump}[3]\n{\n\\xy\n(0,0); (#3,0)**\\crv{(#1,#1)&(#2,#1)};\n\\endxy\n}\n\\newcommand{\\varnbump}[3]\n{\n\\xy\n(0,0); (#3,0)**\\crv{(#1,-#1)&(#2,-#1)};\n\\endxy\n}\nA schematic diagram (think of the line \\(y=x\\) as drawn\nhorizontally) of these functions might be:\n\\[\n\\xy\n(12,2.6)*{\\varbump{12}{36}{48}};  (12,9)*{b_0};\n(36,2.6)*{\\varbump{12}{36}{48}};  (36,9)*{b_1};\n(60,-9)*{\\varnbump{12}{36}{48}}; (60,-10)*{b_2};\n(-12,-3); (-3.5,-3)**@{-};(-3,-3)*{\\circ}; (-8,-5)*{S_1}; (-3,-5)*{};\n(12,-3); (20.5,-3)**@{-};(21,-3)*{\\circ};  (16,-5)*{S_2}; (21,-5)*{};\n(36,-3); (27.5,-3)**@{-};(27,-3)*{\\bullet};  (32,-5)*{D_1}; (27,-5)*{};\n(36,-3); (44.5,-3)**@{-};(45,-3)*{\\circ};  (40, -1)*{D_3}; (45,-5)*{};\n(60,-3); (51.5,-3)**@{-};(51,-3)*{\\bullet};  (56, -5)*{D_2}; (51,-5)*{};\n(84,-3); (75.5,-3)**@{-};(75,-3)*{\\bullet};  (80,-1)*{S_3};  (75,-5)*{};\n\\endxy\n\\]\nIn this diagram, we have assigned intervals \\(S_i\\) and \\(D_i\\)\nto the ends of the support to each \\(b_i\\) so that\nthe entire collection of intervals is pairwise disjoint. \nOur system of homeomorphisms is assumed to have an additional\ndynamical property reminiscent of the hypothesis of the Ping-Pong Lemma:\n\\begin{itemize}\n\n\\item \\(S_i b_i \\cap D_i = \\emptyset\\);\n\n\\item \\(b_i\\) carries \\(\\operatorname{supt}(b_i) \\setminus S_i\\) into \\(D_i\\);\n\n\\item  \\(b_i^{-1}\\) carries \\(\\operatorname{supt}(b_i) \\setminus D_i\\) into \\(S_i\\) for each \\(i\\).\n\n\\end{itemize}\n\nA special case of our main result\nis that these dynamical requirements on the\n\\(b_i\\)'s are sufficient to characterize the isomorphism type of\nthe group \\(\\gen{b_i \\mid i < 3}\\):\nany triple \\((c_i \\mid i < 3)\\)\nwhich produces this same \\emph{dynamical diagram} and satisfies these dynamical\nrequirements will generate a group isomorphic to \\(\\gen{b_i \\mid  i < 3}\\).\nIn fact the map \\(b_i \\mapsto c_i\\) will extend to an isomorphism.\nIn particular,\n\\[\n\\gen{b_i \\mid  i < 3} \\cong\n\\gen{b_i^{k_i} \\mid  i < 3}\n\\]\nfor any choice of \\(k_i\\geq 1\\) for each \\(i < 3\\).\n\nWe will now return to the general discussion and be more precise.\nRecall that if \\(f\\) is in \\({\\operatorname{Homeo}_+(I)}\\), then its \\emph{support} is defined to\nbe \\(\\operatorname{supt}(f):=\\{ t \\in I \\mid tf \\ne t\\}\\).\nA left (right) transition point of \\(f\\) is a \\(t \\in I \\setminus \\operatorname{supt}(f)\\) such that for every \\(\\epsilon > 0\\),\n\\((t,t+\\epsilon) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)\n(respectively \\((t-\\epsilon,t) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)).  \nAn \\emph{orbital} of \\(f\\) is a component its support.\nAn orbital of \\(f\\) is \\emph{positive} if \\(f\\) moves elements of the orbital to\nthe right;\notherwise it is negative.\nIf \\(f\\) has only finitely many orbitals, then the left (right) transition points\nof \\(f\\) are precisely the left (right) end points of its orbitals.\n\nA precursor to the notion of a \\emph{geometrically fast} generating set is that of a\n\\emph{geometrically proper} generating set.\nA set \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is\n\\emph{geometrically proper} if there is no element of \\(I\\) which is\na left transition point of more than one element of \\(X\\)\nor a right transition point of more than one element of \\(X\\).\nObserve that any geometrically proper generating set with only finitely many transition points\nis itself finite.\nFurthermore, geometrically proper sets are equipped with a canonical ordering\ninduced by the usual ordering on the least transition points of its elements.\nWhile the precise definition of \\emph{geometrically fast}\nwill be postponed until Section \\ref{FastBumpsSec},\nthe following statements describe the key features of the definition:\n\\begin{itemize}\n\n\\item geometrically fast generating sets are geometrically proper.\n\n\\item if \\(\\{a_i \\mid  i < n\\}\\) is geometrically proper, then\nthere is a \\(k \\geq 1\\) such that \n\\(\\{a_i^k \\mid  i < n\\}\\) is geometrically fast.\n\n\\item if \\(\\{a_i^k \\mid  i < n\\}\\) is geometrically fast and \\(k \\leq k_i\\) for \\(i < n\\), then\n\\(\\{a_i^{k_i} \\mid  i < n\\}\\) is geometrically fast.\n\n\\end{itemize}\n\\noindent\nOur main result is that the isomorphism types of groups with geometrically fast generating sets\nare determined by their qualitative dynamics.\nSpecifically, we will associate a \\emph{dynamical diagram} to each geometrically fast set\n\\(\\{a_i \\mid i < n\\} \\subseteq {\\operatorname{Homeo}_+(I)}\\) which has finitely many transition points.\nRoughly speaking, this is a record of the relative order of the orbitals\nand transition points of the various \\(a_i\\), as well as the orientation of their orbitals.\nIn the following theorem \\(M_X\\) is a certain finite set of points chosen from the orbitals\nof elements of \\(X\\) and \\(M \\gen{X} = \\{t g \\mid  t \\in M_X \\textrm{ and } g \\in \\gen{X}\\}\\).\nThese points will be chosen such that any nonidentity element of \\(\\gen{X}\\) moves a point in \\(M\\gen{X}\\).\n\n\\begin{thm} \\label{CombToIso}\nIf two geometrically fast sets \\(X , Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) have only finitely many transition points\nand have isomorphic dynamical diagrams, then the induced\nbijection between \\(X\\) and \\(Y\\) extends to an isomorphism of\n\\(\\gen{X}\\) and \\(\\gen{Y}\\) (i.e. \\(\\gen{X}\\) is \\emph{marked isomorphic} to \\(\\gen{Y}\\)).\nMoreover, there is an order preserving bijection\n\\(\\theta :  M \\gen{X} \\to M \\gen{Y}\\) such that\n\\(f \\mapsto f^\\theta\\) induces the isomorphism \n\\(\\gen{X} \\cong \\gen{Y}\\).\n\\end{thm}\n\nWe will also establish that under some circumstances the map \\(\\theta\\) can be extended to a\ncontinuous order preserving surjection \\(\\hat \\theta : I \\to I\\).\n\n\\begin{thm} \\label{SemiConjThm}\nFor each finite dynamical diagram \\(D\\), there is a geometrically fast \\(X_D \\subseteq {\\operatorname{PL}_+(I)}\\)\nsuch that if \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast and has dynamical diagram \\(D\\),\nthen there is a marked isomorphism \\(\\phi:  \\gen{X} \\to \\gen{X_D}\\) and a continuous\norder preserving surjection \\(\\hat \\theta : I \\to I\\) such that\n\\(f \\hat \\theta = \\hat \\theta \\phi (f)\\) for all \\(f \\in \\gen{X}\\).\n\\end{thm}\n\nTheorem \\ref{CombToIso} has two immediate consequences.\nThe first follows from the readily verifiable fact that any dynamical diagram can\nbe realized inside of \\(F\\) (see, e.g., \\cite[Lemma 4.2]{CFP}).\n\n\\begin{cor} \\label{EmbeddInF}\nAny finitely generated\nsubgroup of \\({\\operatorname{Homeo}_+(I)}\\) which admits a geometrically fast generating set\nembeds into Thompson's group \\(F\\).\n\\end{cor}\n\\noindent\nSince by \\cite{brin+squier} \\(F\\) does not contain nontrivial free produces of groups,\nsubgroups of \\({\\operatorname{Homeo}_+(I)}\\) which admit geometrically fast generating sets are not free products.\nIt should also be remarked that while our motivation comes from studying\nthe groups \\(F\\) and \\({\\operatorname{PL}_+(I)}\\), the conclusion of Corollary \\ref{EmbeddInF} remains valid if\n\\(F\\) is replaced by, e.g. \\(\\operatorname{Diff}^\\infty_+ (I)\\).\n\n\\begin{cor} \\label{AlgFast}\nIf \\(\\{f_i \\mid i < n\\}\\) is geometrically fast, then\n\\(\\gen{f_i \\mid i < n}\\) is marked isomorphic to\n\\(\\gen{f_i^{k_i} \\mid i < n}\\) for any choice of \\(k_i \\geq 1\\).\n\\end{cor}\n\nIt is natural to ask how restrictive having a geometrically fast or geometrically\nproper generating set is.\nThe next theorem shows that many finitely generated subgroups of \\({\\operatorname{PL}_+(I)}\\)\nin fact do have at least a geometrically proper generating set.\n\n\\begin{thm} \\label{GeoProperGen}\nEvery \\(n\\)-generated one orbital subgroup of \\({\\operatorname{PL}_+(I)}\\)\neither contains an isomorphic copy of \\(F\\) or else admits an\n\\(n\\)-element geometrically proper generating set.\n\\end{thm}\n\n\\noindent\nNotice that every subgroup of \\({\\operatorname{Homeo}_+(I)}\\) is contained in a direct product of\none-orbital subgroups of \\({\\operatorname{Homeo}_+(I)}\\). \nThus if one's interest lies in studying the structure of subgroups of\n\\({\\operatorname{PL}_+(I)}\\) which do not contain copies of \\(F\\), then\nit is typically possible to restrict one's attention to groups\nadmitting geometrically proper generating sets.\nThe hypothesis of not containing an isomorphic copy of \\(F\\) in Theorem \\ref{GeoProperGen}\ncan not be eliminated.\nThis is a consequence of the following theorem and the fact that there are finite\nindex subgroups of \\(F\\) which are not isomorphic to \\(F\\) (see \\cite{bleak+wassink}).\n\n\\begin{thm} \\label{NoGeoProperGen}\nIf a finite index subgroup of \\(F\\) is isomorphic to \\(\\gen{X}\\) for some  geometrically proper\n\\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\), then it is isomorphic to \\(F\\).\n\\end{thm}\n\\noindent\nWe conjecture, however, that every finitely generated subgroup of \\(F\\) is bi-embeddable\nwith a subgroup admitting a geometrically fast generating set.\n\nWhile the results of this paper do of course readily adapt to \\({\\operatorname{Homeo}_+(\\R)} \\cong {\\operatorname{Homeo}_+(I)}\\),\nit is important to keep in mind that \\(\\pm \\infty\\) must be allowed as possible\ntransition points when applying the definition of geometric properness and hence\ngeometric fastness.\nFor example, it is easy to establish that \\(\\gen{t + \\sin (t), t + \\cos ( t)}\\) contains\na free group using the \\emph{ping-pong lemma} stated above\n(the squares of the generators generate a free group).\nMoreover, once we define \\emph{geometrically fast} in Section \\ref{FastBumpsSec},\nit will be apparent that the squares of the generators satisfies all of the requirements\nof being geometrically fast except that it is not geometrically proper\n(since, e.g., \\(\\infty\\) is a right transition point of both functions).\nAs noted above, this group does not embed into \\(F\\) and thus\ndoes not admit a geometrically fast (or even a geometrically proper) generating set.\nSee Example \\ref{InfiniteBumpEx} below for a more detailed discussion of a related example.\n\nThe paper is organized as follows.\nWe first review some standard definitions, terminology and notation in Section \\ref{DefSec}.\nIn Section \\ref{FastBumpsSec}, we will give a formal definition of \\emph{geometrically fast}\nand a precise definition of what is meant by a \\emph{dynamical diagram}.\nSection \\ref{GeoFastCriteriaSec} gives a reformulation of \\emph{geometrically fast}\nfor finite subsets of \\({\\operatorname{Homeo}_+(I)}\\) which facilitates algorithmic checking.\nThe proof of Theorem \\ref{CombToIso} is then divided between Sections\n\\ref{PingPongSec} and \\ref{CombToIsoSec}.\nThe bulk of the work is in\nSection \\ref{PingPongSec}, which uses an analog of the \\emph{ping-pong argument}\nto study the dynamics of geometrically fast sets of one orbital homeomorphisms.\nSection \\ref{CombToIsoSec} shows how this analysis implies \nTheorem \\ref{CombToIso} and how to derive its corollaries.\nIn Section \\ref{SemiConjSec}, we will prove Theorem \\ref{SemiConjThm}.\nThe group \\(F_n\\), which is the \\(n\\)-ary analog of Thompson's group \\(F\\), is\nshown to have a geometrically fast generating set in Section \\ref{FnSec}.\nSection \\ref{ExcisionSec} examines when bumps in\ngeometrically fast generating sets are extraneous and\ncan be excised without affecting the marked isomorphism type.\nProofs of Theorems \\ref{GeoProperGen} and \\ref{NoGeoProperGen} are given in Section \\ref{GeoProperSec}.\nFinally, the concept of \\emph{geometrically fast}\nis abstracted in Section \\ref{AbstractPingPongSec},\nwhere a generalization of Theorem \\ref{CombToIso} is stated and proved, as well as corresponding embedding\ntheorems for Thompson's groups \\(F\\), \\(T\\), and \\(V\\).\nThis generalization in particular covers infinite geometrically fast subsets of \\({\\operatorname{Homeo}_+(I)}\\).\nEven in the context of geometrically fast sets \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) with only finitely many\ntransition points, this abstraction gives a new way of understanding \\(\\gen{X}\\)\nin terms of symbolic manipulation.\n\n\n\\section[Preliminaries]{Preliminary definitions, notation and conventions}\n\n\\label{DefSec}\n\nIn this section we collect a number of definitions and conventions\nwhich will be used extensively in later sections.\nThroughout this paper, the letters \\(i,j,k,m,n\\) will be assumed to range\nover the nonnegative integers unless otherwise stated.\nFor instance, we will write \\((a_i \\mid i < k)\\) to denote a sequence\nwith first entry \\(a_0\\) and last entry \\(a_{k-1}\\).\nIn particular, all counting and indexing starts at 0 unless stated otherwise.\nIf \\(f\\) is a function and \\(X\\) is a subset of the domain of \\(f\\), we will write \\(f \\restriction X\\) to\ndenote the restriction of \\(f\\) to \\(X\\).\n\nAs we have already mentioned,\n\\({\\operatorname{Homeo}_+(I)}\\) will be used to denote the\nset of all orientation preserving homeomorphisms of \\(I\\);\n\\({\\operatorname{PL}_+(I)}\\) will be used to denote the set of all piecewise linear elements\nof \\({\\operatorname{Homeo}_+(I)}\\). \nThese groups will act on the right.\nIn particular, \\(tg\\) will denote the result of applying a homeomorphism\n\\(g\\) to a point \\(t\\).\nIf \\(f\\) and \\(g\\) are elements of a group, we will\nwrite \\(f^g\\) to denote \\(g^{-1}fg\\).\n\nRecall that from the introduction that\nif \\(f\\) is in \\({\\operatorname{Homeo}_+(I)}\\), then its \\emph{support} is defined to\nbe \\(\\operatorname{supt}(f):=\\{ t \\in I \\mid tf \\ne t\\}\\).\nThe support of a subset of \\({\\operatorname{Homeo}_+(I)}\\) is the union of the supports of its\nelements. \nA left (right) transition point of \\(f\\) is a \\(t \\in I \\setminus \\operatorname{supt}(f)\\) such that for every \\(\\epsilon > 0\\),\n\\((t,t+\\epsilon) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)\n(respectively \\((t-\\epsilon,t) \\cap \\operatorname{supt}(f) \\ne \\emptyset\\)).  \nAn \\emph{orbital} of \\(f\\) is a component its support.\nAn orbital of \\(f\\) is \\emph{positive} if \\(f\\) moves elements of the orbital to\nthe right;\notherwise it is negative.\nIf \\(f\\) has only finitely many orbitals, then the left (right) transition points\nof \\(f\\) are precisely the left (right) end points of its orbitals.\nAn \\emph{orbital} of a subset of \\({\\operatorname{Homeo}_+(I)}\\)\nis a component of its support.\n\nAn element of \\({\\operatorname{Homeo}_+(I)}\\) with one orbital will be referred to as a \\emph{bump function}\n(or simply a \\emph{bump}).\nIf a bump \\(a\\) satisfies that \\(t a > t\\) on its support,\nthen we say that \\(a\\) is \\emph{positive};\notherwise we say that \\(a\\) is \\emph{negative}.\nIf \\(f \\in {\\operatorname{Homeo}_+(I)}\\), then \\(b \\in {\\operatorname{Homeo}_+(I)}\\) is a \\emph{signed bump of \\(f\\)} if\n\\(b\\) is a bump which agrees with \\(f\\) on its support.\nIf \\(X\\) is a subset of \\({\\operatorname{Homeo}_+(I)}\\), then a bump \\(a\\)\nis \\emph{used in} \\(X\\)\nif \\(a\\) is positive and\nthere is an \\(f\\) in \\(X\\) such that \\(f\\)\ncoincides with either \\(a\\) or \\(a^{-1}\\) on the support of \\(a\\).\nA bump \\(a\\) is used in \\(f\\) if it is used in \\(\\{f\\}\\).\nWe adhere to the convention that only positive\nbumps are used by functions to avoid ambiguities in some statements.\nObserve that if \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is such that\nthe set \\(A\\) of bumps used in \\(X\\) is finite, then\n\\(\\gen{X}\\) is a subgroup of \\(\\gen{A}\\).\n\nIf \\((g_i \\mid i < n)\\) and \\((h_i \\mid i < n)\\) are two generating sequences for groups,\nthen we will say that \\(\\gen{g_i \\mid i < n}\\) is \\emph{marked isomorphic} to\n\\(\\gen{h_i \\mid i < n}\\) if the map \\(g_i \\mapsto h_i\\) extends to an isomorphism\nof the respective groups.\nIf \\(X\\) is a finite geometrically proper subset of \\({\\operatorname{Homeo}_+(I)}\\), then\nwe will often identify \\(X\\) with its enumeration in which the minimum transition\npoints of its elements occur in increasing order.\nWhen we write \\(\\gen{X}\\) is marked isomorphic to \\(\\gen{Y}\\), we are\nmaking implicit reference to these canonical enumerations of \\(X\\) and \\(Y\\).\n\nAt a number of points in the paper it will be important to distinguish between\nformal syntax (for instance words) and objects (such as group elements) to which they refer.\nIf \\(A\\) is a set, then a \\emph{string}\nof elements of \\(A\\) is a finite sequence of elements of \\(A\\).\nThe length of a string \\(\\str{w}\\) will be denoted \\(|\\str{w}|\\).\nWe will use \\(\\varepsilon\\) to denote the string of length 0.\nIf \\(\\str{u}\\) and \\(\\str{v}\\) are two strings,\nwe will use \\(\\str{uv}\\) to denote\ntheir concatenation;\nwe will say that \\(\\str{u}\\) is a \\emph{prefix} of \\(\\str{uv}\\)\nand \\(\\str{v}\\) is a \\emph{suffix} of \\(\\str{uv}\\).\nIf \\(A\\) is a subset of a group, then a \\emph{word} (in \\(A\\))\nis a string of elements of\n\\(A^\\pm := A \\cup A^{-1}\\).\nA \\emph{subword} of a word \\(\\str {w}\\) must preserve the order from \\(\\str {w}\\),\nbut does not have to consist of consecutive symbols from \\(\\str {w}\\).\nWe write \\(\\str {w}^{-1}\\) for the formal inverse of \\(\\str {w}\\): the product of\nthe inverses of the symbols in \\(\\str {w}\\) in reverse order.\n\nOften strings have an associated evaluation (e.g. a word represents an element of a group).\nWhile the context will often dictate whether we are working with a string or\nits evaluation, we will generally use the typewriter font (e.g. \\(\\str{w}\\)) for strings\nand symbols in the associated alphabets and standard math font (e.g. \\(w\\)) for the\nassociated evaluations.\n\nIn Section \\ref{GeoProperSec}, we will use the notion of the \\emph{left (right) germ}\nof a function \\(f \\in {\\operatorname{Homeo}_+(I)}\\) at an \\(s \\in I\\) which is fixed by \\(f\\)\n(left germs are undefined at 0 and right germs are undefined at 1).\nIf \\(0 \\leq s < 1\\), then define the \\emph{right germ of \\(f\\) at \\(s\\)} to be the set of all \\(g \\in {\\operatorname{Homeo}_+(I)}\\) such\nthat for some \\(\\epsilon > 0\\), \\(f \\restriction (s,s+\\epsilon) = g \\restriction (s,s+\\epsilon)\\);\nthis will be denoted by \\(\\gamma_s^+(f)\\).\nSimilarly if \\(0 < s \\leq 1\\), then one defines the \\emph{left germ of \\(f\\) at \\(s\\)};\nthis will be denoted by \\(\\gamma_s^-(f)\\).\nThe collections \n\\[\n\\{\\gamma_s^+(f) \\mid f \\in {\\operatorname{Homeo}_+(I)} \\textrm{ and } sf = s\\}\n\\]\n\\[\n\\{\\gamma_s^-(f) \\mid f \\in {\\operatorname{Homeo}_+(I)} \\textrm{ and } sf = s\\}\n\\]\nform groups and\nthe functions \\(\\gamma_s^+\\) and \\(\\gamma_s^-\\) are homomorphisms defined\non the subgroup of \\({\\operatorname{Homeo}_+(I)}\\) consisting of those functions which fix \\(s\\).\n\n\n \n\\section[Fast collections of bumps]{Fast collection of bumps and their dynamical diagrams}\n\n\\label{FastBumpsSec}\n\nWe are now ready to turn to the definition of \\emph{geometrically fast} in the context of finite subsets of\n\\({\\operatorname{Homeo}_+(I)}\\).\nFirst we will need to develop some terminology.\nA \\emph{marking} of a geometrically proper collection of bumps \\(A\\)\nis an assignment of a \\emph{marker} \\(t \\in \\operatorname{supt}(a)\\) to each \\(a\\) in \\(A\\).\nIf \\(a\\) is a positive bump with orbital \\((x,y)\\) and marker \\(t\\), then we define\nits \\emph{source} to be the interval \\(\\operatorname{src}(a) := (x,t)\\)\nand its \\emph{destination} to be the interval \\(\\operatorname{dest}(a) := [t a,y)\\).\nWe also set \\(\\operatorname{src}(a^{-1}) := \\operatorname{dest}(a)\\) and \\(\\operatorname{dest} (a^{-1}) := \\operatorname{src}(a)\\).\nThe source and destination of a bump are collectively called its \\emph{feet}. \nNote that there is a deliberate asymmetry in this definition:\nthe source of a positive bump is an open interval whereas the destination is half open.\nThis choice is necessary so that for any \\(t \\in \\operatorname{supt} (a)\\), there is a unique \\(k\\) such that\n\\(t a^k\\) is not in the feet of \\(a\\), something which is a key feature of the definition.\n\nA collection \\(A\\) of bumps is \\emph{geometrically fast}\nif it there is a marking of \\(A\\) for which its feet form a pairwise disjoint family\n(in particular we require that \\(A\\) is geometrically proper).\nThis is illustrated in Figure \\ref{GeoFastFig}, where the feet of\n\\(a_0\\) are \\((p,q)\\) and \\([r,s)\\) and the feet of \\(a_1\\) are\n\\((q,r)\\) and \\([s,t)\\).\n\\newcommand{\\hashlabel}[1]\n{\n\\xy\n(0,0); (0,-2)**@{-}; (0,-4)*{\\scriptstyle #1};\n\\endxy\n}\n\\begin{figure}\n\\[\n\\xy\n(0,36); (0,0)**@{-}; (36,0)**@{-}; (0,0); (36,36)**@{-};\n(16,15.6)*{\\dvarbump{4}{16}{19}}; (9,19)*{a_0};\n(22,21.6)*{\\dvarbump{2}{14}{16}}; (17.5,27.5)*{a_1};\n(14,14);(14,20)**@{.};(20,20)**@{.};\n(20,25.5)**@{.};(25.5,25.5)**@{.};\n(6.57,4.1)*{\\hashlabel{p}};\n(14.02,11.55)*{\\hashlabel{q}};\n(20.0,17.55)*{\\hashlabel{r}};\n(25.55,23.15)*{\\hashlabel{s}};\n(30,27.55)*{\\hashlabel{t}};\n\\endxy\n\\]\n\\caption{A geometrically fast set of bumps}\\label{GeoFastFig}\n\\end{figure}\nBeing geometrically fast is precisely the set of dynamical requirements made on the set\n\\(\\{a_i \\mid i < 3\\}\\) of homeomorphisms mentioned in the introduction.\nWe do not require here that \\(A\\) is finite and we will explicitly state finiteness as\na hypothesis when it is needed.\nNotice however that, since pairwise disjoint families of intervals\nin \\(I\\) are at most countable, any geometrically fast set of bumps is\nat most countable.\nThe following are readily verified and can be used axiomatically to\nderive most of the lemmas in Section \\ref{PingPongSec}\n(specifically Lemmas \\ref{LocRedBasics}--\\ref{FellowTraveler3}):\n\\begin{itemize}\n\n\\item for all \\(a \\in A^\\pm\\), \\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(a)\\) and if\n\\(x \\in \\operatorname{supt}(a)\\) then there exists a \\(k\\) such that \\(xa^k \\in \\operatorname{dest}(a)\\);\n\n\\item if \\(a \\ne b \\in A^\\pm\\), then \\(\\operatorname{dest}(a) \\cap \\operatorname{dest}(b) = \\emptyset\\);\n\n\\item if \\(a \\in A^\\pm\\) and \\(x \\in \\operatorname{supt}(a)\\), then\n\\(x a \\in \\operatorname{dest}(a)\\) if and only if \\(x \\not \\in \\operatorname{src}(a) := \\operatorname{dest}(a^{-1})\\).\n\n\\item if \\(a,b \\in A^\\pm\\), then \\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b)\\) or\n\\(\\operatorname{dest}(a) \\cap \\operatorname{supt}(b) = \\emptyset\\).\n\n\\end{itemize}\nThis axiomatic viewpoint will be discussed further in Section \\ref{AbstractPingPongSec}.\n\nA set \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast\nif it is geometrically proper and the set of bumps used in \\(X\\) is geometrically fast.\nNote that while geometric properness is a consequence of the disjointness of the feet\nif \\(X\\) uses only finitely many bumps, it is an additional requirement in general.\nThis is illustrated in the next example.\n\n\\begin{example} \\label{InfiniteBumpEx}\nConsider the following homeomorphism of \\({\\mathbf R}\\):\n\\[\nt \\gamma = \n\\begin{cases}\n3t & \\textrm{ if } 0 \\leq t \\leq 1\/2 \\\\\n(t+4)\/3 & \\textrm{ if } 1\/2 \\leq t \\leq 2 \\\\\nt & \\textrm{ otherwise}\n\\end{cases}\n\\]\nDefine \\(\\alpha , \\beta \\in {\\operatorname{Homeo}_+(\\R)}\\) by \n\\((t+2p) \\alpha = t \\gamma + 2p\\) and \\((t+2p+1) \\beta = t \\gamma + 2p+1\\)\nwhere \\(p \\in {\\mathbf Z}\\) and \\(t \\in [0,2]\\).\nThus the bumps used in \\(\\alpha\\) are obtained by translating \\(\\gamma\\) by even integers;\nthe bumps in \\(\\beta\\) are the translates of \\(\\gamma\\) by odd integers.\nIf we assign the marker \\(1\/2\\) to \\(\\gamma\\) and mark the translation of \\(\\gamma\\) by \\(p\\)\nwith \\(p + 1\/2\\), then it can be seen that the feet of \\(\\alpha\\) and \\(\\beta\\) are the intervals\n\\(\\{(p,p+1\/2) \\mid p \\in {\\mathbf Z}\\} \\cup \\{[p+1\/2,p+1) \\mid p \\in {\\mathbf Z}\\}\\),\nwhich is a pairwise disjoint family.\nThus the bumps used in \\(\\{\\alpha,\\beta\\}\\) are geometrically fast.\nSince \\(\\infty\\) is a right transition point of both \\(\\alpha\\) and \\(\\beta\\), \n\\(\\{\\alpha,\\beta\\}\\) is not geometrically proper and hence not fast.\nIn fact, it follows readily from the formulation of the classical \\emph{ping-pong lemma}\nin the introduction that \\(\\gen{\\alpha,\\beta}\\) is free.\n\\end{example}\n\nObserve that if \\(X\\) is geometrically proper, each of its elements uses only finitely many bumps,\nand the set of transition points of \\(X\\) is discrete, \nthen there is a map \\(f \\mapsto k(f)\\) of \\(X\\) into the positive integers such that\n\\(\\{f^{k(f)} \\mid  f \\in X\\}\\) is geometrically fast.\nTo see this, start with a marking such that the closures of the sources of the bumps used in\n\\(X\\) are disjoint;\npick \\(f \\mapsto k(f)\\) sufficiently large so that all of the feet become disjoint.\nAlso notice that if \\(\\{f^{k(f)} \\mid f \\in X\\}\\) is geometrically fast and if \\(k(f) \\leq l(f)\\) for \\(f \\in X\\),\nthen \\(\\{f^{l(f)} \\mid f \\in X \\}\\) is geometrically fast as well.\n\nIf \\(X\\) is a geometrically fast generating set with only finitely many transition points,\nthen the \\emph{dynamical diagram} \\(D_X\\) of \n\\(X\\) is the edge labeled vertex ordered directed graph defined as follows:\n\\begin{itemize}\n\n\\item the vertices of \\(D_X\\) are the feet of \\(X\\) with the order\ninduced from the order of the unit interval;\n\n\\item the edges of \\(D_X\\) are the signed bumps of \\(X\\) directed\nso that the source (destination) of the edge is the source (destination) of the\nbump;\n\n\\item the edges are labeled by the elements of \\(X\\) that they come from.\n\n\\end{itemize}\n\\noindent\nWe will adopt the convention that dynamical diagrams are necessarily finite.\nThe dynamical diagram of a generating set for the Brin-Navas group \\(B\\)\n\\cite{MR2160570} \\cite{MR2135961}\nis illustrated in the left half of Figure \\ref{BNFigure};\nthe generators are \\(f = a_0^{-1} a_2\\) and \\(g = a_1^{-1}\\),\nwhere the \\((a_i \\mid  i < 3)\\) is the geometrically fast generating sequence illustrated in\nFigure \\ref{TrackingPoint}.\nWe have found that when drawing dynamical diagram \\(D_X\\) of a given \\(X\\), it is more\n{\\ae}sthetic whilst being unambiguous to collapse\npairs of vertices \\(u\\) and \\(v\\) of \\(D_X\\) such that:\n\\begin{itemize}\n\n\\item  \\(v\\) is the immediate successor of \\(u\\) in the order on \\(D_X\\),\n\n\\item \\(u\\)'s neighbor is below \\(u\\), and \\(v\\)'s neighbor is above \\(v\\).\n\n\\end{itemize}\nAdditionally, arcs can be drawn as over or under arcs to indicate their direction,\neliminating the need for arrows.\nThis is illustrated in the right half of Figure \\ref{BNFigure}.\nThe result qualitatively resembles the graphs of the homeomorphisms rotated so that\nthe line \\(y =x\\) is horizontal.\n\nAn isomorphism between dynamical diagrams is a directed graph isomorphism\nwhich preserves the order of the vertices and\ninduces a bijection between the edge labels\n(i.e. two directed edges have equal labels before applying the isomorphism if and only if\nthey have equal labels after applying the isomorphism).\nNotice that such an isomorphism is unique if it exists --- there is at most one order preserving bijection\nbetween two finitely linear orders.\n\n\\begin{figure}\n\\[\n\\xy\n(0,5)*{\n\\xymatrix{\n{\\bullet} &\n{\\bullet} &\n{\\bullet} \\ar@\/^1.5pc\/[ll]^f  &\n{\\bullet} \\ar@\/^1.5pc\/[rr]^f &\n{\\bullet} \\ar@\/^1.5pc\/[lll]^g&\n{\\bullet}\n}};\n(-2,10); (63,10)**@{.};\n(40,27)*{}; \n(50,18)*{};\n\\endxy\n\\quad\n\\quad\n\\xy\n(0,5)*{\n\\xymatrix{\n{\\bullet} &\n{\\bullet} &\n{\\bullet} \\ar@{-}@\/^1.5pc\/[ll]^f \\ar@{-}@\/^1.5pc\/[rr]^f&\n{\\bullet} \\ar@{-}@\/^1.5pc\/[ll]^g&\n{\\bullet} \n}};\n(-2,10); (51,10)**@{.};\n(40,27)*{}; \n(50,18)*{};\n\\endxy\n\\]\n\\caption{The dynamical diagram for the Brin-Navas generators, with an illustration\nof the contraction convention \\label{BNFigure}}\n\\end{figure}\n\\begin{figure}\n\\[\n\\xy\n(0,36); (0,0)**@{-}; (36,0)**@{-}; (0,0); (36,36)**@{-};\n(10,9.6)*{\\dvarbump{2}{14}{16}}; (5,15)*{a_0};\n(18,17.6)*{\\dvarbump{2}{14}{16}}; (13.5,23.5)*{a_1};\n(26,25.6)*{\\dvarbump{2}{14}{16}}; (22,32)*{a_2};\n(10,10);(10,15.25)**@{.};(15.25,15.25)**@{.};(15.25,21)**@{.};(21,21)**@{.};(21,26.2)**@{.};(26.2,26.2)**@{.};\n\\endxy\n\\]\n\\caption{A point is tracked through a fast transition chain}\\label{TrackingPoint}\n\\end{figure}\nObserve that the (uncontracted) dynamical diagram of any geometrically\nfast \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) which has finitely many transition points has the property that\nall of its vertices have total degree 1.\nMoreover, any finite edge labeled vertex ordered directed graph in which each vertex has total\ndegree 1 is isomorphic to the dynamical diagram of some geometrically fast \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\)\nwhich has finitely many transition points (see the proof of Theorem \\ref{SemiConjThm} in Section\n\\ref{SemiConjSec}).\nThus we will write \\emph{dynamical diagram} to mean a finite edge labeled vertex ordered\ndirected graph in which each vertex has total degree 1.\nThe edges in a dynamical diagram will be referred to as \\emph{bumps} and the vertices\nin a dynamical diagram will be referred to as \\emph{feet}.\nTerms such as \\emph{source}, \\emph{destination}, \\emph{left\/right foot} will be given the obvious meaning\nin this context.\n\nNow let \\(A\\) be a geometrically fast set of positive bump functions.\nAn element of \\(A\\) is \\emph{isolated} (in \\(A\\)) if\nits support contains no transition points of \\(A\\).\nIn the dynamical diagram of \\(A\\), this corresponds to a bump whose source and destination are\nconsecutive feet.\nThe next proposition shows that we may always eliminate isolated bumps in \\(A\\) by\nadding new bumps to \\(A\\).\nThis will be used in Section \\ref{SemiConjSec}\n\n\\begin{figure}\n\\[\n\\xy\n\\xymatrix{\n{\\bullet} \\ar@\/^2.5pc\/[rrrrr]^a &\n{\\bullet} \\ar@\/^1.0pc\/[rr]^{b_0} &\n{\\bullet} \\ar@\/^1.0pc\/[rr]^{b_1} &\n{\\bullet} &\n{\\bullet} &\n{\\bullet} &\n}\n\\endxy\n\\]\n\\caption{The bump \\(a\\) is made nonisolated by the addition of bumps \\(b_0\\) and \\(b_1\\).\n\\label{IsolatedFixFig}}\n\\end{figure}\n\n\\begin{prop} \\label{IsolatedFixProp}\nIf \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\) is a geometrically fast set of positive bump functions,\nthen there is a geometrically fast \\(B \\subseteq {\\operatorname{Homeo}_+(I)}\\) such that \\(A \\subseteq B\\)\nand \\(B\\) has no isolated bumps.\nMoreover, if \\(A\\) is finite, then \\(B\\) can be taken to be finite as well.\n\\end{prop}\n\n\\begin{proof}\nIf \\(a \\in A\\) is isolated, let \\(b_0\\) and \\(b_1\\) be a geometrically fast pair of bumps with supports contained in\n\\(\\operatorname{supt}(a) \\setminus (\\operatorname{src}(a) \\cup \\operatorname{dest}(a))\\) such that neither \\(b_0\\) nor \\(b_1\\) is isolated in \\(\\{b_0,b_1\\}\\);\nsee Figure \\ref{IsolatedFixFig}.\nSince the feet of \\(A\\) are disjoint, so are the feet of \\(A \\cup \\{b_0,b_1\\}\\) and \\(a\\) is no longer isolated\nin \\(A \\cup \\{b_0,b_1\\}\\).\nLet \\(B\\) be the result of adding such a pair of bumps for each isolated bump in \\(A\\).\n\\end{proof}\n\n\n\\section[Criteria for geometric fastness]{An algorithmically check-able criteria for geometric fastness}\n\\label{GeoFastCriteriaSec}\n\nIn this section we will consider geometrically proper sets which have finitely many transition points and\ndevelop a characterization of when they are geometrically fast.\nThis characterization moreover allows one to determine algorithmically when such sets are geometrically fast.\nIt will also provide a canonical marking of geometrically fast sets with finitely many transition points.\nWe need the following refinement of the notion of a \\emph{transition chain}\nintroduced in \\cite{MR2466019}.\nLet \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\)\nbe a finite geometrically proper set of positive bump functions. \n\nA sequence \\((a_i \\mid i\\le k)\\) of nonisolated elements of \\(A\\)  is a\n\\emph{stretched transition chain of \\(A\\)} if:\n\\begin{enumerate}\n\n\\item \\label{TransitionChain}\nfor all \\(i<k\\), \\(x_i<x_{i+1}<y_i<y_{i+1}\\), where \n\\((x_i,y_i)\\) is the support of \\(a_i\\);\n\n\\item \\label{StretchedChain}\nno transition point of \\(A\\) is in any interval \\((x_{i+1},y_i)\\).\n\n\\end{enumerate}\nNotice that whether \\(C\\) is a stretched transition chain depends not only on the\nelements of \\(A\\) listed in \\(C\\) but on the entire collection \\(A\\).\nSince the left transition points in a stretched transition chain are strictly\nincreasing, we will often identify such sequences with their range.\nIn particular, a stretched transition chain \\(C\\) is \\emph{maximal} if is maximal with\nrespect to containment when\nregarded as a subset of \\(A\\).\nWe will use \\(\\prod C\\) to denote the composition \\(a_0 \\cdots a_k\\).\nAn element \\(a\\) of \\(A\\) is \\emph{initial} (in \\(A\\)) if either \\(a\\) is isolated\nor else the least transition point of \\(A\\) in the support of \\(a\\) is not\nthe right transition point of some element of \\(A\\).\nThese are precisely the elements of \\(A\\) which are the initial element of any\nstretched transition chain which contains them.\n\nFigure \\ref{ChainDecompFig} shows a dynamical diagram, along with a list of the\nmaximal stretched transition chains.\n\\begin{figure}\n\\[\n\\xy\n\\xymatrix{\n{\\bullet} \\ar@\/^1.5pc\/[rrr]^{a_0} &\n{\\bullet} \\ar@\/^2.5pc\/[rrrrr]^{a_1} &\n{\\bullet} \\ar@\/^3.0pc\/[rrrrrrr]^{a_2} &\n{\\bullet} \\ar@\/^2.0pc\/[rrrr]^{a_3} &\n{\\bullet} \\ar@\/^0.5pc\/[r]^{a_4} &\n{\\bullet} \\ar@\/^1.5pc\/[rrr]^{a_5} &\n{\\bullet} &\n{\\bullet} &\n{\\bullet} &\n{\\bullet} &\n}\n\\endxy\n\\]\n\\caption{The stretched transition chains in this dynamical diagram are\n\\(\\{a_0,a_2\\}\\), \\(\\{a_1,a_5\\}\\), and \\(\\{a_3\\}\\); \\(a_4\\) is isolated\n\\label{ChainDecompFig}}\n\\end{figure}\n\n\\begin{prop}\\label{StretchPartProp}\nIf \\(A\\) is a finite geometrically proper set of positive bump functions in \\({\\operatorname{Homeo}_+(I)}\\),\nthen the maximal stretched transition chains in \\(A\\) partition the nonisolated elements of \\(A\\).\n\\end{prop}\n\n\\begin{remark}\nWe will see in Section \\ref{FnSec} that geometrically fast stretched transition chains of length \\(n\\) generate\n\\(F_n\\), the \\(n\\)-ary analog of Thompson's group \\(F\\).\nThus if \\(A\\) is geometrically fast, then \\(\\gen{A}\\) is a sort of amalgam of copies of the\n\\(F_n\\) and copies of \\({\\mathbf Z}\\).\n\\end{remark}\n\n\\begin{proof}\nFirst observe that any sequence consisting of a single nonisolated element of \\(A\\)\nis a stretched transition chain and thus is a subsequence of\nsome maximal stretched transition chain in \\(A\\).\nNext  suppose that \\(a\\) and \\(b\\) are consecutive members of a stretched transition chain.\nIt follows that \\(b\\)'s left transition point is in the support of \\(a\\) and is\nthe greatest transition point of \\(A\\) in the support of \\(a\\).\nIn particular, \\(b\\) must immediately follow \\(a\\) in any maximal stretched transition chain.\nSimilarly, \\(a\\) must immediately precede \\(b\\) in any maximal stretched transition chain.\nThis shows that every nonisolated element of \\(A\\)\noccurs in a unique maximal stretched transition chain of \\(A\\).\n\\end{proof}\n\nIf \\(C\\) is a stretched transition chain, we define \\(C_{\\min}\\) as the\nleast transition point of \\(A\\) in the support \\(C\\) and\n\\(C_{\\max}\\) as the greatest transition point of \\(A\\) in the support \\(C\\).  \nSince \\(C\\) has no isolated bumps, both \\(C_{\\min}\\) and \\(C_{\\max}\\) are well defined.\n\n\\begin{prop}\\label{FastCriterion}\nIf \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\) is a finite geometrically\nproper set of positive bump functions, then the following are equivalent:\n\\begin{enumerate}\n\n\\item \\label{FastHyp}\n\\(A\\) is geometrically fast;\n\n\\item \\label{FastChainsHyp}\nevery stretched transition chain \\(C\\) of \\(A\\) satisfies\n\\(C_{\\max} \\leq C_{\\min} \\prod C\\).\n\n\\item \\label{MaxFastChainsHyp}\nevery maximal stretched transition chain \\(C\\) of \\(A\\) satisfies\n\\(C_{\\max} \\leq C_{\\min} \\prod C\\).\n\\end{enumerate}\n\\end{prop}\n\n\\begin{remark}\nThis criterion for being geometrically fast was our original motivation for\nthe choice of the terminology: \nthe dynamics of the homeomorphisms are such that\ntransition points can be moved to the right\nthrough transition chains as efficiently as possible. \nThis is illustrated in Figure \\ref{TrackingPoint}.\n\\end{remark}\n\n\\begin{remark}\nThe choice not to allow isolated bumps to be singleton stretched transition chains is somewhat arbitrary,\nalthough it would be necessary to make awkward adjustments to the definitions above if\nwe took the alternate approach.\nIt seems appropriate to omit them since they play no role in determining whether a group\nis fast other than contributing to the collective set of transition points of the set of bumps under consideration.\n\\end{remark}\n\n\n\n\\begin{proof}\nTo see that (\\ref{FastHyp}) implies (\\ref{FastChainsHyp}), let \\(A\\) be given\nequipped with a fixed marking witnessing that it is geometrically fast.\nLet \\(C = (a_k \\mid k \\leq m)\\) and\nlet \\(s_k\\) denote the marker of \\(a_k\\).\nNotice that if \\(t\\) is a transition point of \\(A\\) which is in the support of \\(a_k\\),\nthen \\(s_k \\leq t \\leq s_k a_k\\); otherwise\na foot associated to \\(t\\) would intersect a foot of \\(a_k\\).\nIt follows that \\(s_k a_k\\) is in the support of \\(a_{k+1}\\)\nsince the left transition point of \\(a_{k+1}\\) is in the support of \\(a_k\\) by\n(\\ref{TransitionChain}) in the definition of stretched transition chain.\nTherefore the left foot of \\(a_{k+1}\\) is to the left of the right foot of \\(a_k\\)\nand so \\(s_{k+1} \\leq s_k a_k\\).\nWe now have inductively that \\(s_m \\leq s_0 a_0 \\cdots a_{m-1}\\) and\nhence\n\\[\nC_{\\max} \\leq s_m a_m \\leq s_0 a_0 \\cdots  a_m \\leq C_{\\min}  a_0 \\cdots a_m.\n\\]\n\nIn order to see that (\\ref{FastChainsHyp}) implies (\\ref{FastHyp}),\nlet \\(( a_k \\mid  k \\leq n )\\) be the enumeration of \\(A\\)\nsuch that the left transition points of the \\(a_k\\)'s occur in increasing order.\nLet \\((x_k,y_k)\\) denote the support of \\(a_k\\) for \\(k \\leq n\\).\nBegin by setting \\(s_k\\) to be the least transition point of \\(A\\) in the support of \\(a_k\\);\nif \\(a_k\\) is isolated we let \\(s_k\\)\nbe the midpoint of the support of \\(a_k\\).\nDefine \\(t_k\\) by induction.\nIf \\(a_k\\) is initial, we define \\(t_k := s_k\\).\nIf \\(s_k\\) is the right transition point of \\(a_j\\) for some \\(j < k\\),\nthen define \\(t_k := t_j a_j\\).\nBy induction, \\(t_k < y_k\\).\nObserve that if \\(t_k = t_j a_j\\), then \\(t_k = t_j a_j < s_k\\)\nsince \\(s_k\\) is the right transition point of \\(a_j\\) and in particular is fixed by \\(a_j\\).\n\n\\begin{claim} \\label{ChainClaim}\nIf \\(s\\) is a transition point of \\(A\\) in the support of \\(a_k\\),\nthen \\(s \\leq t_k a_k\\).\n\\end{claim}\n\n\\begin{proof}\nLet \\(C\\) be the stretched transition chain which finishes with \\(a_k\\) and\nwhich is maximal with this property.\nIf \\(C\\) is \\(\\{a_k\\}\\), then \\(t_k = s_k = C_{\\min}\\) and the claim follows\nfrom our hypothesis that\n\\(s \\leq C_{\\max} \\leq C_{\\min} \\prod C \\leq t_k a_k\\).\nIf \\(a_j\\) is the immediate predecessor of \\(a_k\\) in \\(C\\)\nand \\(C\\) starts with \\(a_{i}\\),\nthen \\(C_{\\min} = t_i = s_i\\).\nMoreover, if \\(C'\\) is obtained from \\(C\\) by removing \\(a_k\\), then\nby an inductive argument we have that \\(C_{\\min} \\prod C' = t_i \\prod C' = t_k\\).\nBy hypothesis (\\ref{FastChainsHyp}), we have\nthat \\(C_{\\max} \\leq C_{\\min} \\prod C = t_k a_k\\).\nSince \\(s_k\\) is the right transition point of \\(a_j\\) which is the last entry\nof \\(C'\\), we must\nhave that \\(t_i \\prod C' = t_k < s_k\\).\nFurthermore, since any transition point of \\(A\\) in the support of \\(a_k\\)\nis between \\(s_k\\) and \\(C_{\\max}\\) we have our desired conclusion.\n\\end{proof}\n\nWe now claim that the assignment \\(a_k \\mapsto t_k\\) defines a marking which witnesses that \\(A\\) is geometrically fast.\nWe need to show that if \\(i < j \\leq n\\), then the feet of \\(a_i\\) are disjoint from\nthose of \\(a_j\\).\nBy our assumption on the enumeration, we have that \\(x_i < x_j\\).\nNote that if \\(y_i < x_j\\), the support of \\(a_i\\) is disjoint from the support of\n\\(a_j\\).\nIn particular the feet of \\(a_i\\) and \\(a_j\\) are disjoint.\n\nIf \\(x_i < x_j < y_j < y_i\\),\nthen \\(x_i < t_i \\leq x_j\\).\nIn particular, the left foot \\((x_i,t_i)\\)\nof \\(a_i\\) is disjoint from the support of\n\\(a_j\\) and hence from its feet.\nClaim \\ref{ChainClaim}\nimplies that \\(y_j \\leq t_i a_i \\) and hence that the right foot of \\(a_i\\), which\nis \\([t_i a_i,y_i)\\) is disjoint from the support of \\(a_j\\).\n\nFinally, suppose that \\(x_i < x_j < y_i < y_j\\).\nObserve that the left foot of \\(a_i\\) is disjoint from the support\nof \\(a_j\\) and, by a similar argument as in the previous case,\nthe right foot of \\(a_j\\) is disjoint from the support of \\(a_i\\).\nThus we only need to verify that the right foot of \\(a_i\\) is disjoint\nfrom the left foot of \\(a_j\\).\nBy Claim \\ref{ChainClaim} \n\\[\nx_j < t_j \\leq s_j \\leq t_i a_i < y_i\n\\]\nand hence \\((x_j,t_j)\\) is disjoint from \\([t_ia_i,y_i)\\), as desired.\n\nIt now remains to show that (\\ref{MaxFastChainsHyp}) implies (\\ref{FastChainsHyp});\nwe argue the contrapositive.\nSuppose that \\(C\\) is a stretched transition chain and that, as a sequence, \\(C\\) is the\nconcatenation of \\(C'\\) and \\(C''\\), each of which are stretched transition chains.\nSince we have seen above that every stretched transition chain is an interval within\na maximal stretched transition chain,\nthe desired implication will follow if we can show that\n\\(C_{\\max} > C_{\\min} \\prod C\\) holds if either\n\\(C'_{\\max} > C'_{\\min} \\prod C'\\) or \\(C''_{\\max} > C''_{\\min} \\prod C''\\).\nIf \\(C'_{\\max} > C'_{\\min} \\prod C'\\), then\n\\[\nC_{\\max} \\geq C'_{\\max} > C'_{\\min} \\prod C' =\nC_{\\min} \\prod C' \\prod C'' =C_{\\min} \\prod C\n\\]\nsince \\(C'_{\\max}\\) is the least transition point of \\(C''\\) and hence a lower bound for the support\nof \\(\\prod C''\\).\nIf  \\(C'_{\\max} \\leq C'_{\\min} \\prod C'\\) but \\(C''_{\\max} > C''_{\\min} \\prod C''\\), then\n\\[\nC_{\\max} = C''_{\\max} > C''_{\\min} \\prod C''  \\geq C'_{\\min} \\prod C' \\prod C'' = C_{\\min} \\prod C\n\\]\nsince \\(C''_{\\min}\\) is the greatest transition point of \\(C'\\) and thus\nan upper bound for the support of \\(\\prod C'\\).\n\\end{proof}\n\nObserve that the proof of Proposition \\ref{FastCriterion}\ngives an explicit construction\nof a marking of a family \\(A\\) of positive bumps.\nThis marking has the property that if \\(A\\) is geometrically fast,\nthen it is witnessed as such by the marking.\nWe will refer to this marking as the \\emph{canonical marking} of \\(A\\).\n\nFinally, let us note that Proposition \\ref{FastCriterion} gives us a means\nto algorithmically check whether a set of positive bumps \\(A\\) is fast.\nSpecifically, perform the following sequence of steps:\n\\begin{itemize}\n\n\\item determine whether \\(A\\) is geometrically proper;\n\n\\item if so, partition the non isolated elements of\n\\(A\\) into maximal stretched transition chains;\n\n\\item for each maximal stretched transition chain \\(C\\) of \\(A\\),\ndetermine whether \\(C_{\\max} \\leq C_{\\min} \\prod C\\).\n\n\\end{itemize}\nThis is possible provided we are able to perform the following basic queries:\n\\begin{itemize}\n\n\\item test for equality among the transition points of elements of \\(A\\);\n\n\\item determine the order of the transition points of elements of \\(A\\);\n\n\\item determine the truth of \\(C_{\\max} \\leq C_{\\min} \\prod C\\) whenever \\(C\\) is a stretched transition chain.\n\n\\end{itemize}\n\n\n\n\\section{The ping-pong analysis of geometrically fast sets of bumps}\n\\label{PingPongSec}\n\nIn this section, we adapt the ping-pong argument to the setting of fast families of\nbump functions.\nWhile the culmination will be Theorem \\ref{Faithful} below, the lemmas we will develop will\nbe used in subsequent sections.\nThey also readily adapt to the more abstract setting of Section \\ref{AbstractPingPongSec}.\n\nFix, until further notice, a (possibly infinite) geometrically fast collection \\(A\\) of positive bumps\nequipped with a marking; in particular we will write \\emph{word} to mean \\emph{\\(A\\)-word}.\nCentral to our analysis will be the notion of a \\emph{locally reduced word}.\nA word \\(\\str{w}\\) is \\emph{locally reduced} at \\(t\\) if it is freely reduced and whenever\n\\(\\str{ua}\\) is a prefix of \\(\\str{w}\\) for \\(a \\in A^\\pm\\),\n\\(tua \\ne tu\\).\nIf \\(\\str{w}\\) is locally reduced at every element of a set \\(J \\subseteq I\\), then we write that\n\\(\\str{w}\\) is \\emph{locally reduced on \\(J\\)}.\n\nThe next lemma collects a number of useful observations about locally\nreduced words; we omit the obvious proofs.\nRecall that if \\(\\str{u}\\) and \\(\\str{v}\\) are freely reduced, then the free\nreduction of \\(\\str{uv}\\) has the form \\(\\str{u}_0 \\str{v}_0\\) where \\(\\str{u}=\\str{u}_0 \\str{w}\\),\n\\(\\str{v}=\\str{w}^{-1}\\str{v}_0\\), and \\(\\str{w}\\) is the longest common suffix of\n\\(\\str{u}\\) and \\(\\str{v}^{-1}\\).\nIn particular, if \\(\\str{u}\\), \\(\\str{v}\\), and \\(\\str{w}\\) are freely reduced words and the\nfree reductions of \\(\\str{uv}\\) and \\(\\str{uw}\\) coincide, then \\(\\str{v} = \\str{w}\\).\n\n\\begin{lemma}\\label{LocRedBasics} All of the following are true:\n\\begin{itemize}\n\n\\item For all \\(x\\in I\\) and all words \\(\\str{w}\\), there is a subword\n\\(\\str{v}\\) of \\(\\str{w}\\) which is locally reduced at \\(x\\) so that\n\\(xv=xw\\).\n\n\\item For all \\(x\\in I\\) and words \\(\\str{u}\\) and \\(\\str{v}\\), if \\(\\str{u}\\) is\nlocally reduced at \\(x\\), and \\(\\str{v}\\) is locally reduced at \\(xu\\), then\nthe free reduction of \\(\\str{uv}\\) is locally reduced at \\(x\\).\n\n\\item For all \\(x\\in I\\) and words \\(\\str{w}\\), if \\(\\str{w}\\) is locally\nreduced at \\(x\\) and \\(\\str{w}=\\str{uv}\\), then \\(\\str{u}\\) is locally reduced at\n\\(x\\) and \\(\\str{v}\\) is locally reduced at \\(xu\\).\n\n\\item For all \\(x\\in I\\) and words \\(\\str{w}\\), if \\(\\str{w}\\) is locally\nreduced at \\(x\\), then \\(\\str{w}^{-1}\\) is locally reduced at \\(xw\\).\n\n\\end{itemize}\n\\end{lemma}\n\\noindent\n(Recall here our convention that a \\emph{subword} is not required to consist\nof consecutive symbols of the original word.)\nFor \\(x\\in I\\), we use \\(x\\gen{A}\\) to denote the orbit\nof \\(x\\) under the action of \\(\\gen{A}\\) and for\n\\(S\\subseteq I\\), we let \\(S\\gen{A}\\) be the union of\nthose \\(x\\gen{A}\\) for \\(x\\in S\\).\n\nA marker \\(t\\) of \\(A\\) is \\emph{initial} if whenever \\(s < t\\) is the marker of \\(a \\in A\\),\nthen \\(t \\ne sa\\).\nIf \\(A\\) is finite and we are working with the canonical marking, then the initial markers\nare precisely the markers of the initial intervals.\nLet \\(M_A\\) be the set of initial markers of \\(A\\).\nWe will generally suppress the subscript if the meaning is clear from the context;\nin particular we will write \\(M\\gen{A}\\) for \\(M_A\\gen{A}\\).\nNotice that every marker is contained in \\(M \\gen{A }\\).\n\nAside from developing lemmas for the next section,\nthe goal of this section is to prove that the action of\n\\(\\gen{A}\\) on \\(M\\gen{A}\\) is faithful.\nThe next lemma is the manifestation of the \\emph{ping-pong argument}\nin the context in which we are working.\nIf \\(\\str{w} \\ne \\varepsilon\\) is a word, the \\emph{source} (\\emph{destination}) of\n\\(\\str{w}\\) is the source of the first\n(destination of the last) symbol in \\(\\str{w}\\).\nThe source and destination of \\(\\varepsilon\\) are \\(\\emptyset\\).\n\n\\begin{lemma} \\label{PingPong}\nIf \\(x \\in I\\) and \\(\\str{w} \\ne \\varepsilon\\)\nis a word which is locally reduced at \\(x\\), then\neither \\(x \\in \\operatorname{src}(\\str{w})\\) or \\(xw \\in \\operatorname{dest}(\\str{w})\\).\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the length of \\(\\str{w}\\).\nWe have already noted\nthat if \\(\\str{w} = \\str{a}\\) and \\(t \\in \\operatorname{supt}(a)\\),\nthen \\(ta \\in \\operatorname{dest}(a)= \\operatorname{dest}(\\str{w})\\) if and only if \\(t \\not \\in \\operatorname{src}(a) = \\operatorname{src}(\\str{w})\\).\nNext suppose that \\(\\str{w}\\) has length at least 2, \\(x \\not \\in \\operatorname{src}(\\str{w})\\),\nand let \\(\\str{v}\\) be a (possibly empty) word such that \\(\\str{w} = \\str{vab}\\)\nfor \\(a,b \\in A^\\pm\\).\nSince \\(\\str{w}\\) is locally reduced, \\(b \\ne a^{-1}\\)\nand thus the destination of \\(a\\) is not the source of \\(b\\).\nSince \\(A\\) is geometrically fast, the destination of \\(a\\) is disjoint from the source of\n\\(b\\).\nBy our inductive hypothesis, \\(y = x v a\\) is in\n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b) \\setminus \\operatorname{src}(b)\\).\nThus \\(x w = x v ab  = y b\\) is in the destination of \\(b\\).\n\\end{proof}\n\nIf \\(\\str{w}\\) is a word, define \\(J(\\str{w}) :=\\operatorname{supt}(a)\\setminus \\operatorname{src}(a)\\)\nwhere \\(\\str{a}\\) is the first symbol of\n\\(\\str{w}\\).\nNotice that if \\(\\str{w}\\) is locally reduced at \\(x\\), then ``\\(x \\in J(\\str{w})\\)'' is equivalent to\n``\\(x \\not \\in \\operatorname{src}(\\str{w})\\)''.\nThe following lemma is easily established by induction on the length of \\(\\str{w}\\)\nusing Lemma \\ref{PingPong}.\n\n\\begin{lemma} \\label{threaded}\nIf \\(\\str{w}\\) is a word and \\(x \\in J(\\str{w})\\), then \\(\\str{w}\\) is locally reduced\nat \\(x\\) if and only if \\(\\str{w}\\) is freely reduced and \n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b)\\) whenever \\(\\str{ab}\\) are consecutive symbols in \\(\\str{w}\\).\nIn particular, if \\(\\str{w}\\) is freely reduced, then \\(\\str{w}\\) is locally reduced on \\(J(\\str{w})\\)\nprovided \\(\\str{w}\\) is locally reduced at some element of \\(J(\\str{w})\\).\n\\end{lemma}\n\nWhen applying Lemma \\ref{PingPong}, it will be useful to be able to assume\nthat \\(x\\) is not in \\(\\operatorname{src}(\\str{w})\\).\nNotice that if \\(x\\) is not in the feet of any elements of \\(A\\), then this is automatically true\n(for instance this is true if \\(x \\in M\\)).\nThe next lemma captures an important consequence of Lemma \\ref{PingPong}.\n\n\\begin{lemma}\\label{Collision}\nSuppose \\(x_0,x_1 \\in I\\), \\(\\str{u}_i\\) is locally reduced at \\(x_i\\) and\n\\(x_i \\not \\in \\operatorname{src}(\\str{u}_i)\\).\nIf \\(|\\str{u}_0| \\leq |\\str{u}_1|\\) and \\(x_0 u_0 = x_1 u_1\\),\nthen \\(\\str{u}_0\\) is a suffix of \\(\\str{u}_1\\).\nIn particular if \\(t \\in I\\) is not in any of the feet of \\(A\\)\nand \\(\\str{u}\\) and \\(\\str{v}\\) are words\nthat are locally reduced at \\(t\\) with \\(tu=tv\\), then \\(\\str{u}=\\str{v}\\).\n\\end{lemma}\n\n\\begin{proof}\nThe main part of the lemma is proved by induction on \\(|\\str{u}_0|\\).\nIf \\(\\str{u}_0 = \\varepsilon\\), this is trivially true.\nNext suppose that \\(\\str{u}_i \\str{a}_i\\) is locally\nreduced at \\(x_i\\) and \\(x_i \\not \\in \\operatorname{src}(\\str{u}_i \\str{a}_i)\\).\nIf \\(x_0 u_0 a_0 = x_1 u_1 a_1\\),\nthen by Lemma \\ref{PingPong},\n\\(\\str{a}_0 = \\str{a}_1\\).\nWe are now finished by applying our\ninduction hypothesis to conclude that \\(\\str{u}_0\\) is a suffix of \\(\\str{u}_1\\).\n\nIn order to see the second conclusion, let \\(t\\), \\(\\str{u}\\) and \\(\\str{v}\\) be given such that\n\\(x:= tu = tv\\) and assume without loss of generality that \\(|\\str{u}| \\leq |\\str{v}|\\).\nBy the main assertion of the lemma, \\(\\str{v} = \\str{wu}\\) for some \\(\\str{w}\\).\nSince \\(tw = t\\), Lemma \\ref{PingPong} implies \\(\\str{w} = \\varepsilon\\).\n\\end{proof}\n\nIf \\(x \\in \\operatorname{src}(a)\\) for some \\(a \\in A^\\pm\\), then \\(x \\in \\operatorname{dest}(a^{-1})\\).\nThis suggests that we have ``arrived at'' \\(x\\) by applying a locally reduced word to some other point.\nMoreover  \\(a^{-1}\\) is the unique element \\(b\\) of \\(A^\\pm\\) such that \\(x \\in \\operatorname{dest}(b)\\).\nThus we may attempt to ``trace back'' to where \\(x\\) ``came from.''\nThis provides a recursive definition of a sequence which starts at\n\\(\\str{a}^{-1}\\) and grows to the left, possibly infinitely far.\nThis gives rise to the notion of a \\emph{history} of a point \\(x \\in I\\), which\nwill play an important role in the\nproof of Theorem \\ref{Faithful} below and also in Section \\ref{AbstractPingPongSec}.\nIf \\(t \\in I\\) is not in \\(\\operatorname{dest}(a)\\) for any \\(a \\in A^\\pm\\), then we say that\n\\(t\\) has \\emph{trivial history} and define \\(\\tilde t := \\{a \\in A : t \\in \\operatorname{supt}(a)\\}\\).\nIf \\(x \\in I\\), define \\(\\eta(x)\\) to be the set of all strings of the following form:\n\\begin{itemize}\n\n\\item words \\(\\str{u}\\) such that for some \\(t \\in I\\),\n\\(tu = x\\), \\(\\str{u}\\) is locally reduced at \\(t\\), and \\(t \\not \\in \\operatorname{src}(\\str{u})\\);\n\n\\item strings \\(\\str{\\tilde t u}\\) such that \\(t\\) has trivial history, \\(\\str{u}\\) is locally\nreduced at \\(t\\), and \\(tu = x\\).\n\n\\end{itemize}\n\\noindent\nNotice that if \\(\\str{w}\\) is a word, then \\(\\str{w}^{-1}\\) is in \\(\\eta(x)\\) if and only if\n\\(\\str{w}\\) is locally reduced at \\(x\\) and \\(xw \\not \\in \\operatorname{dest}(\\str{w})\\).\n\nWe will refer to elements of \\(\\eta(x)\\) as \\emph{histories} of \\(x\\).\nWe will say that \\(x\\) has \\emph{finite history} if \\(\\eta(x)\\) is finite.\nThe following easily established using Lemmas \\ref{LocRedBasics} and \\ref{Collision};\nthe proof is omitted.\n\n\\begin{lemma} \\label{BasicHistory}\nThe following are true for each \\(x \\in I\\):\n\\begin{itemize}\n\n\\item\n\\(\\eta(x)\\) is closed under taking suffixes;\n\n\\item\nfor each \\(n\\), \n\\(\\eta(x)\\) contains at most one sequence of length \\(n\\);\n\n\\item\nIf  \\(\\str{v}\\) is a word in \\(\\eta(x)\\), then \\(\\eta(xv^{-1}) = \\{\\str{u} : \\str{uv} \\in \\eta(x)\\}\\).\n\n\\end{itemize}\n\\end{lemma}\n\\noindent\nIt is useful to think of \\(\\eta(x)\\) as the suffixes of a single sequence\nwhich is either finite or grows infinitely to the left.\n\nIn what follows, we will typically use \\(s\\) and \\(t\\) to denote elements of \\(I\\)\nwith finite history and \\(x\\) and \\(y\\) for arbitrary elements of \\(I\\).\nThe following is a key property of having a trivial history.\n\n\\begin{lemma}\\label{DisjointOrbits}\nIf \\(s\\ne t\\) have trivial history, then\n\\(s\\gen{A}\\) and \\(t\\gen{A }\\) are disjoint.\n\\end{lemma}\n\n\\begin{proof}\nIf the orbits intersect, then \\(t=sw\\) for some word \\(\\str{w}\\).\nBy Lemma \\ref{LocRedBasics} we can take \\(\\str{w}\\) to be locally reduced\nat \\(s\\).\nBy Lemma \\ref{PingPong}, \\(sw\\) is in the destination of \\(\\str{w}\\).\nBut \\(t = sw\\) has trivial history, which is impossible.\n\\end{proof}\n\nRecall that the set of\nfreely reduced words in a given generating set has the structure of\na rooted tree with the empty word as root and where ``prefix of'' is\nsynonymous with ``ancestor of.''\nThe \\emph{ping-pong argument} discovers orbits that reflect this structure.\nDefine a labeled directed graph on \\(I\\) by putting an arc with label \\(a\\) from \\(x\\) to\n\\(xa\\) whenever \\(a \\in A^\\pm\\) and \\(xa \\ne x\\).\nThe second part of Lemma \\ref{Collision} asserts that if \\(x\\) is in the orbit of a point \\(t\\) with trivial\nhistory, then there is a unique path in this graph connecting \\(t\\) to \\(x\\).\nIt follows that if there is a path between two elements of \\(I\\) with finite history, it is unique,\nyielding the following lemma.\n\n\\begin{lemma}\\label{TreeAction} If\n\\(s,t \\in I\\) have finite histories,\nthen there is at most one word \\(\\str{w}\\) which is\nlocally reduced at \\(s\\) so that \\(sw=t\\).\n\\end{lemma}\n\nNotice that the assumption of finite history in this lemma is necessary.\nFor instance if we consider the positive bumps \\(a_0\\) and \\(a_1\\) Figure \\ref{TrackingPoint},\nthere must be an \\(x \\in \\operatorname{supt}(a_0) \\cap \\operatorname{supt}(a_1)\\) such that \\(xa_0 a_1^{-1} = x\\).\nThis follows from the observation that if \\(s < t\\) are,\nrespectively, the left transition point of \\(a_1\\)\nand the right transition point of \\(a_0\\), then\n\\[s a_0 a_1^{-1} > s a_1^{-1} =  s \\qquad \\textrm{ and } \\qquad  t a_0 a_1^{-1} = t a_1^{-1} < t\\]\nwhich implies the existence of the desired \\(x\\) by applying the Intermediate Value Theorem\nto \\(t \\mapsto t a_0 a_1^{-1} - t\\).\n\nGiven two points \\(x,y \\in I\\) and a word \\(\\str{w}\\), it will be useful to find a single\nword \\(\\str{w}'\\) which is locally reduced at \\(x\\) and \\(y\\) and which satisfies\n\\(xw' = xw\\) and \\(yw' = yw\\).\nThe goal of the next set of lemmas is to provide a set of sufficient conditions for the existence of such a \\(\\str{w}'\\).\nIt will be convenient to introduce some additional terminology at this point.\nIf \\(x \\in I\\), then\nwe say that \\(\\str{w}\\) is a \\emph{return word} for \\(x\\) if\n\\(xw = x\\) and \\(\\str{w} \\ne \\varepsilon\\);\na \\emph{return prefix} for \\(x\\) is a prefix which is a return word.\nWe will see that ``\\(\\str{w}\\) does not have a return prefix for \\(s\\)'' is a useful hypothesis.\nThe next lemma provides some circumstances under which this is true.\n\n\\begin{lemma} \\label{NoReturn}\nIf \\(s \\in I\\) has finite history,\n\\(\\str{u}\\) is locally reduced at \\(s\\),\nand \\(\\str{w}\\) is a word of length less than \\(\\str{u}\\),\nthen \\(\\str{uw}\\) has no return prefix for \\(s\\).\n\\end{lemma}\n\n\\begin{proof}\nNotice that it suffices to prove that \\(\\str{uw}\\)\nis not a return word for \\(s\\).\nIf it were, then there would be a locally reduced subword\n\\(\\str{v}\\) of \\(\\str{w}^{-1}\\) such that \\(su = sv\\).\nSince \\(|\\str{v}| \\leq |\\str{w}| < |\\str{u}|\\),\nthis would contradict Lemma \\ref{TreeAction}.\n\\end{proof}\n\n\\begin{lemma} \\label{FellowTraveler1}\nSuppose that \\(\\str{w} \\ne \\varepsilon\\) is a word and \\(x \\in J (\\str{w})\\).\nIf \\(\\str{w}\\) has no return prefix for \\(x\\) and \\(\\str{w'}\\) is locally reduced at\n\\(x\\) with \\(xw' = xw\\), then:\n\\begin{itemize}\n\n\\item \\(J(\\str{w}') = J(\\str{w})\\);\n\n\\item \\(\\str{w}'\\) is locally reduced on \\(J(\\str{w})\\);\n\n\\item if \\(y \\in J(\\str{w})\\), then \\(yw' = yw\\).\n\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the length of \\(\\str{w}\\).\nIf \\(\\str{w}\\) has length \\(1\\), then there is nothing to show.\nSuppose now that \\(\\str{w} = \\str{ub}\\) for some \\(b \\in A^\\pm\\) and \\(\\str{u} \\ne \\varepsilon\\).\nLet \\(\\str{u}'\\) be locally reduced at \\(x\\) such that \\(xu' = xu\\).\nBy our inductive assumption, \\(J:= J(\\str{u}') = J(\\str{u}) = J(\\str{w})\\),\n\\(\\str{u}'\\) is locally reduced on \\(J\\) and if \n\\(y \\in J\\), then \\(yu' = yu\\).\nBy our assumption, \\(xu' = xu \\ne x\\) and so \\(\\str{u}' \\ne \\varepsilon\\).\nIf \\(\\str{u}'\\str{b}\\) is not freely reduced, then its free reduction \\(\\str{w}'\\)\nsatisfies that \\(xw' = xu'b = xub = xw \\ne x\\).\nIn particular, \\(\\str{w}' \\ne \\varepsilon\\) and retains the first symbol of \\(\\str{u}'\\).\nFurthermore, since \\(\\str{u}'\\) is locally reduced on \\(J\\) and since\n\\(\\str{w}'\\) is a prefix of \\(\\str{u}'\\), \\(\\str{w}'\\) is also locally reduced on \\(J\\).\nAlso, if \\(y \\in J\\), then \\(yw = yub = yu'b = yw'\\).\n\nSuppose now that \\(\\str{u}'\\str{b}\\) is freely reduced. \nBy Lemma \\ref{PingPong}, \\(Ju = Ju' \\subseteq \\operatorname{dest}(\\str{u}')\\) for all \\(y \\in J\\).\nIf \\(\\operatorname{dest}(\\str{u}')\\) is disjoint from \\(\\operatorname{supt}(b)\\), then\n\\(yu' = yu'b = yub = yw\\) for all \\(y \\in J\\).\nSince \\(\\str{u}'\\) is locally reduced, we are again done in this case.\nIn the remaining case, \\(\\operatorname{dest}(\\str{u}') \\subseteq \\operatorname{supt}(b)\\) in which case\n\\(\\str{w}' = \\str{u}'\\str{b}\\) is locally reduced at all \\(y \\in J\\).\nSince  for all \\(y \\in J\\), \\(yu = yu' \\ne yu'b = yub = yw\\) we have that\n\\(\\str{w}'\\) is locally reduced on \\(J\\).\nClearly \\(J(\\str{w}') = J(\\str{u}') = J(\\str{w})\\) and we are finished.\n\\end{proof}\n\\noindent\nLemma \\ref{FellowTraveler1} has two immediate consequences which will be easier to\napply directly.\n\n\\begin{lemma} \\label{FellowTraveler2}\nIf \\(\\str{w}\\) is a word and\nthere is an \\(s \\in J:=J (\\str{w})\\) with finite history such that\n\\(\\str{w}\\) is a minimal return word for \\(s\\),\nthen \\(w\\) is the identity on \\(J\\).\n\\end{lemma} \n\n\\begin{proof}\nLet \\(\\str{w} = \\str{ua}\\) and let \\(\\str{u}'\\) be locally reduced at \\(s\\) with \\(su' = su\\).\nSince \\(sw = su'a = s\\), it must be that \\(\\str{u}' = \\str{a}^{-1}\\).\nBy Lemma \\ref{FellowTraveler1},\n\\(t u' = tu\\) whenever \\(t \\in J\\).\nThus \\(tw = tu' a = ta^{-1} a = t\\) for all \\(t \\in J\\).\n\\end{proof}\n\n\\begin{lemma} \\label{FellowTraveler3}\nIf \\(s,t \\in I\\) have finite histories and \\(\\eta(s) = \\eta(t)\\), then\nany return word for \\(s\\) is a return word for \\(t\\).\nIf moreover \\(s\\) and \\(t\\) have trivial history and\n\\(\\str{w} \\ne \\varepsilon\\) is not a return word for \\(s\\),\nthen there is an \\(a \\in A^\\pm\\) such that \\(\\{sw,tw\\} \\subseteq \\operatorname{dest}(a)\\).\n\\end{lemma}\n\n\\begin{proof}\nIf there is a minimal return prefix of \\(\\str{w}\\) for \\(s\\), then by Lemma \\ref{FellowTraveler2},\nit is also a return prefix for \\(t\\).\nBy iteratively removing minimal return prefixes for \\(s\\), \nwe may assume \\(\\str{w}\\) has no return prefixes for \\(s\\).\nObserve that since \\(\\eta(s) = \\eta(t)\\), \\(\\{s,t\\} \\subseteq J(\\str{w})\\).\nLemma \\ref{FellowTraveler1} now yields the desired conclusion.\n\\end{proof}\n\n\nThe following theorem shows that the restriction of the action of \\(\\gen{A}\\) to \\(M\\gen{A}\\) is faithful.\n\n\\begin{thm}\\label{Faithful}\nSuppose that \\(A \\subseteq {\\operatorname{Homeo}_+(I)}\\)\nis a (possibly infinite) geometrically fast set of positive bump functions, equipped with a marking.\nIf \\(g \\in \\gen{A}\\) is not the identity, then\nthere is a \\(y \\in M \\gen{A}\\) such that \\(y g \\ne y\\).\n\\end{thm}\n\n\\begin{remark}\nIf \\(A\\) is finite and equipped with the canonical marking, then\nthe cardinality of \\(M\\) is the sum of the number of maximal stretched transition chains in \\(A\\)\nand the number of isolated elements of \\(A\\).\n\\end{remark}\n\n\\begin{proof}\nFirst observe that if there is an \\(x \\in I\\) such that \\(xg \\ne x\\), then by continuity of \\(g\\),\nthere is an \\(x \\in I\\) such that \\(xg \\ne x\\) and \\(x\\) is not in the orbit of a transition point or\nmarker (orbits are countable and neighborhoods are uncountable).\nFix such an \\(x\\) and a word \\(\\str{w}\\) representing \\(g\\) for the duration of the proof.\nThe proof of the theorem breaks into two cases, depending on whether \\(x\\) has\nfinite history.\n\nWe will first handle the case in which \\(x\\) has trivial history;\nthis will readily yield the more general case in which \\(x\\) has finite history.\nSuppose that \\(x \\not \\in \\operatorname{src}(a)\\) for all \\(a \\in A\\).\nLet \\(y < x\\) be maximal such that \\(y\\) is either a transition point or a marker.\n\n\\begin{claim} \\label{NotSplit}\n\\(y\\) has trivial history and\n\\(\\tilde x = \\tilde y\\).\n\\end{claim}\n\n\\begin{proof}\nFirst suppose that \\(y\\) is the marker of some \\(a \\in A\\).\nNotice that by our assumption of maximality of \\(y\\), the right transition point\nof \\(a\\) is greater than \\(y\\).\nIn this case, both \\(x\\) and \\(y\\) are in the support of \\(a\\).\nFurthermore, observe that \\(y\\) is not in the foot of any \\(b \\in A\\).\nTo see this, notice that this would only be possible if \\(y\\) is in the right foot of some \\(b\\).\nHowever since \\(x\\) is not in the right foot of \\(b\\),\nthe right transition point of \\(b\\) would then be less than \\(x\\),\nwhich would contradict the maximal choice of \\(y\\).\nFinally, if \\(b \\in A \\setminus \\{a\\}\\), the maximal choice of \\(y\\) implies that\n\\(x\\) is in the support of \\(b\\) if and only if \\(y\\) is.\n\nIf \\(y\\) is a transition point of some \\(a \\in A\\), then \\(y\\) must be the right transition point\nof \\(a\\) since otherwise our maximality assumption on \\(y\\) would imply that\n\\(x\\) is in the left foot of \\(a\\), contrary to our assumption that \\(x\\) has trivial history.\nIn this case neither \\(x\\) nor \\(y\\) are in the support of \\(a\\).\nIf \\(b \\in A \\setminus \\{a\\}\\), then our maximality assumption on \\(y\\) implies\nthat \\(\\{x,y\\}\\) is either contained in or disjoint from the support of \\(b\\).\nTo see that \\(y\\) has trivial history, observe that the only way a transition point\ncan be in a foot is for it to be the left endpoint of a right foot.\nIf \\(y\\) is the left endpoint of a right foot,\nthen our maximal choice of \\(y\\) would mean that \\(x\\) is also in this foot, which is contrary\nto our assumption.\nThus \\(y\\) must have trivial history.\n\\end{proof}\n\nBy Claim \\ref{NotSplit} and Lemma \\ref{FellowTraveler3}, \\(yw \\ne y\\).\nIf \\(y\\) is a marker, we are done.\nIf \\(y\\) is a transition point of some \\(a \\in A\\), then as noted above it is the right transition point of \\(a\\).\nIf \\(s\\) is the marker of \\(a\\), then \\(sa^k \\to y\\) and by continuity \\(sa^k w \\to yw\\).\nThus for large enough \\(k\\), \\(sa^k w \\ne sa^k\\).\nSince \\(sa^k \\in M\\gen{A}\\), we are done in this case.\n\nNow suppose that \\(x\\) has finite history and let \\(\\str{\\tilde{t}u} \\in \\eta(x)\\) with \\(t \\in I\\) and \\(tu = x\\).\nBy definition of \\(\\eta(x)\\), \\(t\\) has trivial history.\nSince \\(xw \\ne x\\), we have that \\(tuw \\ne tu\\) and hence \\(tuwu^{-1} \\ne t\\).\nIt follows from the previous case\nthat there is an \\(s \\in M\\gen{A}\\) such that \\(s uwu^{-1} \\ne s\\).\nWe now have that \\(y :=su\\) is in \\(M\\gen{A}\\) and satisfies \\(yw \\ne y\\) as desired.\n\nFinally, suppose that \\(\\eta(x)\\) is infinite.\nLet \\(\\str{u} \\in \\eta(x)\\) be longer than \\(\\str{w}\\), let \\(s\\) be the marker\nfor the initial symbol of \\(\\str{u}\\), and set \\(y := xu^{-1}\\) and \\(t := su\\).\nSince \\(\\str{u}\\) is locally reduced at \\(y\\) by assumption,\nLemma \\ref{threaded} implies that \\(\\str{u}\\) is locally reduced at \\(s\\).\nBy Lemma \\ref{NoReturn}, \\(\\str{uw}\\) has no return prefix for \\(s\\).\nLet \\(\\str{v}\\) be locally reduced at \\(s\\) such that \\(suw = sv\\).\nApplying Lemma \\ref{FellowTraveler1} to \\(\\str{uw}\\), \\(s\\) and \\(\\str{v}\\), we\ncan conclude that \\(J(\\str{v}) = J(\\str{uw})\\), that\n\\(\\str{v}\\) is locally reduced at \\(y\\), and \\(yuw = yv\\).\nNotice that since \\(xw \\ne x\\), \\(yuw = yv \\ne yu\\) and in particular \\(\\str{v} \\ne \\str{u}\\).\nBy Lemma \\ref{TreeAction}, we have that \\(t = su \\ne sv = suw = tw\\).\nThis finishes the proof of Theorem \\ref{Faithful}.\n\\end{proof}\n\nWe finish this section with two lemmas which concern multi-orbital homeomorphisms but \nwhich otherwise fit the spirit of this section.\nThey will be needed in Section \\ref{ExcisionSec}.\n\n\\begin{lemma} \\label{FindPrefix}\nSuppose that \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast and equipped with a fixed marking.\nLet \\(A\\) be the set of bumps used in \\(X\\) and \\(s \\in M\\).\nIf \\(\\str{w}\\) is an \\(X\\)-word and \\(\\str{u}\\) is an \\(A\\)-word which is locally reduced at \\(s\\)\nand satisfies \\(su = sw\\), then for every prefix \\(\\str{u}'\\) of \\(\\str{u}\\), there is a prefix \\(\\str{w}'\\) of\n\\(\\str{w}\\) such that \\(su' = sw'\\). \n\\end{lemma}\n\n\\begin{proof}\nLet \\(\\str{w}_i\\) be the prefix of \\(\\str{w}\\) of length \\(i\\) for \\(i \\leq |\\str{w}|\\) and let\n\\(\\str{u}_i\\) be the unique word which is locally reduced at \\(s\\) such that\n\\(su_i = sw_i\\).\nNotice that if \\(\\str{u}_{i+1} \\ne \\str{u}_i\\), then \\(\\str{u}_{i+1}\\)\nis obtained by inserting or deleting a single symbol at\/from the end of \\(\\str{u}_i\\).\nIt follows that all prefixes of \\(\\str{u}\\) occur among the \\(\\str{u}_i\\)'s.\n\\end{proof}\n\nIf \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\), then an element \\(s\\) of \\(I\\) is defined to have finite history with respect to \\(X\\)\nif it has finite history with respect to the set of bumps used in \\(X\\).\nThe meaning of \\emph{return word} is unchanged in the context of \\(X\\)-words.\n\n\\begin{lemma} \\label{X_to_A}\nSuppose that \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast, equipped with a fixed marking, and that \\(A\\)\ndenotes the set of bumps used in \\(X\\).\nLet \\(\\str{w} \\ne \\varepsilon\\) be an \\(X\\)-word and \\(\\str{a} \\in A^\\pm\\) be a signed bump of\nthe first symbol of \\(\\str{w}\\).\nIf \\(s \\in J: = J(\\str{a})\\) and \\(\\str{w}\\) has no proper return prefix for \\(s\\),\nthen there is an \\(A\\)-word \\(\\str{v}\\) which begins with \\(\\str{a}\\) and is such that\n\\(v\\) and \\(w\\) coincide on \\(J\\).\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on the length of \\(\\str{w}\\).\nObserve that the lemma is trivially true if \\(\\str{w}\\) has length at most \\(1\\).\nTherefore suppose that \\(\\str{w} = \\str{u g}\\) with \\(g \\in X^\\pm\\) and \\(\\str{u} \\ne \\varepsilon\\).\nLet \\(\\str{v}'\\) be an \\(A\\)-word which begins with \\(\\str{a}\\) and is such that \\(v'\\) and \\(u\\) agree on \\(J\\).\nSince \\(\\str{u}\\) has no return prefix, Lemma \\ref{FindPrefix} implies that \\(\\str{v}'\\) has no return prefix.\nLet \\(\\str{v}''\\) be a subword of \\(\\str{v}'\\) which is locally reduced at \\(s\\) and satisfies\n\\(sv'' = sv'\\).\nBy Lemma \\ref{FellowTraveler1}, \\(\\str{v}''\\) is locally reduced on \\(J\\) and\n\\(J(\\str{v}'') = J(\\str{v}') = J\\).\nIn particular, \\(J u \\subseteq \\operatorname{dest}(\\str{v}'')\\).\nIf the support of \\(g\\) is disjoint from \\(\\operatorname{dest}(\\str{v}'')\\), set \\(\\str{v}:=\\str{v}'\\).\nOtherwise,\nlet \\(b \\in A^\\pm\\) be the signed bump of \\(g\\) such that \\(\\operatorname{dest}(\\str{v}'') \\subseteq \\operatorname{supt}(b)\\)\nand set \\(\\str{v}:=\\str{v}' \\str{b}\\).\nObserve that \\(\\str{v}\\) satisfies the conclusion of the lemma.\n\\end{proof}\n\n\n\n\\section[The isomorphism theorem]{The isomorphism theorem for geometrically fast generating sets}\n\n\\label{CombToIsoSec}\n\nAt this point we have developed all of the tools needed to prove Theorem \\ref{CombToIso},\nwhose statement we now recall.\n\n\\begin{customthm}{\\ref{CombToIso}}\nIf two geometrically fast sets \\(X , Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) have only finitely many transition points\nand have isomorphic dynamical diagrams, then the induced\nbijection between \\(X\\) and \\(Y\\) extends to an isomorphism of\n\\(\\gen{X}\\) and \\(\\gen{Y}\\) (i.e. \\(\\gen{X}\\) is \\emph{marked isomorphic} to \\(\\gen{Y}\\)).\nMoreover, there is an order preserving bijection\n\\(\\theta :  M \\gen{X} \\to M \\gen{Y}\\) such that\n\\(f \\mapsto f^\\theta\\) induces the isomorphism \n\\(\\gen{X} \\cong \\gen{Y}\\).\n\\end{customthm}\n\nObserve that it is sufficient to prove this theorem in the special case when\n\\(X\\) and \\(Y\\) are finite geometrically fast collections of positive bumps:\nif \\(A\\) and \\(B\\) are the bumps used in \\(X\\) and \\(Y\\) respectively,\nthen the dynamical diagrams of \\(A\\) and \\(B\\) are isomorphic and\nthe isomorphism of \\(\\gen{A}\\) and \\(\\gen{B}\\) restricts to \na marked isomorphism of \\(\\gen{X}\\) and \\(\\gen{Y}\\).\n\nFix, for the moment, a finite geometrically fast set of positive bumps \\(X\\).\nAs we have noted, it is a trivial matter, given a word \\(\\str{w}\\) and a \\(t \\in M\\),\nto find a subword \\(\\str{w}'\\) which is locally reduced at \\(t\\) and satisfies \\(t w = tw'\\).\nTheorem \\ref{CombToIso} will fall out of an analysis of a question of\nindependent interest:\nhow does one determine \\(\\str{w}'\\) from \\(\\str{w}\\)\nand \\(t\\) using only the dynamical diagram of \\(X\\)?\nToward this end, we define \\(\\str{\\tilde{t}w}\\) to be a \\emph{local word} if \\(t \\in M\\) and \\(\\str{w}\\) is a word.\n(Notice that \\(t \\mapsto \\tilde t\\) is injective on \\(M\\);\nthe reason for working with \\(\\tilde t\\) is in anticipation\nof a more general definition in Section \\ref{AbstractPingPongSec}.)\nA local word \\(\\str{\\tilde{t}w}\\) is \\emph{freely reduced} if \\(\\str{w}\\) is.\nIt will be convenient to adopt the convention that \\(\\operatorname{dest}(\\str{\\tilde{t}}) = \\{t\\}\\) if\n\\(t \\in M\\).\nDefine \\(\\Lambda = \\Lambda_X\\) to be the set of all freely reduced local words \\(\\str{\\tilde{t}w}\\)\nsuch that if \\(\\str{ab}\\) are consecutive symbols in \\(\\str{\\tilde{t}w}\\), then\nthe destination of \\(a\\) is between \\(\\operatorname{src}(b)\\) and \\(\\operatorname{dest}(b)\\) in the diagram's ordering.\nNotice that the assertion that \\(\\str{\\tilde{t}w}\\) is in \\(\\Lambda\\) can be formulated as an\nassertion about \\(\\str{w}\\), the element of \\(X\\) which has \\(t\\) as a marker and\nthe dynamical diagram of \\(X\\).\n\nEvery local word \\(\\str{\\tilde{t}w}\\)\ncan be converted into an element of \\(\\Lambda\\) by iteratively\nremoving symbols by the following procedure:\nif \\(\\str{ab}\\) is the first consecutive pair in \\(\\str{\\tilde{t}w}\\) which witnesses that it\nis not in \\(\\Lambda\\), then:\n\\begin{itemize}\n\n\\item if \\(b = a^{-1}\\) then delete the pair \\(\\str{ab}\\);\n\n\\item if \\(b \\ne a^{-1}\\) then delete \\(\\str{b}\\).\n\n\\end{itemize}\nObserve that since the first symbol of a local word is not removed by this procedure,\nthe result is still a local word.\nThe \\emph{local reduction} of a local word \\(\\str{\\tilde{t}w}\\) is the result of applying\nthis procedure to \\(\\str{\\tilde{t}w}\\) until it terminates at an element of \\(\\Lambda\\).\nThe following lemma admits a routine proof by induction, which we omit.\n\n\\begin{lemma} \\label{LambdaLem}\nSuppose that \\(X\\) is a geometrically fast set of positive bumps.\nIf \\(t \\in M\\) and \\(\\str{w}\\) is a word,\nthen \\(\\str{w}\\) is locally reduced at \\(t\\) if and only if \\(\\str{\\tilde{t}w}\\)\nis in \\(\\Lambda\\).\nMoreover, if \\(\\str{w}'\\) is such that \\(\\str{\\tilde{t}w}'\\)\nis the local reduction of \\(\\str{\\tilde{t}w}\\),\nthen \\(tw'\\) and \\(tw\\) coincide.\n\\end{lemma} \n\nNext, order \\(A^\\pm\\) so that if \\(a,b \\in A^\\pm\\), then \\(a\\) is less than \\(b\\) if every element\nof \\(\\operatorname{dest}(a)\\) is less than every element of \\(\\operatorname{dest}(b)\\).\nOrder \\(\\Lambda\\) with the \\emph{reverse lexicographic order}:\nif \\(\\str{uw}\\) and \\(\\str{vw}\\) are in \\(\\Lambda\\) and the last symbol of\n\\(\\str{u}\\) is less than that of \\(\\str{v}\\), then we declare\n\\(\\str{uw}\\) less than \\(\\str{vw}\\).\nRecall that the \\emph{evaluation map} on \\(\\Lambda\\) is the function\nwhich assigns the value \\(tu\\) to each string \\(\\str{\\tilde{t}u} \\in \\Lambda\\).\n(This is well defined since \\(t \\mapsto \\str{\\tilde{t}}\\) is injective on \\(M\\).)\nThis order is chosen so that the following lemma is true.\n\n\\begin{lemma} \\label{RevLex}\nThe evaluation map defined on \\(\\Lambda\\) is order preserving.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that \\(\\str{uw}\\) and \\(\\str{vw}\\) are in \\(\\Lambda\\) and the last symbol of \n\\(\\str{u}\\) is less than the last symbol of \\(\\str{v}\\).\nObserve that by Lemma \\ref{PingPong}, the evaluation of \\(\\str{u}\\)\nis an element of its destination.\nThus if the destination of \\(\\str{u}\\) is less than the destination of \\(\\str{v}\\),\nthen this is true of their evaluations as well.\nSince \\(t \\mapsto tw\\) is order preserving, we are done.\n\\end{proof}\n\nNow we are ready to prove Theorem \\ref{CombToIso}.\n\\begin{proof}[Proof of Theorem \\ref{CombToIso}]\nAs noted above, we may assume that \\(X\\) and \\(Y\\) are geometrically fast families of\npositive bump functions with isomorphic dynamical diagrams.\nBy Theorem \\ref{Faithful}, we know that \\(\\gen{X} \\restriction \\big(M\\gen{X}\\big)\\)\nis marked isomorphic to \\(\\gen{X}\\); similarly \\(\\gen{Y} \\restriction \\big(M\\gen{Y}\\big)\\) is marked\nisomorphic to \\(\\gen{Y}\\).\nIt therefore suffices to define an order preserving bijection \\(\\theta: M\\gen{X} \\to M\\gen{Y}\\)\nsuch that \\(\\theta(sa) = \\theta(s) \\tau (a)\\), where \\(s \\in M\\gen{X}\\) and \\(a \\in X\\) and\nwhere \\(\\tau : X \\to Y\\) is the bijection induced by the isomorphism of the dynamical diagrams of \\(X\\) and \\(Y\\).\n\nDefine \\(\\mu : M_X \\to M_Y\\) by \\(\\mu(s) = t\\) if \\(s\\) is the marker for \\(a \\in X\\) and \\(t\\) is the marker\nfor \\(\\tau(a) \\in Y\\).\nLet \\(\\lambda\\) denote the translation of local \\(X\\)-words into local \\(Y\\)-words\ninduced by \\(\\mu\\) and \\(\\tau\\).\nDefine \\(\\theta : M\\gen{X} \\to M\\gen{Y}\\) so that \\(\\theta(tu)\\) is the evaluation of \\(\\lambda (\\str{tu})\\)\nfor \\(\\str{\\tilde{t}u} \\in \\Lambda_X\\).\nThis is well defined by Lemmas \\ref{DisjointOrbits}, \\ref{TreeAction}, and \\ref{LambdaLem}.\nBy Lemma \\ref{RevLex} and the fact that\n\\(\\lambda\\) preserves the reverse lexicographic order,\n\\(\\theta\\) is order preserving.\n\nNow suppose that \\(s \\in M\\gen{X}\\) and \\(a \\in X^\\pm\\).\nFix \\(\\str{\\tilde{t}u} \\in \\Lambda_X\\) such that \\(s = tu\\) and let\n\\(\\str{\\tilde{t}v} \\in \\Lambda_X\\) be the local reduction of \\(\\str{\\tilde{t}ua}\\).\nObserve that on one hand \\(\\theta(sa) = \\theta (tv)\\) is the evaluation of \\(\\lambda(\\str{\\tilde{t}v})\\).\nOn the other hand \\(\\theta(s)\\tau(a)\\) is the evaluation of \\(\\lambda(\\str{\\tilde{t}ua})\\).\nSince \\(\\lambda\\) is induced by an isomorphism of dynamical diagrams, it satisfies\nthat \\(\\str{w}'\\) is the local reduction of \\(\\str{w}\\) if and only if \\(\\lambda(\\str{w}')\\) is the local\nreduction of \\(\\lambda(\\str{w})\\).\nIn particular, \\(\\lambda(\\str{\\tilde{t}v})\\) is the local reduction of \\(\\lambda(\\str{\\tilde{t}ua})\\).\nBy Lemma \\ref{LambdaLem}, these local \\(Y\\)-words have the same evaluation and\ncoincide with \\(\\theta(sa)\\) and \\(\\theta(s) \\tau(a)\\), respectively.\nThis completes the proof of Theorem \\ref{CombToIso}.\n\\end{proof}\n\nAs we noted in the introduction Theorem \\ref{CombToIso} has two immediate consequences.\nFirst, geometrically fast sets \\(X = \\{f_i \\mid i < n\\} \\subseteq {\\operatorname{Homeo}_+(I)}\\) with finitely\nmany transition points are \\emph{algebraically fast}:\nif \\(1 \\leq k_i\\) for each \\(i < n\\), then\n\\(\\gen{f_i \\mid i < n}\\) is marked isomorphic to \\(\\gen{f^{k_i}_i \\mid i < n}\\).\nThe reason for this is that the dynamical diagrams associated to\n\\(\\{f_i \\mid i < n\\}\\) and \\(\\{f_i^{k_i} \\mid i < n\\}\\) are isomorphic.\nSecond, since the dynamical diagram of any geometrically fast set with finitely many transition points\ncan be realized by a geometrically fast subset of \\(F\\) (see, e.g., \\cite[Lemma 4.2]{CFP}),\nevery group admitting a finite geometrically fast generating set can be embedded into \\(F\\).\nThe notion of \\emph{history} in the previous section is revisited in Section \\ref{AbstractPingPongSec}\nwhere it is used to prove a relative  of Theorem \\ref{CombToIso}.\n\n\nMore evidence of the restrictive nature of geometrically fast generating sets\ncan be found in \\cite{kim+koberda+lodha} where groups generated by stretched\ntransition chains \\(C\\) as defined in Section \\ref{FastBumpsSec} are\nconsidered under the weaker assumption\n that consecutive pairs of elements in \\(C\\) are geometrically fast.\nGroups generated by such a \\(C\\) with \\(n\\) elements are called\n{\\itshape \\(n\\)-chain groups}.\nIt is proven in\n\\cite{kim+koberda+lodha} that every \\(n\\)-generated subgroup of\n\\({\\operatorname{Homeo}_+(I)}\\) is a subgroup of an \\((n+2)\\)-chain group.\nAnother result of \\cite{kim+koberda+lodha} is that for each\n\\(n\\ge3\\), there are uncountably many isomorphism types of\n\\(n\\)-chain groups. \nBy contrast, Theorem \\ref{CombToIso} (with Corollary \\ref{Excision} below) implies that\nthe number of isomorphism types of groups with finite, geometrically fast generating sets is\ncountable because the number of isomorphism types of dynamical\ndiagrams is countable.\n\n\n\n\\section[semi-conjugacy]{Minimal representations of\ngeometrically fast groups and topological semi-conjugacy}\n\n\\label{SemiConjSec}\n\nTheorem \\ref{CombToIso} partitions the subgroups of \\({\\operatorname{Homeo}_+(I)}\\)\ngenerated by geometrically fast sets with finitely many transition points: two such sets are\nconsidered equivalent if their dynamical diagrams are isomorphic.\nIn this section we show that each class contains a (nonunique)\nrepresentative \\(Y\\) so that for each \\(X\\) in the class there is a\nmarked isomorphism \\(\\phi : \\gen{X} \\to \\gen{Y}\\) which is\ninduced by a semi-conjugacy on \\(I\\).\nSpecifically, the bijection \\(\\theta : M\\gen{X} \\to M\\gen{Y}\\) of Theorem \\ref{CombToIso}\nextends to a continuous order preserving surjection \\(\\hat{\\theta}:I \\to I\\)\nso that for all \\(f\\in \\gen{X}\\) we have \\(f \\hat{\\theta} =\\hat{\\theta}\\phi(f)\\).\nNotice that in this situation, the graph of \\(\\phi(f)\\) is the image of the graph of \\(f\\) under the\ntransformation \\((x,y) \\mapsto (x\\hat\\theta,y\\hat\\theta)\\).\nWe will refer to such a \\(Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) as \\emph{terminal}.\nTheorem \\ref{SemiConjThm} can now be stated as follows.\n\n\\begin{thm}\nEach dynamical diagram \\(D\\) can be realized by a terminal \\(X_D \\subseteq {\\operatorname{PL}_+(I)}\\).\n\\end{thm}\n\n\\begin{proof}\nAs in the proof of Theorem \\ref{CombToIso}, it suffices to prove the theorem\nunder the assumption that all bumps in \\(D\\) are positive and all labels are distinct.\nFurthermore, by Proposition \\ref{IsolatedFixProp}, we may assume that \\(D\\) has no isolated bumps.\nLet \\(n\\) denote the number of bumps of \\(D\\), set \\(\\ell := 1\/(2n)\\) and \n\\[\n\\mathcal{J} : = \\{ [i \\ell, (i+1) \\ell) \\mid 0 \\leq i < 2n \\},\n\\]\nobserving that \\(\\mathcal{J}\\) has the same cardinality as the set of feet of \\(D\\).\nOrder \\(\\mathcal{J}\\) by the order on the left endpoints of its elements.\nIf \\(i < 2n\\), we will say that the \\(i{}^{\\textrm{th}}\\) interval in \\(\\mathcal{J}\\) \\emph{corresponds} to the \\(i{}^{\\textrm{th}}\\) foot\nof \\(D\\).\n\nFor \\(i < n\\), let \\((x_i,s_i)\\) and \\((t_i,y_i)\\) be the intervals in \\(\\mathcal{J}\\) which correspond to the left and right feet of\nthe \\(i{}^{\\textrm{th}}\\) bump of \\(D\\), respectively.\nNote that since \\(D\\) has no isolated bumps, \\(s_i < t_i\\).\nDefine \\(b_i\\) to be the bump which has support \\((x_i,y_i)\\), maps \\(s_i\\) to \\(t_i\\) and is linear on\n\\((x_i,s_i)\\) and \\((s_i,y_i)\\) --- see Figure \\ref{PLBumpFig}.\n\\begin{figure}\n\\[\n\\xy\n(0,0); (30,30)**@{-};\n(3,3); (11,19)**@{-}; (27,27)**@{-}; (11,11); (11,19)**@{.};\n(19,19)**@{.};\n(3,3); (3,1)**@{-}; (3,-1)*{x_i};\n(11,11); (11,9)**@{-}; (11,7)*{s_i};\n(19,19); (19,17)**@{-}; (19,15)*{t_i};\n(27,27); (27,25)**@{-}; (27,23)*{y_i};\n\\endxy\n\\]\n\\caption{The function \\(b_i\\).\\label{PLBumpFig}}\n\\end{figure}\nIf we assign \\(b_i\\) the marker \\(s_i\\), then \nthe feet of \\(b_i\\) are either in \\(\\mathcal{J}\\) or are the interior of an element of \\(\\mathcal{J}\\);\nin particular, the feet of \\(X_D = \\{b_i \\mid i < n\\}\\) are disjoint.\nThus \\(X_D\\) is geometrically fast and has dynamical diagram isomorphic to \\(D\\).\n\nNotice that the feet \\((x_i,s_i)\\) and \\([t_i,y_i)\\) of \\(b_i\\)\nare each intervals of length \\(\\ell\\) contained in\n\\(I\\) while the middle interval \\([s_i,t_i)\\) is of length \\(m\\ell\\) for some positive integer \\(m\\).\nMoreover, since \\(D\\) has no isolated bumps,\nthere is an interval of \\(\\mathcal{J}\\) between \\(s_i\\) and \\(t_i\\);\nin particular, \\(t_i - s_i \\geq \\ell\\).\nIt follows that the slope of the graph of \\(b_i\\)\non its source \\((x_i,s_i)\\) and the slope of the graph of \\(b_i^{-1}\\)\non its source \\([t_i,y_i)\\) are both at least 2.\n\n\n\\begin{claim}\\label{OrbitDensity}\nIf \\(X_D\\) is the set of positive bumps constructed above, then \\(M\\gen{X_D}\\) is dense in \\(I\\).\n\\end{claim}\n\n\\begin{remark}\nNote that we are working under the assumption that \\(D\\) has no isolated elements.\nIf \\(D\\) has isolated bumps, then \\(M\\gen{X_D}\\) can not be dense.\n\\end{remark}\n\n\\begin{proof} \nSince every transition point of \\(X_D\\) is in the closure of \\(M\\gen{X_D}\\),\nit suffices to show that if \\(0 \\leq p < q \\leq 1\\), then \n\\((p,q)f\\) contains an endpoint of an interval in \\(\\mathcal{J}\\) for some \\(f \\in \\gen{X_D}\\).\nThe proof is by induction on the minimum \\(k \\geq 0\\) such that \\(\\ell 2^{-k} < q-p\\).\nObserve that if \\(k = 0\\), then \\(q-p > \\ell\\) and thus \\(q-p\\) must contain an endpoint of\nan interval in \\(\\mathcal{J}\\).\n\nNext observe that if \\((p,q)\\) does not contain an endpoint of an element of \\(\\mathcal{J}\\),\nthen \\((p,q)\\) is contained in the foot of some \\(b_i\\) for \\(i < n\\).\nIf \\((p,q) \\subseteq \\operatorname{src}(b_i)\\), then since the derivative of \\(b_i\\) is at least 2 on it source,\nit follows that  \\((p,q)b_i\\) is at least twice as long as \\((p,q)\\).\nBy our induction hypothesis, there is an \\(f \\in \\gen{X_D}\\) such that\n\\((p,q)b_i f\\) contains an endpoint of \\(\\mathcal{J}\\).\nSimilarly, if \\((p,q) \\subseteq \\operatorname{src}(b_i^{-1})\\), then \\((p,q)b_i^{-1}\\) is at least twice\nas long as \\((p,q)\\) and we can find an \\(f \\in \\gen{Y}\\) such that\n\\((p,q)b_i^{-1} f\\) contains an endpoint of \\(\\mathcal{J}\\).\n\\end{proof}\n\nIn order to see that \\(X_D\\) is terminal, let\n\\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) be geometrically fast, have finitely many transition points, and have a dynamical diagram\nisomorphic to \\(D\\).\nLet \\(\\theta : M\\gen{X} \\to M\\gen{X_D}\\) be order preserving and satisfy that\n\\(t f \\theta = t \\theta \\phi(f)\\) for all \\(f \\in M\\gen{X}\\),\nwhere \\(\\phi : \\gen{X} \\to \\gen{X_D}\\) is the marked isomorphism.\nDefine \\(\\hat \\theta : I \\to I\\) by\n\\[\nx \\hat \\theta := \\sup \\{t\\theta \\mid t \\in M\\gen{X} \\textrm{ and } t \\leq x\\}\n\\]\nwhere we adopt the convention that \\(\\sup \\emptyset = 0\\).\nClearly \\(\\hat \\theta : I \\to I\\) is order preserving and extends \\(\\theta\\).\nIn particular its range contains \\(M\\gen{X_D}\\), which by\nClaim \\ref{OrbitDensity} is dense in \\(I\\).\nIt follows that \\(\\hat \\theta\\) is a continuous surjection (\\emph{any} order preserving map from \\(I\\) to \\(I\\) with\ndense range is a continuous surjection).\nThat \\(x f \\hat \\theta = x \\hat \\theta \\phi (f)\\) follows from\nthe fact that this is true for \\(x \\in M\\gen{X}\\) and from the continuity of \\(f\\) and \\(\\phi(f)\\).\nThis completes the proof that \\(X_D\\) is terminal.\n\\end{proof}\n\n\\section{Fast generating sets for the groups \\(F_n\\)}\n\\label{FnSec}\n\nIn this section we will give explicit generating sets\nfor some well known variations of Thompson's group \\(F\\).\nFirst notice that since \\((0,1)\\) is homeomorphic to \\({\\mathbf R}\\) by an order preserving map,\nall of the analysis of geometrically fast subsets of \\({\\operatorname{Homeo}_+(I)}\\) transfers to \\({\\operatorname{Homeo}_+(\\R)}\\)\n(with the caveat that \\(\\pm \\infty\\) must be considered as possible\ntransition points of elements of \\({\\operatorname{Homeo}_+(\\R)}\\); see Example \\ref{x+1_x^3_ex} below).\nFix an integer \\(n\\ge2\\).\nFor \\(i < n\\), let \\(g_i\\) be a homeomorphism from \\({\\mathbf R}\\) to itself defined by:\n\\[\ntg_i := \\begin{cases}\nt & \\textrm{if } t\\le i \\\\\ni+n(t-i) & \\textrm{if } i\\le t\\le i+1 \\\\\nt+(n-1) & \\textrm{if } i+1 \\le t.\n\\end{cases}\n\\]\nIn words, \\(g_i\\) is the identity below \\(i\\), has constant slope\n\\(n\\) on the interval \\([i,i+1]\\), and is translation by \\(n-1\\)\nabove \\(i+1\\).\nWe will use \\(F_n\\) to denote \\(\\gen{g_i \\mid i < n}\\).\nThe group \\(F_2\\) is one of the standard\nrepresentations of Thompson's group \\(F\\).\n(The more common representation of \\(F\\) is as a set of piecewise linear\nhomeomorphisms of the unit interval \\cite[\\S1]{CFP}.)\n\nThe groups \\(F_n\\) \\((n \\ge 2)\\)  are discussed in\n\\cite[\\S4]{brown:finiteprop} where \\(F_n\\) is denoted\n\\(F_{n,\\infty}\\), and in \\cite[\\S2]{brin+fer} where \\(F_n\\) is\ndenoted \\(F_{n,0}\\).\nThe standard infinite presentation of \\(F_n\\) is given in \\cite[Cor. 2.1.5.1]{brin+fer}.\nIt follows easily from that\npresentation that the commutator quotient of \\(F_n\\) is a free abelian group of rank \\(n\\).\nIn particular the \\(F_n\\)'s are pairwise nonisomorphic.\n\nWe now describe an alternate generating set for \\(F_n\\) which \nconsists of \\(n\\) positive  bump functions and is geometrically fast.\nFor \\(i<n-1\\), set \\(h_i : = g_i g_{i+1}^{-1}\\) and let\n\\(h_{n-1}\\) denote \\(g_{n-1}\\).\nIt is clear that \\(C = \\{h_i \\mid i<n\\}\\) generates \\(F_n\\).\nWe claim that \\(C\\) is a geometrically fast stretched transition chain.\nIf \\(i < n-1\\), then \\(h_i\\) is the identity outside \\([i,i+2]\\).\nOn that interval it is a positive, one bump function since chain\nrule considerations show that \\(h_i\\) has slope \\(n\\) on\n\\([i,i+\\frac 1 n]\\), slope one on \\([i+\\frac 1 n, i+1]\\) and slope\n\\(1\/n\\) on \\([i+1, i+2]\\). \nThus the support of \\(h_i\\) is \\((i,i+2)\\) and\nthe support of \\(h_{n-1}\\) is \\((n-1, \\infty)\\).\nIn particular, \\(C\\) forms a stretched transition chain.\n\nNext we will show that \\(C\\) is geometrically fast.\nObserve that for \\(0 < j \\leq n\\) that we have:\n\\[ \n\\left( i+\\frac{j}{n} \\right) h_i\n= \n(i+1) + \\frac{j-1}{n}.\n\\]\n\nFor \\(0\\le i\\le n\\) set\n\\(t_i := i+ \\frac{n-i}{n}\\).\nIt follows from the above computation that for \\(0\\le i<n-1\\), we\nhave \\(t_ih_i = t_{i+1}\\) so that \\(t_0=1\\) and \\(t_n=n\\).  This\nmakes \\(t_0\\) the leftmost transition point in the support of\n\\(h_0\\) and \\(t_n\\) the rightmost transition point in the support of\n\\(h_{n-1}\\).\nBy Proposition \\ref{FastCriterion}, \\(C\\) is geometrically fast.\nCombining this with Theorem \\ref{CombToIso}, we now have that\nany geometrically fast stretched transition chain generates a copy of \\(F_n\\).\n\n\\begin{remark}\nThe above proof that \\(F_n\\) is isomorphic to the group generated by a geometrically fast\nstretched transition chain of length \\(n\\) is not an efficient\nway to reach this conclusion.\nThere is a more straightforward\nargument based on the standard presentation of \\(F_n\\) as indicated\nin the proof of \\cite[Proposition 1.11]{kim+koberda+lodha}.\nWe included this example as an introduction to the\nvariety of isomorphism classes in groups generated by geometrically fast sets.  \n\\end{remark}\n\n\\section{Excision of extraneous bumps in fast generating sets}\n\n\\label{ExcisionSec}\n\nSometimes fast generating sets use bumps which do not affect the marked isomorphism type\nof the resulting group.\nThis section gives a sufficient condition for when those bumps\ncan be excised while preserving the marked isomorphism type.\nWe make no finiteness assumptions in this section.\n\nIf \\(f \\in {\\operatorname{Homeo}_+(I)}\\) and \\(E\\) is a set of positive bumps, then we define \\(f\/E \\in {\\operatorname{Homeo}_+(I)}\\) to be the function\nwhich agrees with \\(f\\) on  \n\\[\nI \\setminus \\bigcup \\{\\operatorname{supt} (a) : a \\in E \\textrm{ and } a \\textrm{ is used in } f\\}\n\\]\nand is the identity elsewhere.\nIf \\(X\\) is a fast generating set and \\(E\\) is a set of positive bumps, we \ndefine \\(X\/E := \\{f\/E : f \\in X\\}\\).\n\nNow let \\(X\\) be a fast generating set and let \\(J \\subseteq I\\) be an interval.\nWe say a set \\(E\\) of bumps is \\emph{extraneous in \\(X\\) as witnessed by \\(J\\)} if\nfor some \\(f \\in X\\):\n\\begin{itemize}\n\n\\item every element of \\(E\\) is an isolated bump used in \\(f\\) whose\nsupport is contained in \\(J\\);\n\n\\item there is a bump \\(a\\) used in \\(f\\) not in \\(E\\) such that \\(J\\) contains a foot of \\(a\\);\n\n\\item \\(J\\) is disjoint from the feet of all \\(g \\in X \\setminus \\{f\\}\\).\n\n\\end{itemize}\nObserve that if \\(E\\) is extraneous in \\(X\\), then \\(g \\mapsto g\/E\\) preserves the canonical ordering of \\(X\\).\n\n\\begin{thm} \\label{Excision}\nIf \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is a (possibly infinite) geometrically fast set and\n\\(E\\) is an extraneous set of bumps used in \\(X\\),\nthen the map \\(g \\mapsto g\/E\\) extends to an isomorphism between \\(\\gen{X}\\) and \\(\\gen{X\/E}\\).\n\\end{thm}\n\n\n\\begin{proof}\nFix \\(X\\) and \\(E\\) as in the statement of the theorem and\nlet \\(J\\) be an interval witnessing that \\(E\\) is extraneous\nand \\(f \\in X\\) be the element of \\(X\\) such that the elements of \\(E\\) occur in \\(f\\).\nLet \\(A\\) denote the set of bumps used in \\(X\\),\n\\(M\\) denote \\(M_X\\), and \\(\\widetilde M\\) denote \\(M_{X\/E}\\).\nSet \\(S :=M \\gen{X} = M \\gen{A}\\) and\n\\(\\widetilde S := \\widetilde M \\gen{X\/E} = \\widetilde M \\gen{A \\setminus E}\\).\nObserve that since the elements of \\(E\\) are isolated in \\(A\\) and since\nevery element of \\(\\widetilde M \\gen{A \\setminus E} \\setminus \\widetilde M\\) is in \\(\\operatorname{dest}(b)\\) for some\n\\(b \\in (A \\setminus E)^{\\pm}\\), it follows that \\(\\widetilde M \\gen{A \\setminus E} = \\widetilde M \\gen{A}\\).\nIn particular, \\(\\widetilde S\\) is \\(\\gen{A}\\)-invariant and hence \\(g \\mapsto g \\restriction \\widetilde S\\)\ndefines a homomorphism of \\(\\gen{X}\\) to \\(\\gen{X} \\restriction \\tilde S\\).\n\nRecall that, by Theorem \\ref{Faithful}, \\(g \\mapsto g \\restriction S\\) defines an isomorphism\nbetween \\(\\gen{X}\\) and \\(\\gen{X} \\restriction S\\).\nSimilarly, \\(g\/E \\mapsto g\/E \\restriction \\widetilde S\\) defines an isomorphism between \\(\\gen{X\/E}\\)\nand \\(\\gen{X\/E} \\restriction \\widetilde S\\).\nSince \\(g \\restriction \\widetilde S = (g\/E) \\restriction \\widetilde S\\), \nit suffices to show that \\(g \\mapsto g \\restriction \\widetilde S\\) is injective on \\(\\gen{X}\\).\nThat is, if \\(\\str{w}\\) is an \\(X\\)-word, \\(s\\) is a marker for an \\(a \\in E\\),\nand \\(suw \\ne su\\) for some \\(X\\)-word \\(\\str{u}\\),\nthen there is a \\(t \\in \\widetilde S\\) such that \\(tuw \\ne tu\\).\nEquivalently we need to show that for every marker \\(s\\) of an \\(a \\in E\\) and \\(X\\)-word \\(\\str{w}\\),\nif  \\(sw \\ne s\\), then there is a \\(t \\in \\widetilde S\\) with \\(tw \\ne t\\).\nTo this end let such \\(s\\), \\(a\\), and \\(\\str{w}\\) be given with \\(sw \\ne s\\).\n\n\\begin{claim}\nIf there is an \\(x \\in \\operatorname{supt}(a)\\) such that \\(xw \\not \\in \\operatorname{supt}(a)\\), then there is a \\(t \\in \\widetilde S\\)\nsuch that \\(tw \\ne t\\).\n\\end{claim}\n\n\\begin{proof}\nSuppose that \\(xw \\not \\in \\operatorname{supt}(a)\\) for some \\(x \\in \\operatorname{supt}(a)\\).\nLet \\(\\str{uv}\\) be a locally reduced at \\(x\\) such that \\(xuv = xw\\) with \\(\\str{u}\\) maximal\nsuch that \\(xu \\in \\operatorname{supt}(a)\\);\nnotice that \\(\\str{v} \\ne \\varepsilon\\).\nSince \\(a\\) is isolated, \\(xu \\not \\in \\operatorname{src}(\\str{v})\\) and thus by Lemma \\ref{PingPong}\n\\(xuv \\in \\operatorname{dest}(\\str{v})\\).\nIt follows that \\(w\\) must move an endpoint of \\(\\operatorname{dest}(\\str{v})\\), both of which are in\nthe closure of \\(\\widetilde S \\cap \\operatorname{dest}(\\str{v})\\).\nHence \\(w\\) moves an element of \\(\\widetilde S\\). \n\\end{proof}\n\nWe may therefore assume that \\(sw \\ne s\\) but that \\(w\\) maps \\(\\operatorname{supt}(a)\\) to \\(\\operatorname{supt}(a)\\).\nLet \\(\\str{w} = \\prod_{i < n} \\str{w}_i\\) be a factorization of \\(\\str{w}\\) into minimal\npositive length words which define maps from \\(\\operatorname{supt}(a)\\) into \\(\\operatorname{supt}(a)\\).\nLet \\(b\\) be the bump used in \\(f\\) with \\(b \\not \\in E\\) and such that\n\\(J\\) contains a foot of \\(b\\).\nLet \\(t \\in J \\cap \\widetilde S\\) be sufficiently close to the transition point of \\(b\\) which is in \\(J\\)\nsuch that for all \\(k \\in {\\mathbf Z}\\) with \\(|k| \\leq |\\str{w}|\\), \\(t f^k \\in J\\).\n\n\\begin{claim}\nFor all \\(i < n\\), there is a \\(p \\in \\{1,0,-1\\}\\) such that\n \\(w_i \\restriction J = f^p \\restriction J\\).\n\\end{claim}\n\n\\begin{proof}\nIf \\(\\str{w}_i\\) begins with \\(f\\) or \\(f^{-1}\\), then it must have length \\(1\\) and there is nothing to\nshow.\nIf \\(\\str{w}_i\\) begins with \\(\\str{g} \\ne \\str{f}^{\\pm 1}\\),\nthen either \\(J\\) is contained in or disjoint from the support of\n\\(g\\) and it is disjoint from all of the feet of \\(g\\).\nIf \\(g \\restriction J\\) is the identity, then again \\(\\str{w}_i\\) has length 1.\n\nNow suppose that \\(J\\) is contained in the support of \\(g\\) where \\(g\\) is neither \\(f\\) nor \\(f^{-1}\\) and\nhas \\(s\\) in its support.\nNotice that since \\(J\\) is disjoint from the feet of \\(g\\),\n\\(J\\) is contained in the support of a single bump \\(c \\in A^\\pm\\) of \\(g\\).\n\nWe will now show that \\(sw_i = s\\).\nLet \\(\\str{gv}\\) be a minimal prefix of \\(\\str{w}_i\\) such that \\(sgv \\in \\operatorname{supt}(a)\\).\nBy Lemmas \\ref{FellowTraveler1} and \\ref{X_to_A}, there is an \\(A\\)-word \\(\\str{u}\\) which \nis locally reduced on \\(Jc = Jg\\) and such that \\(u\\) and \\(v\\) coincide on \\(Jg\\).\nSince \\(\\str{c}\\) is locally reduced at \\(s\\) and \\(\\str{u}\\) is locally reduced at \\(sc\\),\nthe free reduction of \\(\\str{cu}\\) is locally reduced at \\(s\\).\nIf this free reduction is not \\(\\varepsilon\\), then it must be that \\(\\str{cu}\\) was freely reduced and\n\\(scu \\in \\operatorname{dest}(\\str{cu})\\).\nSince \\(scu = sgv \\in \\operatorname{supt}(a)\\), \\(\\str{cu}\\) would have to end in \\(\\str{a}\\) or \\(\\str{a}^{-1}\\).\nBut then if \\(\\str{u}'\\) is the result of deleting the last symbol of \\(\\str{u}\\),\nthen \\(scu'\\) is in \\(\\operatorname{supt}(a)\\) and, by Lemma \\ref{FindPrefix}, coincides with\n\\(sgv'\\) for some proper prefix \\(\\str{v}'\\) of \\(\\str{v}\\).\nThis contradicts the minimal choice of \\(\\str{gv}\\).\nIt must therefore have been that \\(\\str{u} = \\str{c}^{-1}\\).\nIt follows that \\(c^{-1}\\) and \\(v\\) coincide on \\(Jg\\) and thus\n\\(gv\\) is the identity on \\(J\\).\nBy minimality of \\(\\str{w}_i\\), \\(\\str{w}_i = \\str{v}\\) and thus \\(\\str{w}_i\\) is the identity on \\(J\\). \n\\end{proof}\n\nApplying the claim, there are \\(p_i \\in \\{1,0,-1\\}\\) for \\(i < n\\) such that:\n\\[\nsw = s\\prod_{i < n} f^{p_i}\n\\qquad \\qquad \ntw = t \\prod_{i < n} f^{p_i}.\n\\]\nIn particular, since \\(sw \\ne w\\), it follows that \\(tw \\ne t\\).\n\\end{proof}\n\n\\begin{cor}\nIf \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\) is geometrically fast and finite, then there is a\ngeometrically fast \\(Y \\subseteq {\\operatorname{Homeo}_+(I)}\\) which has finitely many transition points such that\n\\(\\gen{X}\\) is marked isomorphic to \\(\\gen{Y}\\).\n\\end{cor}\n\n\\begin{proof}\nLet \\(X\\) be given and let \\(\\mathcal{J}\\)  consist\nof the maximal intervals \\(J \\subseteq I\\) such that for some \\(f \\in X\\) (dependent on \\(J\\)):\n\\begin{itemize}\n\n\\item \\(J\\) contains at least one transition point of \\(f\\);\n\n\\item \\(J\\) contains no transition points of \\(X \\setminus \\{f\\}\\);\n\n\\item \\(J\\) is disjoint from the feet of \\(X \\setminus \\{f\\}\\).\n\n\\end{itemize}\nObserve that since \\(X\\) is geometrically proper, \\(\\mathcal{J}\\) is finite.\nThe proof is by induction on the number of elements of \\(\\mathcal{J}\\) which contain infinitely many\ntransition points of \\(X\\).\nSuppose that \\(J \\in \\mathcal{J}\\) contains infinitely many transition points of some \\(f \\in X\\).\nLet \\(E\\) consist of all but one of the isolated bumps of \\(f\\) with support contained in \\(J\\).\nNotice that \\(J\\) witnesses that \\(E\\) is extraneous in \\(X\\).\nBy Theorem \\ref{Excision}, \\(\\gen{X\/E}\\) is marked isomorphic to \\(\\gen{X}\\).\nWe are now finished by our induction hypothesis.\n\\end{proof}\n\n\\section[Geometrically proper generating sets]{The existence of geometrically proper generating sets}\n\n\\label{GeoProperSec}\n\nIn this section we consider the question of when finitely generated subgroups of\n\\(F\\) admit geometrically proper generating sets.\nOur first task will be to prove:\n\n\\begin{customthm}{\\ref{GeoProperGen}}\nEvery \\(n\\)-generated one orbital subgroup of \\({\\operatorname{PL}_+(I)}\\)\neither contains an isomorphic copy of \\(F\\) or else admits an\n\\(n\\)-element geometrically proper generating set.\n\\end{customthm}\n\\noindent\nIn this section we will use the standard embedding of Thompson's group \\(F\\) in \\({\\operatorname{PL}_+(I)}\\)\nand we will make use of the homomorphism \\(\\pi:{\\operatorname{PL}_+(I)} \\rightarrow {\\mathbf R} \\times {\\mathbf R}\\)\ndefined by \\(\\pi(f) := \\big(\\log_2 \\left(f'(0)\\right),\\log_2\\left(f'(1)\\right)\\big)\\).\nIf \\(G\\) is a subgroup of \\({\\operatorname{PL}_+(I)}\\), we will write \\(\\pi_G\\) for \\(\\pi \\restriction G\\).\nIt is well known that \\(F'\\) is exactly\nthe kernel of \\(\\pi_F\\) and that \\(F'\\) is simple.\nWe will need the following lemma which combines the main result of \\cite{brin:ubiq} and\nLemma 3.11 of \\cite{MR2383051}.\n\n\\begin{lemma} \\label{CyclicGerm}\nLet \\(G\\) be a finitely generated subgroup of \\({\\operatorname{PL}_+(I)}\\) with connected support into which \n\\(F\\) does not embed.\nThe image of the homomorphism \\(\\pi_G\\) is either trivial or cyclic.\n\\end{lemma}\n\n\n\\begin{proof}[Proof of Theorem \\ref{GeoProperGen}]\nLet \\(G\\) be a one orbital subgroup of \\({\\operatorname{PL}_+(I)}\\) which\nis generated by a finite set \\(X\\).\nBy conjugating by an appropriate affine homeomorphism of \\({\\mathbf R}\\),\nwe may assume that the support of \\(G\\) is \\((0,1)\\).\nWe assume that \\(F\\) does not embed in \\(G\\).\nBy Lemma \\ref{CyclicGerm}, the image of\n\\(\\pi_G\\) must then be isomorphic to \\({\\mathbf Z}\\).\nLet \\(\\varpi\\) be the composition of \\(\\pi_G\\) with this isomorphism.\nIf there is more than one element of \\(X\\) not in the\nkernel of \\(\\varpi\\), then we apply the Euclidean algorithm and\nrepeatedly reduce \\(\\sum \\left\\{ |\\varpi(f)| \\mid f \\in X\\right\\}\\) by replacing at each stage some\n\\(g\\in X\\) by \\(gh^\\epsilon\\), for some suitably chosen \\(h \\in X\\) and \\(\\epsilon\\in\\{-1,1\\}\\), where  \\(|\\varpi(h)|\\le |\\varpi(g)|\\).\nNote that this\nreplacement does not change the cardinality of \\(X\\).  Thus we can\nassume that there is only one element \\(f\\) of \\(X\\) not in the\nkernel of \\(\\varpi\\).\n\nIf \\(X = \\{g_i \\mid i < n\\}\\), define \\(N(X)\\) to be the number of\npairs \\((j,x)\\) such that for some \\(i < j\\) either \\(x\\) is left\ntransition point of both \\(g_i\\) and \\(g_j\\) or a right transition\npoint of both \\(g_i\\) and \\(g_j\\).\nObserve that \\(N(X)\\) is finite since \\(X\\) has only\nfinitely many transition points and that \\(X\\) is geometrically\nproper precisely if \\(N(X) = 0\\).\nNotice also that if \\((j,x)\\) is a pair counted by \\(N(X)\\),\nthen \\(x\\) can be neither 0 nor 1.\n\nAssume that \\(N(X) > 0\\), and let \\(x\\) be a point that contributes\nto \\(N(X)\\) with \\(g\\ne h\\) two functions in \\(X\\) with \\(x\\) a\ncommon left transition point or common right transition point of\nboth \\(g\\) and \\(h\\).\nSince \\(x\\in (0,1)\\), our assumption on the support of \\(G\\)\nimplies that there is an \\(f\\in X\\) with \\(xf \\ne x\\).\nLet \\(T\\) be the set of transition points of\n\\(g\\) that are moved by \\(f\\).\nSince each point in \\(T\\) is moved to infinitely\nmany places by powers of \\(f\\) and since the set of\ntransition points of elements of \\(X\\) is finite, we can find a\npower \\(f^k\\) of \\(f\\) so that no element of \\(T{f^k}\\) is a\ntransition point of an element of \\(X\\). \nLet \\(X'\\) be obtained from \\(X\\) by replacing \\(g\\) by \\(g^{f^k}\\).\nWe have arranged that \\(|X'|=|X|\\),\nthat \\(\\gen{X'} = \\gen{X}=G\\),\nthat there is only one element of \\(X'\\) outside the kernel of \\(\\varpi\\),\nand that \\(N(X')<N(X)\\).\nBy induction \\(\\gen{X'} = \\gen{X}\\)\nadmits a geometrically proper generating set.\n\\end{proof}\n\nThe assumption that the support of \\(G\\) be connected in Theorem \\ref{GeoProperGen}\nturns out to be necessary as the next example shows.\nBefore proceeding we will develop some notation which will be helpful in proving\nTheorem \\ref{NoGeoProperGen} as well.\nFor the remainder of the section, fix two elements \\(a\\) and \\(b\\) of \\(F\\) which are\none bump functions whose supports are, respectively, \\(\\left(0,\\frac{1}{2}\\right)\\)\nand \\(\\left(\\frac{1}{2}, 1\\right)\\) and which satisfy \\(a'(0)=b'(1)=2\\).\nNotice that \\(a\\) is a positive bump and \\(b\\) is a negative bump.\n\n\\begin{example}\nThe group\n\\(\\gen{ ab, a^{-1} b }\\) is isomorphic to \\({\\mathbf Z}\\times{\\mathbf Z}\\) but has\nno geometrically proper generating set.\nWe leave the details as an exercise for the reader.\n\\end{example}\n\n\\begin{example} \\label{x+1_x^3_ex}\nIt was shown in \\cite{x+1_x^p_free} that\nsubgroup \\(\\gen{t+1,t^3}\\) of \\({\\operatorname{Homeo}_+(\\R)}\\)\nis free.\nSince any minimal generating set for a free group is a free basis for the group,\nany minimal generating set is algebraically fast.\nOn the other hand this group is not embeddable\ninto \\(F\\) by \\cite{brin+squier} and hence by Corollary \\ref{EmbeddInF}\nis not isomorphic to a group with a geometrically proper generating set.\n\\end{example}\n\nNext we will prove Theorem \\ref{NoGeoProperGen},\nwhich shows that the ``\\(F\\)-less'' hypothesis in Theorem \\ref{GeoProperGen} can not be removed.\nNote that no assumption is made in this theorem concerning the number of bumps used in \nthe geometrically proper generating set.\n\n\\begin{customthm}{\\ref{NoGeoProperGen}}\nIf a finite index subgroup of \\(F\\) is isomorphic to \\(\\gen{X}\\) for some  geometrically proper\n\\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\), then it is isomorphic to \\(F\\).\n\\end{customthm}\n\n\\begin{proof}\nLet \\(G\\) be a subgroup of \\(F\\) of finite index such that\n\\(G \\cong \\gen{X}\\) for some finite geometrically proper \\(X \\subseteq {\\operatorname{Homeo}_+(I)}\\).\nLet \\(H\\) denote \\(\\gen{X}\\) and \\(\\phi : G \\to H\\) be a fixed isomorphism.\nWe must show that \\(G\\) is isomorphic to \\(F\\).\n\nBy \\cite{bleak+wassink} we know that, since \\(G \\leq F\\) is of\nfinite index, it is\nof the form \\(\\pi^{-1}(K)\\) for some finite\nindex subgroup \\(K\\) of \\({\\mathbf Z}\\times{\\mathbf Z}\\).\nFurthermore, also by \\cite{bleak+wassink},\n\\(\\pi^{-1}(K)\\) is isomorphic to \\(F\\) if and only if \\(K\\) admits a generating set of the form \\(\\left\\{\\left(p,0\\right),\\left(0,q\\right)\\right\\}\\) for\n\\(p,\\,q\\in{\\mathbf Z}\\setminus\\{0\\}\\).\n\nObserve in any case that \\(K\\) admits a two element generating\nset \\(\\{(p_i,q_i) \\mid  i < 2\\}\\), as it is finite index in \\({\\mathbf Z}\\times{\\mathbf Z}\\).\nWe will first derive some properties of \\(G\\) which come from viewing it\nas a subgroup of \\(F\\).\nDefine \\(f_i := a^{p_i} b^{q_i}\\).\nIt is shown in \\cite{bleak+wassink} that \\(G\\) can be generated by\n\\(\\{f_ i \\mid i < 2\\} \\cup F'\\).\nFor \\(0\\le x<y\\le 1\\) with both \\(x\\) and \\(y\\) in\n\\({\\mathbf Z}[1\/2]\\) we let \\(F_{[x,y]}\\) denote the subgroup of \\(F\\)\nconsisting of those elements of \\(F\\) whose support is contained in\n\\([x,y]\\).  It is standard that \\(F_{[x,y]}\\cong F\\) and that if\n\\(x<w<y<z\\), then \\(F_{[x,y]}\\cup F_{[w,z]}\\) generates\n\\(F_{[x,z]}\\).\n\nNext we claim that the kernel of \\(\\pi_G\\) is \\(G' = F'\\).\nIt is trivial that \\(F'\\subseteq G\\) and since \\(F'\\) is simple and not abelian\nit follows that we also have \\(F'\\subseteq G'\\).\nOn the other hand, since \\(G\\subseteq F\\) we have \\(G'\\subseteq F'\\).\nSince the image of \\(\\pi_G\\) is abelian, we have\n\\(G'\\) is contained in the kernel of \\(\\pi_G\\).\nLastly, \\(\\ker(\\pi_G) \\subseteq \\ker(\\pi) \\subseteq F'\\). \n\n\\begin{claim} \\label{BothGermsNontrivial}\nIf \\(g \\in G\\) is such that neither coordinate of \\(\\pi (g)\\)\nis \\(0\\), then there is a finite subset \\(T\\) of \\(G'\\) with\n\\(G' \\subseteq \\gen{T \\cup\\{g\\}}\\).\n\\end{claim}\n\n\\begin{proof}\nNotice that there is an\n\\(\\epsilon>0\\) so that \\(g\\) moves all points in \\((0,\\epsilon)\\)\nand \\((1-\\epsilon,1)\\).  It follows that for some \\(x<y\\) in\n\\({\\mathbf Z}[1\/2]\\) the images of \\((x,y)\\) under integral powers of \\(g\\)\ncover \\((0,1)\\).\nHence the conjugates of \\(F_{[x,y]}\\) under integral powers of \\(g\\) generate \\(F'=G'\\).\nThe claim now follows from the fact that\n\\(F_{[x,y]} \\cong F\\) is generated by two elements.\n\\end{proof}\n\nIn what follows, we will refer to the closure of the support of an \\(f\\in{\\operatorname{Homeo}_+(I)}\\)\nas the \\emph{extended support of \\(f\\)}.\n\n\\begin{claim} \\label{EquivCommute}\nSuppose \\(g \\in G \\setminus G'\\) has connected extended support and \\(h \\in G\\).\nIf \\(g\\) commutes with \\(g^h\\) then \\(g\\) commutes with \\(h\\).\n\\end{claim}\n\n\\begin{proof}\nWe know that 0 or 1 is in the extended support of \\(g\\) and both 0 and 1 are fixed by \\(h\\). \nSince \\(g\\) has only finitely many bumps, it has finitely many transition points.\nIf any of these are moved by \\(h\\), then at least one fixed point of\n\\(g^h\\) is moved by \\(g\\) or at least one fixed point of \\(g\\) is\nmoved by \\(g^h\\).\nIn either case this implies that \\(g\\) does not commute with \\(g^h\\).\nIf no transition point of \\(g\\) is moved by \\(h\\), then the orbitals of \\(g^h\\) are\nthose of \\(g\\).\nIt follows from the chain rule that on each\norbital \\(J\\) of \\(g\\), \\(g\\) and \\(g^h\\) agree on a small neighborhood of the\nendpoints of \\(J\\).\nBy \n\\cite[\\S 4]{MR1855112} two elements of \\({\\operatorname{PL}_+(I)}\\) commute over a\ncommon orbital of support only if they admit a common root over that orbital.\nIn particular, if \\(g \\restriction J\\) and \\(g^h \\restriction J\\)\nagree in a neighborhood of the endpoints of \\(J\\)\nand they commute, then they must be equal.  Hence \\(h\\) commutes with \\(g\\) over each orbital of support of \\(g\\), so \\(h\\) and \\(g\\) commute.\n\\end{proof}\n\nWe now turn our attention to our representation of \\(G\\) as\na subgroup \\(H=\\gen{X}\\) of \\({\\operatorname{Homeo}_+(I)}\\) where \\(X\\) is geometrically proper.\nIf \\(H\\) has more than one component of support, then the\nrestriction to each is a quotient of \\(H\\).\nSince no non-trivial element of \\(F\\) commutes with every element of \\(F'\\) and since \\(F'=G'\\) is simple,\nit follows that every\nnontrivial normal subgroup of \\(G\\) contains \\(G'\\).\nConsequently every proper quotient of \\(H\\cong G\\) is abelian.\nThus if no restriction were an isomorphism, \\(H\\) would be abelian (which is absurd).\nWe can thus replace \\(H\\) by its restriction to a component of its support on which the\nrestriction is faithful, and further, \nwe can conjugate by a homeomorphism of \\({\\mathbf R}\\)\nso that the support is \\((0,1)\\).  Note that this new embedding of \\(G\\) in \\({\\operatorname{Homeo}_+(I)}\\) is geometrically proper if the original embedding is geometrically proper.\n\nLet \\(\\pi_0\\) denote the restriction of the germ homomorphism \\(\\gamma_0^+\\) to \\(H\\) and \n\\(\\pi_1\\) denote the restriction of \\(\\gamma_1^-\\) to \\(H\\);\ndefine \\(\\pi_H(h) := (\\pi_0(h),\\pi_1(h))\\).\nObserve that since \\(X\\) is geometrically proper we have that\nfor each \\(i < 2\\), there is at most one\nelement of \\(X\\) which is not in the kernel of \\(\\pi_i\\).\nObserve that this implies the image of \\(\\pi_H\\) is abelian and hence\n\\(H' \\subseteq \\ker (\\pi_H)\\).\n\n\\begin{claim} \\label{NonCyclicImage}\nThe image of \\(\\pi_H\\) is not cyclic.\n\\end{claim}\n\n\\begin{proof}\nRecall that \\(G' = \\ker (\\pi_G)\\) and \\(H' \\subseteq \\ker(\\pi_H)\\).\nIn particular, \\(\\phi\\) induces a well defined homomorphism\nfrom \\(G\/G'\\) to \\(H\/\\ker (\\pi_H)\\).\nIf the image of \\(\\pi_H\\) were cyclic, then this homomorphism would\nhave a nontrivial kernel. \n\nPick \\(g:=f_0^m f_1^n \\in G\\setminus G'\\) in the kernel of \\(\\pi_H\\).\nThe element \\(g\\) will have connected extended support.\nAs \\(\\phi(g)\\in\\ker(\\pi_H)\\) we must have the support of \\(\\phi(g)\\) is contained in \\([x,y]\\) for\nsome \\(0 < x < y < 1\\).  As the support of \\(H\\) is \\((0,1)\\) there is an \\(h \\in H\\) so that \\(xh>y\\), and for this \\(h\\) we have \n\\(\\phi(g)\\) commutes with \\(\\phi(g)^h\\), but not with \\(h\\).\nNow, \\(g\\) commutes with \\(\\phi^{-1} (\\phi(g)^h)\\), but not with \\(\\phi^{-1}(h)\\), however, \\(\\phi^{-1}(\\phi(g)^h) = g^{\\phi^{-1}(h)}\\),\ncontradicting Claim \\ref{EquivCommute}.\n\\end{proof}\n\nAt this point we know that, by the geometric properness of \\(X\\), for each coordinate \\(i\\in\\{0,1\\}\\)\nthere is a unique element \\(h_i\\in X\\) so that\n\\(i\\) is in the extended support of \\(h_i\\) and so that \\(h_0\\ne h_1\\).\nThe fact that \\(h_0\\) and \\(h_1\\) are distinct and unique implies that\nthe image of \\(\\pi_H\\) is the product of the images of \\(\\pi_0\\) and \\(\\pi_1\\).\n\n\\begin{claim}\\label{claim_noFiniteGenSet}\nFor each element \\(h\\) of \\(X\\) whose extended support contains 0 or 1,\nthere is no finite subset \\(T\\) of the kernel of \\(\\pi_{H}\\) so that\n\\(\\ker(\\pi_{H}) \\subseteq \\gen{T\\cup \\{h\\}}\\).\n\\end{claim}\n\n\\begin{proof}\nLet \\(h\\) be given and suppose without loss of generality that \\(0\\) is\nin the extended support of \\(h\\).\nAs noted above, \\(1\\) can not be in the extended support of \\(h\\). \nSince \\(\\gen{X}'\\subseteq \\ker(\\pi_{H})\\),\nthere are nontrivial elements in \\(\\ker(\\pi_{H})\\).\nBecause the orbital of \\(H\\) is \\((0,1)\\),\nthere are points arbitrarily close to 0 and 1 moved by elements of\n\\(\\ker (\\pi_{H})\\).\nIt follows that if \\(T\\subseteq\n\\ker(\\pi_{H})\\) is finite, then there is a neighborhood of \\(1\\)\nfixed by all elements of \\(\\gen{T\\cup \\{h\\}}\\) and thus\n\\(\\gen{T\\cup \\{h\\}}\\) cannot contain all of \\(\\ker(\\pi_{H})\\).\n\\end{proof}\n\nIn order to finish the proof, it suffices to show that \\(\\pi_G\\left(\\phi^{-1}(h_0)\\right)\\) and\n\\(\\pi_G\\left(\\phi^{-1}(h_1)\\right)\\) generate the image of \\(\\pi_G\\) and that\neach have exactly one (necessarily different) nonzero coordinate. \nSince, by the proof of Claim \\ref{NonCyclicImage},\n\\(\\phi\\) induces an isomorphism \\(G\/G' \\cong H\/\\ker (\\pi_H)\\), it follows that\n\\(\\{\\pi_G(\\phi^{-1}(h_i)) \\mid i < 2\\}\\) must generate \\(G\/G'\\).\nOn the other hand, by Claims \\ref{BothGermsNontrivial} and \\ref{claim_noFiniteGenSet},\nit must be that one coordinate\nof \\(\\pi_G(\\phi^{-1}(h_i))\\) must be 0 for each \\(i \\in \\{0,1\\}\\).\nThis shows that these two elements, which generate the group \\(K\\), together make a set of the form \\(\\left\\{(p,0),(0,q)\\right\\}\\) for some non-zero \\(p\\) and \\(q\\), and therefore  \\(G \\cong F\\).\n\\end{proof}\n\n\\begin{remark} \nThe group \\(E=\\{(p,q)\\in{\\mathbf Z}\\times{\\mathbf Z}\\mid p+q \\equiv0 \\mod2\\}\\)\nis a subgroup of \\({\\mathbf Z} \\times {\\mathbf Z}\\) which is not of the form \\(P \\times Q\\) and hence\n\\(\\pi^{-1}_F (E)\\) is a finite index subgroup of \\(F\\) which is not isomorphic to \\(F\\).\nIn particular, there are finite index subgroups of \\(F\\) which do not admit\ngeometrically proper generating sets.\n\\end{remark}\n\n\\section{Abstract Ping-Pong Systems}\n\n\\label{AbstractPingPongSec}\n\nIn this section we will abstract the analysis of geometrically\nfast systems of bumps in previous sections to\nthe setting of permutations of a set \\(S\\).\n(By \\emph{permutation} of \\(S\\) we simply mean a bijection from \\(S\\) to \\(S\\).)\nOur goal will be to state the analog of Theorem \\ref{CombToIso} and its consequences.\nThe proofs are an exercise for the reader.\n\nSuppose now that \\(A\\) is a collection of permutations of a set \\(S\\) such that\n\\(A \\cap A^{-1} = \\emptyset\\).\nA \\emph{ping-pong system} on \\(A\\) is an assignment \\(a \\mapsto \\operatorname{dest}(a)\\) of sets\nto each element of \\(A^\\pm\\) such that whenever \\(a\\) and \\(b\\) are in \\(A^\\pm\\) and \\(s \\in S\\):\n\\begin{itemize}\n\n\\item \\label{basic_Dp}\n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(a)\\) and if \\(s \\in \\operatorname{supt}(a)\\), then\nthere is an integer \\(k\\) such that \\(s a^k \\in \\operatorname{dest} (a)\\);\n\n\\item \\label{ping-pong_cond}\nif \\(s \\in \\operatorname{supt}(a)\\), then \\(s a  \\in \\operatorname{dest}(a)\\) if and only if \\(s \\not \\in \\operatorname{src}(a) : = \\operatorname{dest}(a^{-1})\\);\n\n\\item if \\(a \\ne b\\), then \\(\\operatorname{dest}(a) \\cap \\operatorname{dest}(b)\\) is empty;\n\n\\item if \\(\\operatorname{dest}(a) \\cap \\operatorname{supt}(b) \\ne \\emptyset\\), then \\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b)\\).\n\n\\end{itemize}\n\\noindent\nThe following lemma summarizes some immediate consequences of this definition.\n\\begin{lemma}\nGiven a set \\(S\\) and  a collection \\(A\\) of permutations of \\(S\\)\nequipped with a ping pong system, the following are true:\n\\begin{itemize}\n\n\\item if \\(a \\in A\\) and \\(s \\in \\operatorname{supt} (a)\\), then there is a unique \\(k \\in {\\mathbf Z}\\) such that:\n\\[\ns a^k \\in \\operatorname{supt}(a) \\setminus (\\operatorname{src}(a) \\cup \\operatorname{dest}(a))\\]\n\n\\item if \\(a \\in A\\), then \\(\\operatorname{dest}(a) a \\subseteq \\operatorname{dest}(a)\\);\n\n\\end{itemize}\nIn particular, all elements of \\(A\\) have infinite order.\n\\end{lemma}\nAs remarked in Section \\ref{FastBumpsSec}, geometrically fast sets of bumps admit\na ping-pong system.\nThe meanings of \\emph{source}, \\emph{destination}, and \\emph{locally reduced word}\nall readily adapt to this new context.\nFurthermore, the proofs of Lemmas \\ref{LocRedBasics}--\\ref{FellowTraveler3} given in Section \\ref{PingPongSec}\nuse only the axiomatic properties of a ping-pong system and thus these lemmas \nare valid in the present context.\nThe next example is simplistic, but it will serve to illustrate a number of points in this section.\n\n\\begin{example} \\label{PSL2Example}\nView the real projective line \\(\\mathbf{P}\\) as \\({\\mathbf R} \\cup \\{\\infty\\}\\) and \\(\\operatorname{PSL}_2({\\mathbf Z})\\) as a group of\nfractional linear transformations of \\(\\mathbf{P}\\).\nThe homeomorphisms \\(\\alpha\\) and \\(\\beta\\) of \\(\\mathbf{P}\\) defined by\n\\[\nt\\alpha := t + 1 \\qquad t \\beta := \\frac{t}{1-t}\n\\]\ngenerate \\(\\operatorname{PSL}_2({\\mathbf Z})\\).\nIf we take \\(A = \\{\\alpha^2,\\beta^2\\}\\), then\n\\[\n\\operatorname{src}(\\alpha^2) := (-\\infty,-1)  \\qquad \\qquad \\operatorname{dest}(\\alpha^2) := [1,\\infty)\n\\]\n\\[\n\\operatorname{src}(\\beta^2) := (0,1) \\qquad \\qquad \\operatorname{dest}(\\beta^2) := [-1,0) \n\\]\ndefines a ping-pong system.\nIt is well known that \\(\\gen{\\alpha^2,\\beta^2}\\) is free;\nin fact this is one of the classical applications of the Ping-Pong Lemma.\n\\end{example}\n\nIn order to better understand \\(\\gen{A}\\) when \\(A\\) is a set of permutations admitting a\nping-pong system, it will be helpful to represent \\(\\gen{A}\\) as a family of homeomorphisms\nof a certain space \\(K_A\\).\nThis compact space can be thought of as a space of \\emph{histories} in the sense\nof Section \\ref{PingPongSec}.  \nIf \\(S\\) is the underlying set which elements of \\(A\\) permute,\nlet \\(M = M_A\\) denote the collection of all sets of the form\n\\[\n\\tilde s: = \\{a \\in A \\mid s \\in \\operatorname{supt}(a)\\}\n\\]\nwhere \\(s \\in S \\setminus \\bigcup \\{\\operatorname{dest}(a) \\mid a \\in A^\\pm\\}\\).\nElements of \\(M\\) will play the same role as the \ninitial markers of a geometrically fast collection of bumps.\n\n\\begin{example} \\label{PSL2Marker}\nContinuing with the Example \\ref{PSL2Example}, \\(M\\) consists of two points:\n\\(\\widetilde 0 = \\{\\alpha\\}\\) and \\(\\widetilde \\infty = \\{\\beta\\}\\).\nWe can also restrict the action of \\(\\operatorname{PSL}_2({\\mathbf Z})\\) on \\(\\mathbf{P}\\) to the irrationals.\nIn this case \\(M\\) is empty.\n\\end{example}\n\nIt will be convenient to define \\(\\operatorname{dest}(\\tilde s) := \\bigcap \\{\\operatorname{supt}(a) \\setminus \\operatorname{src}(a) \\mid a \\in \\tilde s\\}\\) and\n\\(\\operatorname{supt}(\\tilde s) := \\emptyset\\).\nDefine \\(K_A\\) to be all \\(\\eta\\) such that:\n\\begin{itemize}\n\n\\item \\(\\eta\\) is a suffix closed family of finite \nstrings in the alphabet \\(A^\\pm \\cup M\\);\n\n\\item if \\(\\str{ab}\\) are consecutive symbols of an element of \\(\\eta\\), then\n\\(\\operatorname{dest}(a) \\subseteq \\operatorname{supt}(b) \\setminus \\operatorname{src}(b)\\);\n\n\\item for each \\(n\\), there is at most one element of \\(\\eta\\) of length \\(n\\);\n\n\\item if \\(\\str{w} \\in \\eta\\) and \\(\\eta\\) does not contain a symbol from \\(M\\),\nthen \\(\\str{w}\\) is a proper suffix of an element of \\(\\eta\\).\n\n\\end{itemize}\n\\noindent\nThe second condition implies that elements of \\(\\eta\\) are freely reduced since if \\(b = a^{-1}\\),\nthen \\(\\operatorname{src}(b) = \\operatorname{dest}(a)\\).\nObserve that if \\(\\str{w}\\) is in \\(\\eta\\), the only occurrence of an element of \\(M\\) in \\(\\str{w}\\) must\nbe as the first symbol of \\(\\str{w}\\) (and there need not be any occurrence of an element of \\(M\\) in \\(\\str{w}\\)).\n\nNotice that every \\(\\eta \\in K_A\\) has at least one element other than \\(\\varepsilon\\) and that all elements\nof \\(\\eta\\) of positive length must have the same final symbol.\nWe define \\(\\operatorname{dest}(\\eta) := \\operatorname{dest}(a)\\) where \\(\\str{a}\\) is the final symbol of every\nelement of \\(\\eta\\) other than \\(\\varepsilon\\).\nWe topologize \\(K_A\\) by declaring that\n\\([\\str{w}] : = \\{\\eta \\in K_A \\mid \\str{w} \\in \\eta\\}\\) is closed and open.\nNotice that if \\(\\eta\\) is finite, it is an isolated point of \\(K_A\\).\n\n\\begin{prop}\n\\(K_A\\) is a Hausdorff space and if \\(A\\) is finite, then \\(K_A\\) is compact.\n\\end{prop}\n\n\\noindent\nEach \\(a \\in A^\\pm\\) defines a homeomorphism \\(\\hat a : K_A \\to K_A\\) by:\n\\[\n\\eta \\hat a := \n\\begin{cases}\n\\{\\str{ua} \\mid \\str{u} \\in \\eta\\} \\cup \\{\\varepsilon\\} & \\textrm{ if } \\operatorname{dest}(\\eta) \\subseteq \\operatorname{supt}(a) \\setminus \\operatorname{src}(a) \\\\\n\\{\\str{u} \\mid \\str{ua}^{-1} \\in \\eta\\}  & \\textrm{ if } \\operatorname{dest} (\\eta) = \\operatorname{src}(\\str{a}) \\\\\n\\eta & \\textrm{ if } \\operatorname{dest}(\\eta) \\cap \\operatorname{supt}(\\str{a}) = \\emptyset\n\\end{cases}\n\\]\nThus \\(\\eta \\hat a\\) is obtained by \nappending \n\\(\\str{a}\\) to the end of every element of \\(\\eta\\),\ncollecting the local reductions, and possibly including \\(\\varepsilon\\).\nSet \\(\\hat A = \\{\\hat a: a \\in A\\}\\).\n\nWe say that a ping-pong system on \\(A\\) is \\emph{faithful} if\n\\(\\Lambda_A : = \\{\\eta \\in K_A : \\eta \\textrm{ is finite}\\}\\) is dense in \\(K_A\\) (i.e.\nwhenever \\(\\str{w}\\) is in some \\(\\eta \\in K_A\\), there is a finite \\(\\eta' \\in K_A\\)\nwhich has \\(\\str{w}\\) as an element).\n\n\\begin{example} \\label{NonfaithfulExample}\nAs noted above, if we restrict the elements of \\(\\operatorname{PSL}_2({\\mathbf Z})\\) to the irrationals,\nthen  \\(M = \\emptyset\\) and in particular the system is not faithful.\nOn the other hand, \n\\[\n\\operatorname{dest}(\\alpha^4) := (2,\\infty) \\cap S \\qquad \\qquad \\operatorname{dest}(\\alpha^{-4}) := (-\\infty,-2) \\cap S\n\\]\n\\[\n\\operatorname{dest}(\\beta^4) := (-1\/2,0) \\cap S \\qquad \\qquad \\operatorname{dest}(\\beta^{-4}) := (0,1\/2) \\cap S.\n\\]\ndefines a ping-pong system in which \\(M\\) contains a single element \\(\\{\\alpha,\\beta\\}\\).\n\\end{example}\n\n\\noindent\nWhile not every ping-pong system is faithful,\nthe reader is invited to verify that if \\(A\\) admits a ping-pong system, then\n\\(\\{a^2 \\mid a \\in A\\}\\) admits a faithful ping-pong system.\n\n\\begin{thm}\nIf \\(A\\) is a set of permutations which admits a ping-pong system, then\n\\(a \\mapsto \\hat a\\) extends to an epimorphism of \\(\\gen{A}\\) onto \\(\\gen{\\hat A}\\).\nIf the ping-pong system is faithful, then the epimorphism is an isomorphism.\n\\end{thm}\n\nThe map \\(x \\mapsto \\eta(x)\\) defined in Section \\ref{PingPongSec} adapts \\emph{mutatis mutandis}\nto define a map \\(s \\mapsto \\eta(s)\\) from \\(S\\) into \\(K_A\\).\nIt is readily verified that if \\(a \\in A\\) and \\(s \\in S\\), then \\(\\eta(sa) = \\eta(s)\\hat a\\).\nThe existence of a faithful ping-pong system also has the following structural consequence\nwhich follows readily from the abstract form of Lemma \\ref{TreeAction}.\n\n\\begin{prop}\nIf \\(A\\) admits a faithful ping-pong system, then \\(\\gen{A}\\) is torsion free.\n\\end{prop}\n\\noindent\nThis shows in particular that \\(\\operatorname{PSL}_2({\\mathbf Z}) = \\gen{\\alpha,\\beta}\\) --- which contains elements of finite order such as\n\\(t \\mapsto -1\/t\\) --- does not admit a ping-pong system.\n\nA \\emph{blueprint} for a ping-pong system is a pair \\(\\mathfrak{B} = (\\str{B},\\str{supt})\\)\nsuch that:\n\\begin{itemize}\n\n\\item \\(\\str{B}\\) is a set and \\(\\str{supt}\\) is a binary relation on \\(\\str{B}\\)\nwhich is interpreted as a set-valued function:\n\\(\\str{b} \\in \\str{supt}(\\str{a})\\) if \\((\\str{a}, \\str{b}) \\in \\str{supt}\\);\n\n\\item if \\(\\str{a} \\in \\str{B}\\) and \\(\\str{supt}( \\str{a})\\) is nonempty, then \\(\\str{a} \\in \\str{supt}(\\str{a})\\);\n\n\\item if \\(\\str{a} \\in \\str{A}\\), then there is a unique \\(\\str{a}^{-1} \\in \\str{A} \\setminus \\{\\str{a}\\}\\) with\n\\(\\str{supt} (\\str{a}^{-1}) = \\str{supt} (\\str{a})\\).\n\n\\end{itemize}\nAdditionally, setting \n\\(\\str{A} : = \\{\\str{a} \\in \\str{B} \\mid \\str{supt}(\\str{a}) \\ne \\emptyset\\}\\) we require that:\n\\begin{itemize}\n\n\\item if \\(\\str{b} \\ne \\str{c} \\in \\str{B} \\setminus \\str{A}\\), then \\(\\str{\\tilde b} \\ne \\str{\\tilde c}\\) where\n\\({\\str{\\tilde b}} := \\{\\str{a} \\in \\str{A} \\mid \\str{b} \\in \\str{supt}(\\str{a}) \\}\\).\n \n\\end{itemize}\nTwo blueprints are isomorphic if they are isomorphic as structures.\nIf \\(A\\) is a set of permutations which admits a ping-pong system, then the blueprint\n\\(\\mathfrak{B}_A = (\\str{B}_A,\\str{supt}_A)\\)\nfor the system is defined by\n\\(\\str{B}_A := \\{\\str{a} \\mid a \\in A^\\pm\\} \\cup \\{{\\str{\\tilde  s}} \\mid \\tilde s \\in \\widetilde S\\}\\)\nwith \\(\\str{b} \\in \\str{supt}_A(\\str{a})\\) if \\(\\operatorname{dest}(b) \\subseteq \\operatorname{supt}(a)\\).\nAlso, if \\(\\mathfrak{B} = (\\str{A},\\str{supt})\\) is a blueprint for a ping-pong system, then\none defines \\(K_{\\mathfrak{B}}\\) and homeomorphisms \\({\\str{\\hat a}} : K_{\\mathfrak{B}} \\to K_{\\mathfrak{B}}\\)\nfor \\(\\str{a} \\in \\str{A}^{\\mathfrak{B}}\\) by a routine adaptation of the construction above.\nIn fact \\(K_A\\) is can be regarded as factoring though its blueprint in the sense that\n\\(K_A = K_{\\mathfrak{B}_A}\\) modulo identifying \\(a\\) and \\(\\str{a}\\).\nIt is routine to check that the following theorem holds as well.\n\n\\begin{thm} \\label{AbsCombToIso}\nIf \\(\\mathfrak{B}_0\\) and \\(\\mathfrak{B}_1\\) are isomorphic blueprints for ping-pong systems,\nthen the isomorphism induces an homeomorphism \\(\\theta:K_{\\mathfrak{B}_0} \\to K_{\\mathfrak{B}_1}\\)\nsuch that \\(g \\mapsto g^\\theta\\) defines an isomorphism between\n\\(\\gen{\\hat {\\str{a}} \\mid \\str{a} \\in \\str{A}_0}\\) and \\(\\gen{\\hat {\\str{a}} \\mid \\str{a} \\in \\str{A}_1}\\).\n\\end{thm}\n\nIf \\(A\\) is a set of permutations which admits a faithful ping-pong system,\nthen the blueprint for \\(A\\) and for \\(\\{a^{k(a)} \\mid a \\in A\\}\\) are canonically\nisomorphic whenever \\(a \\mapsto k(a)\\) is an assignment of a positive integer to each element of \\(A\\).\n\n\\begin{cor}\nAny set of permutations \\(A\\) which admits a faithful ping-pong system is an algebraically\nfast generating set for \\(\\gen{A}\\).\n\\end{cor}\n\nIf the system is not faithful, then \\(\\{a^2 \\mid a \\in A^2\\}\\) may contain new markers, as\nwas illustrated in Example \\ref{NonfaithfulExample}.\nNotice that in this example,\nboth \\(\\gen{\\alpha^2,\\beta^2}\\) and \\(\\gen{\\alpha^4,\\beta^4}\\) are free --- and hence marked isomorphic ---\neven though the blueprints associated to the ping-pong systems are not isomorphic.\n\nA blueprint \\(\\mathfrak{B}\\) is (cyclically) orderable if there is a (cyclic) ordering on \\(\\str{B}\\)\nsuch that for all \\(\\str{a} \\in \\str{A}\\), \n\\(\\str{supt}(\\str{a})\\) is an interval in the (cyclic) ordering with endpoints \\(\\operatorname{src}(\\str{a})\\) and \\(\\operatorname{dest}(\\str{a})\\).\nIt is readily checked that the (cyclic) order on \\(\\str{B}\\)\ninduces a reverse lexicographic (cyclic) order on \\(K_{\\mathfrak{B}}\\) which is preserved by the\nhomeomorphisms \\(\\hat {\\str{a}}\\) for \\(\\str{a} \\in \\str{A}\\).\n\n\\begin{cor}\nIf \\(A\\) is a finite set of permutations which admits a faithful ping-pong system, then\n\\(\\gen{A}\\) embeds into Thompson's group \\(V\\).\nIf the blueprint of \\(A\\) is cyclically orderable, then \\(\\gen{A}\\) embeds into\n\\(T\\).\nIf the blueprint of \\(A\\) is orderable, then \\(\\gen{A}\\) embeds into \\(F\\). \n\\end{cor}\n\n\\begin{proof}\nIt is readily verified that any finite blueprint can be realized as the blueprint of\na ping-pong system of a finite subset of \\(V\\).\nMoreover, if the blueprint is cyclically orderable (or orderable), then it can be realized\nby elements of \\(T\\) (respectively \\(F\\)). \nThe corollary follows by Theorem \\ref{AbsCombToIso}.\n\\end{proof}\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\n\\providecommand{\\MRhref}[2]{\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nHypertensive disorders of pregnancy are a class of high blood pressure disorders that occur during the second half of pregnancy, which include gestational hypertension, preeclampsia and severe preeclampsia. They are characterized by a diastolic blood pressure higher than 90 mm Hg and\/or a systolic blood pressure higher than 140 mm Hg and they are often accompanied by proteinuria. These disorders affect about 10\\% of pregnant women around the world, with preeclampsia occurring in 2--8\\% of all pregnancies \\citep{timokhina2019}. These disorders represent one of the leading causes \nof maternal and fetal morbidity and mortality, contributing to 7--8\\% of maternal death worldwide \\citep{dolea2003,shah2009,mcclure2009}. The World \nHealth Organization estimates that the incidence of preeclampsia is seven \ntimes higher in developing countries than in developed countries. However, the occurrence  of these diseases appears under-reported in low and middle income countries, implying that the true incidence is unknown \\citep{igberase2006,malik2018}. While there is evidence that hypertensive disorders of pregnancy are related with the development of cardiac dysfunctions both in the mother and in the child \\citep{bellamy2007,davis2012, ambrovzic2020dynamic, garcia2020maternal, aksu2021cardiac, demartelly2021long}, there is no common agreement on the relation between the severity of hypertension and cardiac dysfunction \\citep{tatapudi2017} and echocardiography is not included in baseline evaluation of hypertensive disorders of pregnancy. Further investigations on these disorders are needed, especially for developing countries, where women often give birth at a younger age with respect to developed countries. \n\nThe goal of this work is to detect which cardiac function is altered and \nunder which hypertensive disorders by relying on a principled Bayesian nonparametric approach. An interesting case-control study to explore the relation between cardiac dysfunction and hypertensive disorders is provided by \\citet{data}, where the measures of ten different cardiac function indexes were recorded in four groups of pregnant women in India. Groups of women are characterized by different hypertensive disorder diagnoses, that are naturally ordered based on the severity of the diagnosed disorder: healthy (C), gestational hypertension (G), mild preeclampsia (M) and severe preeclampsia (S). \nHypertensive diagnoses are used as identifiers for what we call populations of patients and we refer to cardiac function indexes also with the term response variables. For each response variable we want to determine a partition of the four populations of patients. This amounts to identifying similarities between different hypertensive disorders, with respect to each cardiac index. Supposing, for instance, that the selected partition assigns all the populations to the same cluster, one can conclude that no \nalteration is shown for the corresponding  cardiac index across different hypertensive diseases.\n\nOur goal of identifying a partition of the four patients' populations for each of the ten responses can be rephrased as a problem of multiple model selection: we want to select the most plausible partition for each cardiac index. \nFrequentist hypothesis testing does not allow to deal with more than two \npopulations in a straightforward way\nand pairwise comparisons may lead \nto conflicting conclusions. Conversely, a Bayesian approach yields the posterior distribution on the space of partitions, which can be used for simultaneous comparisons. \nMoreover, the presence of $M=10$ jointly tested cardiac indexes requires to perform model selection repeatedly ten times. Once again, a Bayesian approach seems to be preferred, because, as observed for instance by \\citet{scott2006}, it does not require the introduction of a penalty term for multiple comparison, thanks to the prior distribution build-in penalty. \n\nHere we design\na Bayesian nonparametric model, that is tailored to deal with both a collection of ordered populations and the multivariate information of the response variables, while preserving the typical flexibility of nonparametric models and producing easily interpretable results.\nWhen applied to the dataset on transthoracic echocardiography results for a cohort of Indian pregnant women in Section \\ref{s:results}, our model effectively identifies \nmodified cardiac functions in hypertensive patients compared to healthy subjects and progressively increased alterations with the severity of the disorder, in addition to other more subtle findings. The observed data $X_{i,j,m}$ represent the measurement of the $m$-th response variable (cardiac index) on the $i$-th individual (pregnant woman) in the $j$-th population (hypertensive disorder) \nand, as in standard univariate ANOVA models, they are assumed to be partially exchangeable across disorders.\nThis means that for every $m\\in\\{1,\\ldots,M\\}$,  \nthe law of $(\\,(X_{i,1,m})_{i\\ge 1},\\ldots,(X_{i,J,m})_{i\\ge 1})$ is invariant with respect to permutations within each sequence of random variables, namely for any positive integers $n_1,\\ldots,n_J$ \n\\[\n(\\,(X_{i,1,m})_{i=1}^{n_1},\\ldots,(X_{i,J,m})_{i=1}^{n_J})\\stackrel{\\mbox{\\scriptsize{d}}}{=} \n(\\,(X_{{\\sigma_1(i)},1,m})_{i=1}^{n_1},\\ldots,(X_{{\\sigma_J(i)},J,m})_{i=1}^{n_J})\n\\]\nfor all permutations $\\sigma_j$ of $(1,\\ldots,n_j)$, with $j=1,\\ldots,J$. This is a natural generalization of exchangeability to tackle heterogeneous data and, by de Finetti's representation theorem, it amounts to assuming the existence of a collection of (possibly dependent) random probability measures $\\{\\pi_{j,m}:\\:\\: j=1,\\ldots,J\\;\\: m=1,\\ldots,M\\}$ such that\n\\[\nX_{i,j,m} \\mid \\pi_{j,m} \\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim} \\pi_{j,m} \\qquad i=1,\\ldots,n_j\n\\]  \nHence, for any two populations $j\\ne j'$, homogeneity corresponds to $\\pi_{j,m}=\\pi_{j',m}$ (almost surely). However, a reliable assessment of this type of homogeneity is troublesome when having just few patients per \ndiagnosis, as it happens in the mild preeclampsia subsample.  \nWithout relying on simplifying parametric assumptions, a small sub-sample size may not be sufficiently informative to infer equality of entire unknown distributions. \nTo overcome this issue, without introducing parametric assumptions, we resort to an alternative weaker notion of homogeneity between populations $j$ and $j'$: we only require the conditional means of the two populations to (almost surely) coincide\n\\begin{equation}\n\t\\label{eq:equality_means}\n\t\\mathbb{E}(X_{i,j,m}\\mid\\pi_{j,m})=\\mathbb{E}(X_{i,j',m}\\mid\\pi_{j',m}).\n\\end{equation}\nAccording to this definition, the detection of heterogeneities in cardiac function indexes amounts to inferring which cardiac indexes have means that differ \nacross diagnoses, as it is done in standard parametric ANOVA models.\nBesides clustering populations according to \\eqref{eq:equality_means}, it is also of interest to cluster patients, both within and across different groups, once the effect of the specific hypertensive disorder is taken \ninto account. This task may be achieved by assuming a model that decomposes the observations as \n\\begin{equation}\n\t\\label{eq:decomp_x}\n\tX_{i,j,m}=\\theta_{j,m}+\\varepsilon_{i,j,m}\\qquad\n\t\\varepsilon_{i,j,m}|(\\xi_{i,j,m},\\sigma_{i,j,m}^2)\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} \\mbox{N}(\\xi_{i,j,m},\\sigma_{i,j,m}^2)\n\\end{equation} \nand the $\\xi_{i,j,m}$ have a symmetric distribution around the origin, in order to ensure $E(\\xi_{i,j,m})=0$. In view of this decomposition, we \nwill let $\\theta_{j,m}$ govern the clustering of populations while the  $(\\xi_{i,j,m},\\sigma^2_{i,j,m})$'s determine the clustering of individuals, namely patients, \nafter removing the effect of the specific hypertensive disorder. In order to pursue this, for each cardiac index $m$, we will specify a\nhierarchical process prior for $(\\xi_{i,j,m},\\sigma^2_{i,j,m})$ that is suited to infer the clustering structure both within and across different \nhypertensive disorders for a specific cardiac index. In particular, we will deploy a novel instance of hierarchical Dirichlet process, introduced in \\cite{teh2006}, that we name \\textit{symmetric}, to highlight its centering in $0$.\n\n\nEarly examples of Bayesian nonparametric models for ANOVA can be found in \\cite{cifreg78} and \\cite{mulierepetrone}, while the first popular proposal, due to \\citet{de2004}, uses the dependent Dirichlet process (DDP) \\citep{maceachern2000dependent} and is therefore termed ANOVA-DDP. This model is mainly tailored to estimate populations' probability distributions, while we draw inferences over clusters of populations' means and obtain estimates of the unknown distributions as a by-product. Moreover, the ANOVA-DDP of \\citet{de2004} was not introduced as a model selection procedure. A popular Bayesian nonparametric model, that does cluster \npopulations and can be used for model selection, is the nested Dirichlet process of \\citet*{rodriguez2008}. As shown in \\citet{camerlenghi2019}, such a prior is biased towards homogeneity, in the sense that even a single \ntie between populations $j$ and $j'$, namely $X_{i,j,m}=X_{i',j',m}$ for some $i$ and $i'$, entails $\\pi_{j,m}=\\pi_{j'm}$ (almost surely). In order to overcome such a drawback, a novel class of nested, and more flexible, priors has been proposed in   \n\\citet{camerlenghi2019}. See also \\cite{soriano_ma} for related work. Interesting alternatives that extend the analysis to more than two populations can be found in \\cite{ChristMa}, \\cite*{Rebaudo} and in \\cite*{Beraha}. Another similar proposal is the one by \\citet{gutierrez2019bayesian}, whose model identifies differences over cases' distributions and the control group. These models imply that two populations belong to the same cluster if they share the entire distribution. However, as already mentioned, distribution-based clustering is not ideal when dealing with scenarios as the one of hypertensive dataset. Further evidence will be provided in Section~\\ref{ss:sim}, through simulation \nstudies. In addition, note that all these contributions deal with only one response variable and would need to be suitably generalized to fit \nthe setup of this paper. As far as the contributions treating multiple response variables are concerned, uses of nonparametric priors for multiple \ntesting can be found, for instance, in \\citet{gopalan1998}, \\citet*{do2005bayesian}, \\citet{dahl2007multiple}, \\citet*{guindani2009}, \\citet{martin2012} and more recently in \\citet*{cipolli2016bayesian}, who propose an approximate finite P\\'olya tree multiple testing procedure to compare two-samples' locations, and in \\citet{dentitwo}. However, in all these contributions, models are developed directly over summaries of the original data (e.g. averages, z-scores) and, as such, do not allow to draw any inference on the entire distributions and clusters of subjects. \n\nThe outline of the paper is as follows. In Section~\\ref{s:model} we introduce the model, which makes use of an original hierarchical prior structure for symmetric distributions (Section~\\ref{ss:s-HDP}). In Section~\\ref{s:partition} we derive the prior law of the random partitions induced by \nthe model, key ingredient for the Gibbs sampling scheme devised in Section~\\ref{s:inference}. In Section~\\ref{s:results}, we first present a series of simulation studies that highlight the behaviour of the model before applying it to obtain our results on cardiac dysfunction in hypertensive disorders. Section~\\ref{s:conclusion} contains some concluding remarks. As Supplementary Material we provide the datasets and Python codes, some further background material and details about the derivation of \nthe posterior sampling scheme as well as additional simulation studies and results on the application, including an analysis of prior sensitivity.\n\n\\section{The Bayesian nonparametric model}\n\\label{s:model}\n\nThe use of discrete nonparametric priors for Bayesian model-based clustering has become standard practice. The Dirichlet process (DP) \\citep{ferguson1973} is the most popular instance, and clustering is typically addressed by resorting to a mixture model, which with our data structure amounts to\n\\begin{equation*}\n\t\\label{eq:mixture_kernel}\n\tX_{i,j,m}|\\psi_{i,j,m}\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} k(X_{i,j,m};\\psi_{i,j,m}),\\qquad\n\t\\psi_{i,j,m}|\\tilde p_{j,m}\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} \\tilde p_{j,m} \n\\end{equation*}\nfor $m=1,\\ldots,M$, $j=1,\\ldots,J$ and $i=1,\\ldots,n_j$. Here $k(\\,\\cdot\\,;\\,\\cdot\\,)$ is some kernel and the $\\tilde p_{j,m}$'s are discrete random probability measures. Hence, the $\\psi_{i,j,m}$'s may exhibit ties. The model specification for $\\tilde p_{j,m}$ will be tailored to address the following goals: (i) cluster the $J$ probability distributions based on their means; (ii) cluster the observations $X_{i,j,m}$ according to the ties induced on the $\\psi_{i,j,m}$'s by the $\\tilde p_{j,m}$'s for \na given fixed $j$ and across different $j$'s. These two issues will be targeted separately: we first design a clustering scheme for the populations, through the specification of a prior on the means of the $X_{i,j,m}$'s and, then, we cluster the data using a hierarchical DP having a specific invariance structure that is ideally suited to the application at hand. \n\n\\subsection{The prior on disease-specific locations}\n\\label{ss:like}\nAs a model for the observations we consider a nonparametric mixture of Gaussian distributions specified as\n\\begin{equation} \n\t\\label{eq:model}\n\tX_{i,j,m}\\,|\\,(\\bm{\\theta}_m,\\bm{\\xi}_m,\\bm{\\sigma}^2_m)\n\t\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim} \\mathcal{N}(\\theta_{j,m}+\\xi_{i,j,m},\\sigma_{i,j,m}^2)\n\\end{equation}\nwhere $\\bm{\\theta}_m=(\\theta_{1,m},\\ldots,\\theta_{J,m})$, $\\bm{\\xi}_m=(\\xi_{1,1,m},\\ldots,\\xi_{1,n_1,m},\\xi_{2,1,m},\\ldots,\n\\xi_{n_J,J,m})$, with a similar definition for the vector $\\bm{\\sigma}_m^2$, and $\\mathcal{N}(\\,\\mu,\\,\\sigma^2\\,)$ denotes a normal distribution with mean $\\mu$ and variance $\\sigma^2$. The assumption in \\eqref{eq:model} clearly reflects \\eqref{eq:decomp_x}. Moreover, in order to account for the two levels of clustering we are interested in, we will assume that \n\\begin{equation}\n\t\\label{eq:model_parameters}(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})\\sim P,\\qquad\n\t(\\xi_{i,j,m},\\sigma_{i,j,m}^2)\\,|\\,\\tilde{q}_{j,m}\\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim} \\tilde q_{j,m}\\quad (i=1,\\ldots,n_j)\n\\end{equation}\nwhere $\\tilde q_{1,m},\\ldots,\\tilde q_{J,m}$ are discrete random probability measures independent from  $(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})$. Thus, the likelihood corresponds to\n\\begin{equation}\\label{eq1}\n\t\\prod_{m=1}^M\\,\\prod_{j=1}^J \\,\n\t\\prod_{i=1}^{n_j}\n\t\\frac{1}{\\sigma_{i,j,m}}\\:\n\t\\varphi\\Big(\\frac{x_{i,j,m}-\\theta_{j,m}-\\xi_{i,j,m}}{\\sigma_{i,j,m}}\\Big)\n\t\\:\\tilde{q}_{j,m}(\\mathrm{d}\\xi_{i,j,m},\n\t\\mathrm{d}\\sigma_{i,j,m})\n\\end{equation}\nwith $\\varphi$ denoting the standard Gaussian density. Relevant inferences can be carried out if one is able to marginalize this expression with respect to both $(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})$ and $(\\tilde q_{1,m},\\ldots,\\tilde{q}_{J,m})$ for each \n$m=1,\\ldots,M$. \n\nThis specification allows to address the model selection problem in the following way. If $\\mathcal{M}^m$ stands for the space of all partitions of the $J$ populations for the $m$-th cardiac function index, then $\\mathcal{M}^m=\\{M_b^m \\,:\\, b =1,\\ldots,\\mbox{card}(\\mathcal{P}_J)\\}$ where $\\mathcal{P}_J$ is the collection of all possible partitions of $[J]=\\{1,\\ldots,J\\}$. In our specific case, $J=4$ and card$(\\mathcal{P}_J)=15$, thus we have 15 competing models per cardiac index. Each competing model corresponds to a specific partition in $\\mathcal{M}^m$. In particular, the partition arises from ties between the population specific means in $\\bm{\\theta}_m$ and, hence, the distribution $P$ in \\eqref{eq:model_parameters} needs to associate positive probabilities to ties between the parameters within the vector $\\bm{\\theta}_m$, for each $m=1,\\ldots,M$. \n\n\nLet us start considering as distribution $P$ a well-known effective clustering prior, i.e. a mixture of DPs in the spirit of \\cite{antoniak1974mixtures}, namely\n\\begin{equation}\n\t\\label{eq2}\n\t\\begin{aligned}\n\t\t\\theta_{j,m}\\mid \\tilde p_m \\enskip&\\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim}\\enskip \\tilde p_m \\qquad &j=1,\\ldots,J \\\\\n\t\t\\tilde p_m\\mid\\omega\\enskip &\\stackrel{\\mbox{\\scriptsize{\\rm iid}}}{\\sim} \\enskip \\mbox{DP}(\\omega,G_{m})\\qquad& m=1,\\ldots,M \\\\\n\t\t\\omega \\enskip &\\sim \\enskip p_\\omega&\n\t\\end{aligned}\n\\end{equation}\nwhere DP$(\\omega,G_m)$ denotes the DP with concentration parameter $\\omega$ and non-atomic baseline probability measure $G_m$ and $p_{\\omega}$ is a probability measure on $\\mathds{R} ^+$. The discreteness of the DP implies the presence (with positive probability) of ties within the vector of locations $\\bm{\\theta}_m$ associated to a certain cardiac index $m$, as desired. The ties give rise to a random partition: as shown in \\cite{antoniak1974mixtures}, the probability of observing a specific partition of the elements in $\\bm{\\theta}_m$ consisting of $k\\le J$ distinct values with respective frequencies $n_1,\\ldots,n_k$ coincides with\n\\begin{equation}\n\t\\label{eq:eppf_dir}\n\t\\Pi_k^{(J)}(n_1,\\ldots,n_k)=\\frac{\\omega^k}{(\\omega)_J}\\,\\prod_{i=1}^k (n_i-1)!\n\\end{equation}\nwhere $(\\omega)_J=\\Gamma(\\omega+J)\/\\Gamma(\\omega)$. The use of a shared concentration parameter over \\eqref{eq:eppf_dir} to address multiple model selection has been already successfully employed in \\cite{moser2021multiple}, where they cluster parameters in a probit model.  When there is no pre-experimental information available on competing partitions, the use of \\eqref{eq:eppf_dir} as a prior for model selection has some relevant benefits. Indeed, it induces borrowing of strength across diagnoses and, being $\\omega$ random, it generates borrowing of information also across cardiac indexes, thus improving the Bayesian learning mechanism. These two features can also be given a frequentist interpretation in terms of desirable penalties.\nAs a matter of fact, the procedure penalizes for the multiplicity of the model selections that are performed. The penalty has to be meant \nin the following way: while $J$ and\/or $M$ increase, the prior odds change in favor of less complex models. For more details on this, see \\cite{scott2010bayes}. Summing up, the mixture of DPs automatically induces a prior distribution on $\\{\\mathcal{M}^m:m=1,\\ldots,M\\}$, that arises from \\eqref{eq:eppf_dir} combined with the prior $p_\\omega$ on $\\omega$, and it presents desirable properties for model selection \nthat can be interpreted either in terms of borrowing of information or in terms of penalties. \n\nHowever, in the analysis of hypertensive disorders, some prior information on competing models is available, and this is not yet incorporated in \\eqref{eq:eppf_dir}. In fact, as already mentioned, there is a natural order of the diagnoses, which is given by the severity of the disorders, i.e.\\ C, G, M, S. Partitions that do not comply with this ordering, e.g. $\\{\\{C, S\\}\\{G\\},\\{M\\}\\}$, should be excluded from the support of the prior. Thus, we consider a prior over $\\mathcal{M}^m$ that associates zero probability to partitions that do not respect the natural order of the diagnoses and a probability proportional to that in \\eqref{eq:eppf_dir} for the remaining partitions, i.e.\\\n\\begin{equation}\\label{eq:prior2}\n\t\\mathbb{P}(M_b^m\\mid \\omega) \\propto \\begin{cases}\n\t\\Pi_k^{(J)}(n_1,\\ldots,n_k)&\\text{if $M_b^m$ is compatible with the natural order}\\\\\n0 & \\text{otherwise}\\end{cases}\n\\end{equation} \nThis amounts to a distribution $P$ for $(\\bm{\\theta}_{1},\\ldots,\\bm{\\theta}_{M})$ given by \n\\begin{equation}\n\t\\label{eqnew}\n\t\\begin{aligned}\n\t\t(\\theta_{1,m},\\ldots,\\theta_{J,m})\\mid \\omega \\enskip&\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim}\\enskip P_{\\omega, G_m} \\qquad m=1,\\ldots,M \\\\\n\t\t\\omega \\enskip &\\sim \\enskip p_\\omega&\n\t\\end{aligned}\n\\end{equation}\nwhere $P_{\\omega, G_m}$ is the distribution obtained sampling a partition according to \\eqref{eq:prior2} and associating to each cluster a unique value sampled from $G_m$. \nUsing \\eqref{eqnew} as a prior for the disease-specific locations, we preserve the desirable properties of the mixture of DPs mentioned before, while incorporating prior information on the severity of the diseases.\n\nAs detailed in the next section, we further define random probability measures $\\tilde q_{j,m}$ that satisfy the symmetry condition\n\\begin{equation}\n\t\\label{eq:symmetry}\n\t\\tilde q_{j,m}(A\\times B ) \n\t= \\tilde \n\tq_{j,m}((-A)\\times B )\\qquad\\qquad  \\text{a.s.}\n\\end{equation}\nfor any $A$ and $B$. This condition ensures that the parameters $\\theta_{j,m}$, for $j=1,\\ldots,J$ and $m=1,\\ldots,M$, in \\eqref{eq:model} are identified, namely $\\mathbb{E}(X_{i,j,m}\\mid \\boldsymbol{\\theta}_{m}, \\, \\tilde q_{j,m} ) = \\theta_{j,m}$ with probability one. This identifiability property is crucial to make inference over the location parameters $\\bm{\\theta}_m$'s. Similar model specifications for discrete \nexchangeable data have been proposed and studied in \\cite{dalal1979nonparametric},  \\cite{doss1984}, \\cite{diaconis1986} and \\cite*{ghosal1999}, of which \\eqref{eq1} represents a generalization to \ndensity functions and partially exchangeable data. \n\n\\subsection{The prior for the error terms}\n\\label{ss:s-HDP}\nWhile the clustering of populations is governed by \\eqref{eq:prior2}, we use a mixture of hierarchical discrete processes for the error terms. This has the advantage of modeling the clustering of the observations, both within and across different samples, once the disease-specific effects are account for. This clustering structure allows to model heterogeneity across patients in a much more realistic way with respect to standard ANOVA models based on assumption of normality. Cardiac indexes may be influenced by a number of factors that are not directly observed in the study, such as pre-existing conditions \\citep{hall2011heart} and psychosocial factors \\citep{pedersen2017psychosocial}. These unobserved relevant factors may be shared across patients with the same or a different diagnosis and may also result in outliers. To take into account this latent heterogeneity of the data, we introduce the hierarchical\nsymmetric DP that satisfies the symmetry condition in \\eqref{eq:symmetry} and, moreover, allows to model heterogeneous data similarly to the \nhugely popular hierarchical DP \\citep{teh2006}.\n\nThe basic building block of the proposed prior is the invariant Dirichlet \nprocess, which was introduced for a single population ($J=1$) in an exchangeable framework by \\cite{dalal1979}. Such a modification of the DP satisfies a symmetry condition, in the sense that it is a random probability measure that is invariant with respect to a chosen group of transformations $\\mathcal{G}$. A more formal definition and detailed description of the invariant DP can be found in Section A of the Supplement. For our purposes it is enough to consider the specific case of the symmetric Dirichlet process, \nwhich can be constructed through a symmetrization of a Dirichlet process. Consider a non-atomic probability measure $P_0$ on $\\mathbb{R}$\nand let $\\tilde Q_0 \\sim \\text{DP}(\\alpha,P_0)$. \nIf \n\\begin{equation}\n\t\\label{eq:inv_DP}\n\t\\tilde Q(A) = \\frac{\\tilde Q_0(A) + \\tilde Q_0(-A)}{2}\\quad \\quad \\forall A \\in \\mathcal{B}(\\mathbb{R})\n\\end{equation}\nwhere $-A=\\{x\\in \\mathbb{R}: -x \\in A\\}$, then $\\tilde Q$ is symmetric about 0 (almost surely) and termed symmetric DP, in symbols $\\tilde Q\\sim\\text{s-DP}(\\alpha\\,,\\,P_0)$. For convenience and without loss of generality, we assume that $P_0$ is symmetric: this implies that $P_0$ is the expected value of $\\tilde Q$ making it an interpretable parameter.\nThe random probability measure $\\tilde Q$ is the basic building block of the hierarchical process that we use to model the heterogeneity of the error terms across different populations, $j=1,\\ldots,J$, in such a way that clusters identified by the unique values can be shared within and across populations. \nThis prior is termed \\textit{symmetric hierarchical Dirichlet process} (s-HDP) and described as\n\\begin{equation}\n\t\\label{eq:s-hdp_def}\n\t\\begin{split}\n\t\t\\tilde q_{j,m}\\mid\\gamma_{j,m}, \\, \n\t\t\\tilde \n\t\tq_{0,m}\\enskip\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim}\\enskip\\text{s-DP}(\\gamma_{j,m},\\tilde q_{0,m})\\\\\n\t\t\\tilde q_{0,m}\\mid\\alpha_{m}\\enskip\\stackrel{\\mbox{\\scriptsize{\\rm ind}}}{\\sim}\\enskip\\text{s-DP}(\\alpha_{m},P_{0,m})\n\t\\end{split}\n\\end{equation}\nwhere $\\gamma_{j,m}$ and $\\alpha_{m}$ are positive parameters and $P_{0,m}$ is a non-atomic probability distribution symmetric about 0. We use the \nnotation $(\\tilde q_{1,m},\\ldots,\\tilde q_{J,m})\\sim \\mbox{s-HDP}(\\bm{\\gamma}_m,\\alpha_m,P_{0,m})$, where $\\bm{\\gamma}_m=(\\gamma_{1,m},\\ldots,\\gamma_{j,m})$. This definition clearly ensures the validity of \\eqref{eq:symmetry}. \nA graphical model representation of the over-all proposed model is displayed in Figure~\\ref{fig:figure1}. \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{tikzpicture}[-latex ,auto ,node distance =2 cm and 2cm ,on grid ,\n\t\t\tsemithick ,\n\t\t\tstate\/.style ={ circle ,top color =white , bottom color = white ,\n\t\t\t\tdraw,black , text=black , minimum width =1 cm},\n\t\t\ttransition\/.style = {rectangle, draw=black!50, thick, minimum width=6.1cm, minimum height = 5.8cm},\n\t\t\ttransition3\/.style = {rectangle, draw=black!50, thick, minimum width=4cm, minimum height = 3.8cm}, \n\t\t\ttransition2\/.style = {rectangle, draw=black!50, thick, minimum width=6.5cm, minimum height = 8cm}, scale=0.8]\n\t\t\t\\node[state] (C){$\\boldsymbol{\\theta_{m}}$};\n\t\t\t\\node[draw=none] (A) [below =of C] { };\n\t\t\t\\node[draw=none] (B) [below =of A] {};\n\t\t\t\\node[circle ,top color =white , bottom color = white ,\n\t\t\tdraw,black , text=black , minimum width =1 cm]  (D) [left =2 cm of C] {${\\omega}$};\n\t\t\t\\node[circle ,top color =white , bottom color = ashgrey ,\n\t\t\tdraw,black , text=black , minimum width =1 cm] (E) [below = of B] {${X_{i,j,m}}$};\n\t\t\t\\node[state] (F) [ right =of B] {$\\varepsilon_{i,j,m}$};\n\t\t\t\\node[state] (G) [above =of F] {${\\tilde q_{j,m}}$};\n\t\t\t\\node[state] (Z) [above =of G] {${\\tilde q_{0,m}}$};\n\t\t\t\\node[state] (X) [right =of Z] {${\\alpha_{m}}$};\n\t\t\t\\node[state] (Y) [right =of G] {${\\gamma_{j,m}}$};\n\t\t\t\\node[transition] (H)[ below =0.02cm of F] { };\n\t\t\t\\node[transition3] (J)[  above  right =1.5cm of E] { };\n\t\t\t\\node[transition2] (L)[ below =1cm of G] { };\n\t\t\n\t\t\t\\path (D) edge node[right] {} (C);\n\t\t\t\\path (C) edge node[below] {} (E);\n\t\t\t\\path (G) edge node[below] {} (F);\n\t\t\t\\path (F) edge node[below] {} (E);\n\t\t\t\\path (X) edge node[below] {} (Z);\n\t\t\t\\path (Y) edge node[below] {} (G);\n\t\t\t\\path (Z) edge node[below] {} (G);\n\t\t\\end{tikzpicture}\n\t\t\\caption{\\small Graphical representation of the model. Each node represents a random variable and each rectangle denotes conditional i.i.d. replications of the model within the rectangle.}\n\t\t\\label{fig:figure1}\n\t\\end{center}\n\t\\vspace{-0.5 cm}\n\\end{figure}\n\nStill referring to the decomposition of the observations into disease-specific locations and an error term, i.e. $X_{i,j,m}=\\theta_{j,m}+\\varepsilon_{i,j,m}$, it turns out that the $\\varepsilon_{i,j,m}$'s are from a symmetric hierarchical DP mixture (s-HDP mixture) with a normal kernel. Hence, the patients' clusters are identified through the $\\varepsilon_{i,j,m}$, which, according to \\eqref{eq:model}, are conditionally independent from a $\\mathcal{N}(\\xi_{i,j,m},\\sigma_{i,j,m}^2)$ given $(\\xi_{i,j,m},\\sigma_{i,j,m}^2)$. The choice of the specific invariant DP is aimed at ensuring that $\\mathbb{E}(\\varepsilon_{i,j,m}|\\tilde \nq_{j,m})=0$. \nThe clusters identified by the s-HDP mixture can be interpreted as representing common unobserved factors across patients, once the disease-specific locations have been accounted for. Indeed, for any pair of patients, we may consider the decomposition $X_{i,j,m}-X_{i',j',m}=\\,\\Delta^{(m)}_{\\theta} + \\Delta^{(m)}_{\\xi} + (e_{i,j,m}-e_{i',j',m})$ where $\\Delta^{(m)}_{\\theta}=\\theta_{j,m} - \\theta_{j',m}$, $\\Delta^{(m)}_{\\xi}=\\xi_{i,j,m} - \\xi_{i',j',m}$ and $e_{i,j,m}$ and $e_{i',j',m}$ are independent and normally distributed random variables with zero mean and variances $\\sigma_{i,j,m}^2$ and $\\sigma_{i',j',m}^2$, respectively. \n\nHence, patients' clustering reflects the residual heterogeneity that is not captured by the disease-specific component $\\Delta^{(m)}_{\\theta}$ and are related to the subject-specific locations $\\Delta^{(m)}_{\\xi}$ and to the zero-mean error component $(e_{i,j,m}-e_{i',j',m})$. \nIn view of this interpretation, using a s-HDP mixture over error terms offers a three-fold advantage. Firstly, the presence of clearly separated clusters of patients within and across populations will indicate the presence of unobserved relevant factors which affect the cardiac response variables. Secondly, single patients with very low probabilities of co-clustering \nwith all other subjects will have to be interpreted as outliers. \nFinally, the estimated clustering structure can also be used to check whether the relative effect of a certain disease (with respect to another) is fully explained by the corresponding $\\Delta^{(m)}_{\\theta}$. To clarify this last point consider two diseases: if the posterior co-clustering probabilities among patients sharing the same disease are different between the two populations, this will indicate that different diagnoses not only have an influence on disease-specific locations (which is measured by $\\Delta^{(m)}_{\\theta}$), but they also have \nan impact on the shape of the distribution of the corresponding cardiac index. More details on this can be found in Section D of the Supplement.\n\n\n\\section{Marginal distributions and random partitions}\n\\label{s:partition}\nAs emphasized in the previous sections, ties among the $\\theta_{j,m}$'s and the $(\\xi_{i,j,m},\\sigma_{i,j,m}^2)$'s are relevant for inferring the clustering structure both among the populations (hypertensive diseases) and among the individual units (patients). Indeed, for each $m$ (cardiac index) they induce a random partition that emerges as a composition of two partitions generated respectively by the prior in \\eqref{eqnew} and the s-HDP. The laws of these random partitions are not only crucial to understand the clustering mechanism, but also necessary in order to derive posterior sampling schemes. In this section such a law is derived and used to compute the predictive distributions that, jointly with the likelihood, determine the full conditionals of the Gibbs sampler devised in Section~\\ref{s:inference}. \nTo reduce the notational burden, in this and the following section, we remove the dependence of observations and parameters on the specific response variable $m$ and denote with $\\phi_{i,j}$ the pair $(\\xi_{i,j},\\sigma_{i,j}^2)$ and with $\\boldsymbol{\\phi}$ the collection $(\\phi_{1,1},\\ldots,\\phi_{1,n_1},\\phi_{2,1},\\ldots \\phi_{n_J,J})$.  \n \nConditionally on $\\omega$, the law of the partition in \\eqref{eq:prior2} leads to the following predictive distribution for the disease-specific locations\n\\[\n\\theta_j\\,|\\,\\omega,\\theta_1,\\ldots,\\theta_{j-1}\\:\\sim\\:\na_j(\\omega,\\theta_1,\\ldots,\\theta_{j-1})\\,\\delta_{\\theta_{j-1}}+\n\\left[1-a_j(\\omega,\\theta_1,\\ldots,\\theta_{j-1})\\right] G\n\\]\nwhere \n\\begin{equation}\n\\label{eq:a}\na_j(\\omega,\\theta_1,\\ldots,\\theta_{j-1}) = \\frac{\\sum_{(*_j)}\\Pi^{(J)}_{k}(n_1,\\ldots,n_k)}{\\sum_{(\\Delta_j)}\\Pi^{(J)}_{k}(n_1,\\ldots,n_k)}\n\\end{equation}\nwhere the sum at the denominator runs over the set of partitions consistent with the one generated by\n \t$(\\theta_1,\\ldots,\\theta_{j-1})$ and the one at the numerator runs over a subset of those \n \tpartitions where one further has $\\theta_j=\\theta_{j-1}$. For $j=4$, the predictive equals\n\\[\n\\theta_{4}\\mid \\omega,\\theta_{1}, \\theta_{2},\\theta_{3} \\begin{cases} \n\t\\frac{3}{\\omega+3}\\delta_{\\theta_{3}} + \n\t\\frac{\\omega}{\\omega+3}G &\\mbox{if }\\theta_{1}=\\theta_{2}=\\theta_{3}\\\\\n\t\\frac{2}{\\omega+2}\\delta_{\\theta_{3}} + \n\t\\frac{\\omega}{\\omega+2}G &\\mbox{if }\\theta_{1}\\neq\\theta_{2}=\\theta_{3}\\\\\n\t\\frac{1}{\\omega+1}\\delta_{\\theta_{3}} + \n\t\\frac{\\omega}{\\omega+1}G&\\mbox{otherwise}\n\t\\end{cases}\n\\]\nExplicit expressions for the function $a$, for $j=1,2,3$, can be easily computed using \\eqref{eq:a} and \\eqref{eq:prior2} and are provided in Section B of the Supplement.\n\nMoving to second-level partitions induced by the s-HDP, we recall that the key concept for studying random partitions on multi-sample data is the \\textit{partially exchangeable partition probability function} (pEPPF). See, e.g., \\cite*{lnp2014} and \\citet{camerlenghi2019b}. The pEPPF returns the probability of \na specific multi-sample partition and represents the appropriate generalization of the well-known single-sample EPPF, which in the DP case corresponds to \\eqref{eq:eppf_dir}.\nDiscreteness of the s-HDP $(\\tilde q_1,\\ldots,\\tilde q_m)$ in \\eqref{eq:s-hdp_def} induces a partition of the elements of \n$\\boldsymbol{\\phi}$ into equivalence classes identified by the distinct values. Taking into account the underlying partially exchangeable structure, such a random partition is characterized by the pEPPF\n\\begin{equation}\n\t\\label{eq:peppf}\n\t\\tilde \\Pi_k^{(N)}(\\bm{n}_1,\\ldots,\\bm{n}_J)=\\mathbb{E}\\left(\\int_{\\Phi^k}\n\t\\prod_{j=1}^J \\prod_{h=1}^k\\tilde q_{j,m}^{n_{j,h}}(\\mathrm{d}\\phi_i)\\right)\n\\end{equation}\nwhere $\\bm{n}_{j}=(n_{j,1},\\ldots,n_{j,k})$ are non-negative integers, for any $j=1,\\ldots,J$, such that $n_{j,h}$ is the number of elements in $\\bm{\\phi}$ corresponding to population $j$ and belonging to cluster $h$. Thus $\\sum_{j=1}^J n_{j,h}\\ge 1$ for any $h=1,\\ldots,k$, $\\sum_{h=1}^kn_{j,h}=n_j$ and $\\sum_{h=1}^k \\sum_{j=1}^J n_{j,h}=N$.\nThe determination of probability distributions of this type is challenging and only recently the first explicit instances have appeared in the literature. See e.g., \\citet{lnp2014}, \\citet{camerlenghi2019} and \\citet{camerlenghi2019b}. With respect to the hierarchical case considered in \\citet{camerlenghi2019b}, the main difference is that here we have to take into account the specific structure \\eqref{eq:inv_DP} of the $\\tilde q_{j,m}$. The almost sure symmetry of the process generates a natural random matching between sets in the induced partition. Therefore, instead of studying the marginal law in \\eqref{eq:peppf}, we derive the joint law of the partition and of the random matching. Formally, consider a specific partition $\\{A_1^{+},A_1^{-},\\ldots,A_k^{+},A_k^{-}\\}$ of $\\boldsymbol{\\phi}$, such that, for $h=1,\\ldots,k$, all the elements in $A_h^+$ belong to $\\mathbb{R}^+\\times\\mathbb{R}^+$, all the elements in $A_h^-$ belong to $\\mathbb{R}^-\\times\\mathbb{R}^+$ and, if $\\phi_{i,j}\\in A_h^+$ and $\\phi_{i',j'}\\in A_h^-$, then the element-wise absolute values of $\\phi_{i,j}$ and $\\phi_{i',j'}$ are equal. Denote with $n^+_{j,h}$ the number of elements in $A_h^{+}\\cap\\{\\phi_{i,j},i=1,\\ldots,n_j\\}$ and with $n^-_{j,h}$ the number of elements in $A_h^{-}\\cap\\{\\phi_{i,j},i=1,\\ldots,n_j\\}$. The probability of observing  $\\{A_1^{+},A_1^{-},\\ldots,A_k^{+},A_k^{-}\\}$ is\n\\begin{equation}\n\t\\label{eq:peppfsym}\n\t\\ddtilde{\\Pi}_{k}^{(N)}(\\boldsymbol{n_1}^+,\\boldsymbol{n_1}^-,\\ldots,\\boldsymbol{n_J}^+,\\boldsymbol{n_J}^-) = \\mathbb{E}\\left(\\int_{\\Phi^k}\\prod_{j=1}^J\\prod_{h=1}^k{\\tilde q_{j,m}}^ {n^{+}_{j,h}+n^{-}_{j,h}}(d\\phi)\\right)\n\\end{equation} \nwith $\\boldsymbol{n_j^+}=(n_{j,1}^+,\\ldots,n_{j,k}^+)$. As for the determination of \\eqref{eq:peppfsym}, a more intuitive understanding may be gained if one considers its corresponding Chinese restaurant franchise (CRF) metaphor, which displays a variation of both the standard Chinese restaurant franchise of \\cite{teh2006} and the skewed Chinese restaurant process of \\cite*{iglesias2009nonparametric}. Figure~\\ref{fig:figure2} provides a graphical representation. \n\\begin{figure}\n\t\\centering\n\t\\resizebox{14cm}{!}{\n\t\t\\begin{tikzpicture}[-latex, auto, node distance = 2.4 cm and 3.4cm, \n\t\ton grid,\n\t\tsemithick,\n\t\tstate\/.style ={circle, top color = white, bottom color = white,\n\t\t\tdraw, black, text=black, minimum width = 0.5 cm},\n\t\tfill fraction\/.style n args={2}{path picture={\n\t\t\t\t\\fill[#1] (path picture bounding box.south west) rectangle\n\t\t\t\t($(path picture bounding box.north west)!#2!(path picture bounding box.north east)$);}},\n\t\ttransition\/.style = {rectangle, draw=black!50, thick, minimum width=17cm, minimum height = 6.7cm} ]\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (A)\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_1}$\\quad $-\\boldsymbol{\\phi^{**}_1}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (B) [right =of A]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_2}$\\quad $-\\boldsymbol{\\phi^{**}_2}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (C) [right =of B] \t{\\Large$\\boldsymbol{\\phi^{**}_3}$\\quad $-\\boldsymbol{\\phi^{**}_3}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (D) [right =of C] \t{\\Large$\\boldsymbol{\\phi^{**}_1}$\\quad $-\\boldsymbol{\\phi^{**}_1}$};\n\t\t\\node[circle, fill, draw, scale=0.3] (M) [right =1.5 cm of D] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (N) [right =0.5 cm of M] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (O) [right =0.5 cm of N] { };\n\t\t\\node[draw=none,fill=none] (E) [above left =1.5 cm of A] {\\large$\\phi_{1,1}$};\n\t\t\\node[draw=none,fill=none] (F) [above left =1.55 cm of B] {\\large$\\phi_{2,1}$};\n\t\t\\node[draw=none,fill=none] (G) [below left =0.60 cm of E] {\\large$\\phi_{3,1}$};\n\t\t\\node[draw=none,fill=none] (H) [left =1.56 cm of A] {\\large$\\phi_{4,1}$};\n\t\t\\node[draw=none,fill=none] (I) [above left =1.55 cm of C] {\\large$\\phi_{5,1}$};\n\t\t\\node[draw=none,fill=none] (L) [above right=1.55 cm of B] {\\large$\\phi_{6,1}$};\n\t\t\\node[draw=none,fill=none] (P) [below left =1.35 cm of A] {  };\n\t\t\\node[draw=none,fill=none] (Q) [above right=1.50 cm of A] {\\large$\\phi_{7,1}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (AA) \n\t\t[below =3.2cm of A]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_3}$\\quad $-\\boldsymbol{\\phi^{**}_3}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (BB) \n\t\t[below =3.2cm of B]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_1}$\\quad $-\\boldsymbol{\\phi^{**}_1}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (CC) \n\t\t[below =3.2cm of C] \t{\\Large$\\boldsymbol{\\phi^{**}_4}$\\quad $-\\boldsymbol{\\phi^{**}_4}$};\n\t\t\\node[circle, draw, scale=0.7, fill fraction={darkgray}{0.5}] (DD) \n\t\t[below =3.2cm of D]\n\t\t{\\Large$\\boldsymbol{\\phi^{**}_5}$\\quad $-\\boldsymbol{\\phi^{**}_5}$};\n\t\t\\node[circle, fill, draw, scale=0.3] (MM) [right =1.5 cm of DD] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (NN) [right =0.5 cm of MM] { };\n\t\t\\node[circle, fill, draw, scale=0.3] (OO) [right =0.5 cm of NN] { };\n\t\t\\node[draw=none,fill=none] (EE) [above left =1.5 cm of AA] {\\large$\\phi_{4,2}$};\n\t\t\\node[draw=none,fill=none] (FF) [above left =1.55 cm of BB] {\\large$\\phi_{2,2}$};\n\t\t\\node[draw=none,fill=none] (GG) [below left =0.60 cm of EE] {\\large$\\phi_{3,2}$};\n\t\n\t\t\\node[draw=none,fill=none] (II) [above left =1.55 cm of CC] {\\large$\\phi_{5,2}$};\n\t\t\\node[draw=none,fill=none] (LL) [above right=1.55 cm of CC] {\\large$\\phi_{6,2}$};\n\t\t\\node[draw=none,fill=none] (PP) [below left =1.35 cm of AA] {  };\n\t\t\\node[draw=none,fill=none] (QQ) [above right=1.5 cm of AA] {\\large$\\phi_{7,2}$};\n\t\t\\node[transition] (R)  [above  left=2.2 cm of CC]{ };\n\t\\end{tikzpicture}\n}\n\\caption{{Chinese restaurant franchise representation of the symmetric hierarchical DP for $J=2$ populations. Each circle represents a table.}}\n\\label{fig:figure2}\n\\end{figure}\nThe scheme is as follows: there are $J$ restaurants\nsharing the same menu and the customers are identified by their choice of \n$\\phi_{i,j}$ but, unlike in the usual CRF, at each table two \\textit{symmetric dishes} are served. Denote with $\\phi^*_{t,j}=(\\xi^*_{t,j},\\, \\sigma_{t,j}^{2*})$ and $-\\phi^*_{t,j}=(-\\xi^*_{t,j},\\, \\sigma_{t,j}^{2*})$ the two dishes served at table $t$ in restaurant $j$, with \n$\\phi^{**}_{h}=(\\xi^{**}_{h},\\, \\sigma^{**2}_{h})$ and $-\\phi^{**}_{h}=(-\\xi_{h},\\, \\sigma_{h}^{**2})$ the $h$-th pair of dishes in the menu and with $n_{j,h}^{+}$ and $n_{j,h}^{-}$ the number of customers in restaurant $j$ eating dish $\\phi^{**}_{h}$ and $-\\phi^{**}_{h}$, respectively. This means that two options are available to a customer entering restaurant \n$j$: she\/he will either sit at an already occupied table, with probability proportional to the number of customers at that table or will sit at a new table with probability proportional to the concentration parameter $\\gamma_j$. In the former case, the customer will choose the dish $\\phi^*_{t,j}$ with probability $1\/2$ and $-\\phi^*_{t,j}$ otherwise. In the latter \ncase, the customer will eat a dish served at another table of the franchise with probability proportional to half the number of tables that serve that dish, or will make a new order with probability proportional to the concentration parameter $\\alpha$. In view of this scheme, the probability in \\eqref{eq:peppfsym} turns out to be\n\\[\n\\ddtilde{\\Pi}_{k}^{(N)}(\\boldsymbol{n_1^+},\\ldots,\\boldsymbol{n_J^-}) = 2 ^ {-N}\\bar\\Pi_{k}^{(N)}(\\boldsymbol{n_1^+}+\\boldsymbol{n_1^-},\\ldots,\\boldsymbol{n_J^+}+\\boldsymbol{n_J^-}) \n\\]\nand $\\bar{\\Pi}_k^{(N)}$ on the right-hand-side is the pEPPF of the hierarchical DP derived in \\citet{camerlenghi2019b}, namely\n\\[\n\\bar{\\Pi}_k^{(N)}(\\bm{n}_1,\\ldots,\\bm{n}_k)=\\Biggl(\\prod_{j=1}^J \\frac{\\prod_{i=1}^k (\\gamma_j)_{n_{j,h}}}{(\\gamma_j)_{n_j}}\\Biggr)\\:\n\\sum\\limits_{\\boldsymbol{\\ell}}\\frac{\\alpha^k}{(\\alpha)_{|\\boldsymbol{\\ell}|}}\\:\n\\prod_{h=1}^k (\\ell_{\\bullet,h} -1)!\\prod_{j=1}^J P(K_{n_{j,h}}=\\ell_{j,h})\n\\] \nwhere each sums runs over all $\\ell_{j,h}$ in $\\{1,\\ldots,n_{j,h}\\}$, if $n_{j,h}\\ge 1$, and equals $1$ if $n_{j,h}=0$, whereas $\\ell_{\\bullet,h}=\\sum_{j=1}^J \\ell_{j,h}$ and $|\\boldsymbol{\\ell}| = \\sum_{j=1}^J \\sum_{h=1}^k \\ell_{j,h}$. Note that the latent variable $\\ell_{j,h}$ is the number of tables in restaurant $j$ serving the $h$-th pair of dishes. Moreover, $K_{n_{j,h}}$ is a random variable denoting the number of distinct clusters, out of $n_{j,h}$ observations generated by a DP with parameter $\\gamma_j$ and diffuse baseline $P_0$ and it is well-known that  \n\\[\n\\mathbb{P}(K_{n_{j,h}}=\\ell_{j,h})=\\frac{\\gamma_j^{\\ell_{j,h}}}\n{(\\gamma_j)_{n_{j,h}}}\\:\n|\\mathfrak{s}(n_{j,h},\\ell_{j,h})|\n\\] \nwhere $|\\mathfrak{s}(n_{j,h},\\ell_{j,h})|$ is the signless Stilring number of the first kind. \nIn view of this, one can deduce the predictive distribution\n\\begin{equation*}\n\\begin{split}\n\t\\mathbb{P}(\\phi_{n_j+1,j}\\in \\cdot&\\,|\\, \\boldsymbol{\\phi}) =\n\t\\frac{\\gamma_j}{i-1+\\gamma_j}\n\t\\sum\\limits_{\\boldsymbol{\\ell}}\n\t\\frac{\\alpha}{|\\boldsymbol{\\ell}| + \\alpha}\\pi(\\boldsymbol{\\ell}\\,|\\,\\boldsymbol{\\phi})P_0(\\cdot)\\\\\n\t&+\\sum\\limits_{h=1}^{k}\n\t\\left[\\frac{n_{j,h}^+ + n_{j,h}^-}{n_j+\\gamma_j} + \n\t\\frac{\\gamma_j}{n_j+\\gamma_j}\n\t\\sum\\limits_{\\boldsymbol{\\ell}}\n\t\\frac{\\ell_{\\bullet,h}}{|\\boldsymbol{\\ell}| + \\alpha}\\pi(\\boldsymbol{\\ell}\\,|\\,\\boldsymbol{\\phi})\\right]\n\t\\Bigg(\\frac{\\delta_{\\phi^{**}_{h}}(\\cdot)+\\delta_{-\\phi^{**}_{h}}(\\cdot)}{2}\t\\Bigg)\n\\end{split}\n\\end{equation*}\nwhere \n\\begin{equation*}\n\\pi(\\boldsymbol{\\ell}\\,|\\,\\boldsymbol{\\phi})\\propto\n\\frac{\\alpha^k}{(\\alpha)_{|\\boldsymbol{\\ell}|}}\n\\prod_{h=1}^k (\\ell_{\\bullet,h} -1)!\\prod_{j=1}^J\n\\frac{\\gamma_j^{\\ell_{j,h}}}\n{(\\gamma_j)_{n_{j,h}^+ + n_{j,h}^-}}\\:\n|\\mathfrak{s}(n_{j,h}^+ + n_{j,h}^-,\\ell_{j,h})|\\mathds{1}_{\\{1,\\ldots,n_{j,h}^+ + n_{j,h}^-\\}}(\\ell_{j,h})\n\\end{equation*}\nis the posterior distribution of the latent variables $\\ell_{j,h}$'s, and $\\mathds{1}_A$ is the indicator function of set $A$.\n\n\\section{Posterior inference}\n\\label{s:inference}\nThe findings of the previous section are the key ingredients to perform posterior inference with a marginal Gibbs sampler.\nThe output of the sampler is structured into three levels: the first produces posterior probabilities on partitions of disease-specific locations; the second generates density estimates; the third provides clusters of patients. \nFor notational simplicity, we omit\nthe dependence on $m$, except for the description of the sampling step that generates $\\omega$.\nRecall that ${\\boldsymbol{\\theta}}=(\\theta_{1},\\ldots,\\theta_{J}) $ and \n$\\boldsymbol{\\phi}=\\{(\\phi_{1,j},\\ldots,\\phi_{n_j,j}):\\,j=1,\\ldots,J\\}$, with $\\phi_{i,j}=(\\xi_{i,j},\\sigma_{i,j}^2)$. The target distribution of \nthe sampler is the joint distribution of $\\bm{\\theta}$, $\\bm{\\phi}$ and $\\omega$ conditionally on the observed data $\\bm{X}$.\n\n\\textbf{Sampling $\\boldsymbol{\\phi}$.} In view of the CRF representation of the \ns-HDP, $t_{i,j}$ stands for the label of the table where the $i$-th customer in restaurant $j$ sits \nand $h_{t,j}$ for the dish label served at table $t$ in restaurant $j$ and with $\\boldsymbol{t}$ and $\\boldsymbol{h}$ we denote the corresponding arrays. \nMoreover, define the assignment variable\n$s_{i,j}=\\mathds{1}(\\phi_{i,j}=\\phi^*_{t_{i,j},j})-\n\\mathds{1}(\\phi_{i,j}=-\\phi^*_{t_{i,j},j})$ and $\\boldsymbol{s}$ is the corresponding arrays. In order to generate $\\boldsymbol{\\phi}$, we need to sample \n\\begin{itemize}\n\\item[(i)]  $(t_{i,j}, s_{i,j})$ for $i=1,\\ldots,n_j$ and $j=1,\\ldots,J$;\n\\item[(ii)] $h_{t,j}$ for $t \\in \\boldsymbol{t}$ and $j=1,\\ldots,J$;\n\\item[(ii)] $\\phi^{**}_{h}$ for $h \\in \\boldsymbol{h}$. \n\\end{itemize}\n\nNote that, using the latent allocation indicators in $\\boldsymbol{t}$ and $\\boldsymbol{h}$, the sampling scheme is more efficient than sampling directly from the full conditional of each $\\phi_{i,j}$, since the algorithm can update more than one parameter simultaneously \\citep{neal2000}. Define $\\varepsilon_{i,j} = X_{i,j} - \\theta_{j}$ and denote with $h(\\varepsilon_{i,j,m}|\\phi^*)$ the conditional normal density of $\\varepsilon_{i,j}$\ngiven $\\phi^*=(\\xi^*,\\sigma^{2*})$, while the marginal density\nis\n\\[\n\\bar h (\\varepsilon_{i,j}) = \\int h(\\varepsilon_{i,j}|\\phi) P_{0}(d\\phi)\n\\]\n\nTo sample $(t_{i,j}, s_{i,j})$ from their joint full conditional, we first sample $t_{i,j}$ from\n\\[\nP(t_{i,j}=t\\mid \\boldsymbol{t}^{-(i,j)}, \\boldsymbol{h}^{-(i,j)}, \\boldsymbol{\\phi}^{*-({i,j})},\\boldsymbol{\\phi}^{**-({i,j})},\\varepsilon_{i,j}) \\propto\n\\begin{cases} n_{t,j}^{-(i,j)}\\,p_{\\mbox{\\footnotesize old}}(\\varepsilon_{i,j}|\\phi^*_{t,j})\n& \\mbox{if } t\\in \\boldsymbol{t}^{-(i,j)}\n\\\\ \n\\gamma_{j}\\,p_{\\mbox{\\footnotesize new}}(\\varepsilon_{i,j}|{\\boldsymbol{\\phi}^{**-({i,j})}})\n& \\mbox{if } t=t^{\\mbox{new}}\\end{cases}\n\\]\nwhere $\\boldsymbol{t}^{-(i,j)}$, $\\boldsymbol{h}^{-(i,j)}$ $\\boldsymbol{\\phi}^{*-({i,j})}$,$\\boldsymbol{\\phi}^{**-({i,j})}$ coincide with the arrays $\\boldsymbol{t}$, $\\boldsymbol{h}$ $\\boldsymbol{\\phi}^*$,$\\boldsymbol{\\phi}^{**}$ after having removed the entries corresponding to the $i$-th customer in restaurant $j$. Moreover\n\\[\np_{\\mbox{\\footnotesize old}}(\\varepsilon_{i,j} | \\phi^*_{t,j}) \n=\n\\frac{1}{2}h(\\varepsilon_{i,j}|\\phi^*_{t,j})  + \\frac{1}{2}h(\\varepsilon_{i,j}|-\\phi^*_{t,j})\n\\]\nand\n\\[\np_{\\mbox{\\footnotesize new}}(\\varepsilon_{i,j}|{\\boldsymbol{\\phi}^{**-({i,j})}}) \n= \\sum\\limits_{h=1}^{k^{-(i,j)}}\\frac{\\ell_{\\bullet,h}}{|\\boldsymbol{\\ell}|+ \\alpha}\\left\\{\\frac{1}{2}h(\\varepsilon_{i,j}|\\phi^{**}_h)+\\frac{1}{2}h(\\varepsilon_{i,j}|-\\phi^{**}_h)\t\\right\\}+\\frac{\\alpha}{|\\boldsymbol{\\ell}|+ \\alpha} \\bar h (\\varepsilon_{i,j})\n\\]\t\nThen we sample $s_{i,j}$ from its full conditional\n\\[\np(s_{i,j}=s\\mid\\boldsymbol{\\phi}^*, t_{i,j},\\epsilon_{i,j}) \\propto \\begin{cases} h(\\varepsilon_{i,j}|\\phi^*_{t_{i,j}}) & \\mbox{if } s=1 \\\\ h(\\varepsilon_{i,j}|-\\phi^*_{t_{i,j}}) & \\mbox{if } s=-1\\end{cases}\n\\]\nThe conditional distribution of $h_{t,j}$ is\n\\[\np(h_{t,j}=h\\mid \\boldsymbol{t},\\boldsymbol{h}^{-(t,j)},\\boldsymbol{\\phi}^{**-(t,j)},\\boldsymbol{s},\\boldsymbol{\\varepsilon}) \n\\propto\n\\begin{cases} \\ell_{\\bullet,h}^{-(t,j)}\\prod\\limits_{\\{(i,j):\\:t_{i,j}=t\\}}\nh(s_{i,j}\\,\\varepsilon_{i,j}|\\phi_{h})\n& \\mbox{if } h \\in \\boldsymbol{h}^{-(t,j)} \\\\ \n\\alpha\\,\\displaystyle\\int \\prod\\limits_{\\{(i,j):\\:t_{i,j}=t\\}} h(s_{i,j}\\,\\varepsilon_{i,j}|\\phi) P_{0}(d\\phi) & \\mbox{if } h=h^{new}\\end{cases}\n\\]\n\nFinally, when $P_{0}$ is conjugate with respect to the Gaussian kernel, the full conditional distribution of $\\phi_h^{**}$ is obtained in closed form as posterior distribution of a Gaussian model, using as observations the collection $\\{\\,(s_{i,j}\\,\\epsilon_{i,j}):\\: h\t_{t_{i,j},j} = h\\}$. \n\n\\textbf{Sampling $\\boldsymbol{\\theta}$.} When sampling the disease-specific location parameters, one can rely on \na Chinese restaurant process restricted to those partitions that are consistent with the ordering of the diseases.\nThus, in order to generate $\\boldsymbol{\\theta}$, we first sample the labels $\\boldsymbol{t}_{\\theta}=\\{t_{1},\\ldots,t_{J}\\}$, where $t_{j}$ is the label of the table where the $j$-th customer sits. Then, we sample the dish $\\theta^*_{t}$ associated to table $t$ for all $t\\in\\boldsymbol{t}_{\\theta}$. If $z_{i,j} = X_{i,j} - \\xi_{i,j}$, the conditional density of $\\bm{z}_{j}=(z_{1,j},\\ldots,z_{n_j,j})$ associated to the location parameter $\\theta^*$,\ngiven $\\bm{\\sigma}_{j}=(\\sigma_{1,j},\\ldots,\\sigma_{n_j,j})$, is\n\\[\nf_{\\theta^*}(\\bm{z}_{j}|\\bm{\\sigma}_{j})=\\frac{1}{\\sqrt{2\\pi}\\prod\\limits_{i=1}^{n_j} \\sigma_{i,j}}\\:\n\\exp\\left\\{-\\frac{1}{2}\\sum\\limits_{i=1}^{n_j}\\frac{(z_{i,j}-\\theta^*)^2}{\\sigma_{i,j}^2}\n\\right\\}\n\\]\nUnder the prior in \\eqref{eqnew}, the full conditional distribution of $\\boldsymbol{t}_{\\theta}$ is provided by\n\\[\n\\begin{split}\n\tp(t_{j}=t\\mid t_1,\\ldots,&t_{j-1},\\theta_{j-1},\\bm{z}_{j}, \\bm{\\sigma}_{j})\\\\\n\t&\\propto\n\t\\begin{cases} \n\t\ta(\\omega,\\theta_1,\\ldots,\\theta_{j-1})\\,f_{\\theta_{j-1}}(\\bm{z}_{j}|\\bm{\\sigma}_{j}) \n\t\t& \\mbox{if } t=t_j \\\\[4pt] \n\t\t[1 - a(\\omega,\\theta_1,\\ldots,\\theta_{j-1})] \\, \n\t\t\\displaystyle\\int f_{\\theta}(\\bm{z}_{j}|\\bm{\\sigma}_{j})\\, G (d\\theta)& \\mbox{if } t=t^{\\mbox{\\footnotesize new}}\\\\[4pt]\n\t\t0&\\mbox{otherwise}\n\t\\end{cases}\n\\end{split}\n\\]\nFinally, when $G$ is conjugate with respect to the Gaussian kernel, the full conditional distribution of $\\theta_{t}^{*}$, given $\\{\\bm{z}_{j}:\\: t_{j}=t\\}$, is obtained in closed form using conjugacy of the Normal-Normal model.\n\n\\textbf{Sampling the concentration parameter.} Finally, \nthe concentration parameter $\\omega$ can be sampled through an importance sampling step using as importance distribution the prior $p_{\\omega}$ over $\\omega$. Denoting with $M_m$ the selected partition for $\\bm{\\theta}_{m}$ and with $T_m$ the number of clusters in $M_m$, we have \n\\[\np(\\omega\\,|\\,M_m:m=1,\\ldots,M)\\propto p_{\\omega} (\\omega)  \\,\\frac{\\omega^ {\\sum_{m=1}^M T_m - M}}{(\\omega + 2)^{M} (\\omega^2 + \\omega + 3)^{M}}.\n\\]\n\n\\section{Results}\n\\label{s:results}\n\\subsection{Simulation studies}\n\\label{ss:sim}\nWe perform a series of simulation studies with two main goals. First, we aim to highlight the drawbacks of clustering based on the entire distribution with respect to our proposal in the context of small sample sizes. Second, we check the model's ability of detecting the presence of underlying relevant factors in the sense described in Section~\\ref{ss:s-HDP}. \n\nTo accomplish the first goal, we compare the results obtained using our model with the nested Dirichlet process (NDP) \\citep{rodriguez2008}, arguably the most popular Bayesian model to cluster populations. Mimicking the real hypertensive dataset, we simulate data for 4 samples, ideally corresponding to four diseases, with respective sample sizes \nof 50, 19, 9 and 22, which correspond to the sample sizes of the real data investigated in Section \\ref{ss:real}. Since the NDP does not allow to treat jointly multiple response variables, we consider only one response variable to ensure a fair comparison. The observations are sampled from the following distributions and 100 simulation studies are performed.\n\\begin{equation*}\n\\begin{split}\n\tX_{i,1}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,0,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,2,\\,0.5\\,) \\qquad&\\text{for}\\enskip i=1,\\ldots,n_1\\\\\n\tX_{i,2}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,2,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,4,\\,0.5\\,)\\qquad&\\text{for}\\enskip i=1,\\ldots,n_2\\\\\n\tX_{i,3}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,4,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,6,\\,0.5\\,) \\qquad&\\text{for}\\enskip i=1,\\ldots,n_3\\\\\n\tX_{i,4}\\,\\overset{iid}{\\sim}0.5\\,\\mathcal{N}(\\,6,\\,0.5\\,)+0.5\\,\\mathcal{N}(\\,8,\\,0.5\\,) \\qquad&\\text{for}\\enskip i=1,\\ldots,n_4\\\\\n\\end{split}\n\\end{equation*}\nNote that the true data generating process corresponds to samples from distinct distributions with pairwise sharing of a mixture component. Alternative scenarios are considered in the additional simulation studies that can be found in Section D of the Supplement.\n\nThe implementation of the NDP was carried out through the marginal sampling scheme proposed in \\cite{zuanetti2018clustering}, which is suitably extended \nto accommodate hyperpriors on the concentration parameters of the NDP. To simplify the choice of the hyperparameters, as suggested by \\citet[p.~535 and p.~551--554]{gelman2013bayesian} we estimate both models over standardized data. For our model, we set $G_m=\\mathcal{N}(0,\\,1)$ and $P_{0,m}=\\text{NIG}(\\mu=0, \\,\\tau=1,\\, \\alpha=2,\\,\\beta=4)$. Here, $\\text{NIG}(\\mu, \\,\\tau,\\, \\alpha,\\,\\beta)$ indicates a normal inverse gamma distribution. The base distribution for the NDP is $\\text{NIG}(\\mu=0, \\,\\tau=0.01,\\, \\alpha=3,\\,\\beta=3)$, as in \\cite{rodriguez2008}. Finally, we use gamma priors with shape 3 and rate 3 for all concentration parameters, which is a common choice.\nFor each simulation study, we perform 10,000 iterations of the MCMC algorithms with the first 5,000 used as burn-in.\n\n\\begin{table}[t]\n\t\t\\caption{Simulation studies summaries.}\n\t\\label{tab:table1}\n\\begin{center}\n\t\t\\begin{tabular}{lcccccc}\n\t\t\t&\\multicolumn{3}{c}{\\textbf{sHDP}}& \\multicolumn{3}{c}{\\textbf{NDP}}\\\\\\hline\\hline\n\t\t\t&MAP&Average&Median&MAP&Average&Median\\\\\n\t\t\tPartitions& count& post. prob.&post. prob.& count& post. prob.& post. prob.\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&2&0.020&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&\\textbf{72}&\\textbf{0.695}&\\textbf{0.860}\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}\\{\\textcolor{burntorange}{2}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3},\\textcolor{burgundy}{4}\\}&0&0.027&0.007&3&0.035&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&0&0.000&0.000&5&0.061&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{burgundy}{4}\\}\\{\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&1&0.054&0.015&0&0.014&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burntorange}{2},\\textcolor{burgundy}{4}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2},\\textcolor{burgundy}{4}\\}\\{\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2},\\textcolor{burgundy}{4}\\}\\{\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1},\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&0&0.004&0.000&18&0.175&0.032\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{burgundy}{4}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{1},\\textcolor{burgundy}{4}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3}\\}&0&0.000&0.000&0&0.000&0.000\\\\\\hline\n\t\t\t\\{\\textcolor{aoe}{1}\\}\\{\\textcolor{burntorange}{2}\\}\\{\\textcolor{bostonuniversityred}{3}\\}\\{\\textcolor{burgundy}{4}\\}&\\textbf{99}&\\textbf{0.915}&\\textbf{0.954}&0&0.000&0.000\n\t\\end{tabular}\n\\end{center}\n\\vspace{-\\baselineskip}\n\\end{table}\n\nTable~\\ref{tab:table1} displays summaries of the results on population clustering, darker rows correspond to partitions that are not consistent with the natural ordering of the diseases. The true clustering structure is given by the finest partition. \nAs already observed in \\cite{rodriguez2008}, the NDP tends to identify fewer, rather than more clusters, due to the presence of small sample sizes. Using the \\textit{maximum a posteriori} estimate, our model correctly \nidentifies the partition in 99 out of 100 simulation studies and a partition with three elements or more in 100 out of 100 simulation studies. The \nsame counts for the NDP are, respectively, 0 out of 100 and 21 out of 100. Analogous conclusions can be drawn looking at posterior probability averages and medians across the 100 simulation studies (see Table~\\ref{tab:table1}) leaving \nno doubt about the model to be preferred under this scenario. \n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{7cm}\n\t\t\\centering\n\t\t\\includegraphics[ width=7cm]{figures\/confmean26}\n\t\t\\caption{95\\% credible intervals for population-specific locations}\n\t\t\\label{fig:fig3a}\n\t\\end{subfigure}\\hspace{0.2cm}%\n\t\\begin{subfigure}{7cm}\n\t\n\t\t\\centering\n\t\t\\includegraphics[width=7cm]{figures\/K_secondlevel_scenario0_shdp26}\n\t\t\\caption{Number of second-level clusters.}\n\t\t\\label{fig:fig3b}\n\t\\end{subfigure}\n\t\\begin{subfigure}{7cm}\n\t\n\t\t\\centering\n\t\t\\includegraphics[ width=7cm, trim={2cm 0 2cm 0}]{figures\/cluster_scenario0_natord26}\n\t\t\\caption{Co-clustering.}\n\t\t\\label{fig:fig3c}\n\t\\end{subfigure}\\hspace{0.2cm}%\n\t\\begin{subfigure}{7cm}\n\t\n\t\t\\centering\n\t\t\\includegraphics[width=7cm,, trim={2cm 0 2cm 0}]{figures\/cluster_scenario0_reord26}\n\t\t\\caption{Co-clustering.}\n\t\t\\label{fig:fig3d}\n\t\\end{subfigure}\n\\caption{ Panel (a): Mean point estimates and 95\\% credible intervals for the four populations, vertical lines correspond to true values. Panel (b): Posterior distribution on the number of second-level clusters. Panels (c) and (d): heatmaps of second-level clustering, darker colors correspond to higher probability of co-clustering; in (c) patients are ordered based on the diagnosis and the four black squares highlight the within-sample probabilities and in (d) patients are reordered based on co-clustering probabilities.}\n\\end{figure}\n\nFinally, we randomly select three simulation studies among the 100 to better understand the performance in estimating the other model parameters. Here we comment on one of the studies, the other two leading to similar results are reported in Section D.1.1 of the Supplement. \nFigure~\\ref{fig:fig3a} shows point estimates and credible intervals for the population-specific location parameters $\\theta_1,\\theta_2,\\theta_3,\\theta_4$. The true means belong to the 95\\% credible intervals.\n\nMoreover, it turns out that the model is able to detect the presence of two clusters of subjects leading to a posterior distribution for the number of clusters that is \nrather concentrated on the true value, see Figure~\\ref{fig:fig3b}--\\ref{fig:fig3d}. Moreover, the point estimate for the subject partition, obtained minimizing the Binder loss function, also contains two clusters, proving the ability of the model to detect the underlying relevant factor. In Section D of the Supplement, a number of additional simulation studies are conducted, both using alternative specifications over the disorder-specific parameters and different data generating mechanisms: the results highlight a good performance of the model, which appears also able to detect outliers, to highlight non-location effects of the disorders and to produce reliable outputs even under deviation from symmetry.\n\n\\subsection{Impact of hypertensive disorders on maternal cardiac dysfunction}\n\\label{ss:real}\nOur analysis is based on the dataset of \\cite{data}, which can be obtained from https:\/\/data.mendeley.com\/datasets\/d72zr4xggx\/1. The dataset contains observations for $10$ cardiac function measurements collected through a prospective case-control study on women in the third semester of pregnancy divided in $n_1=50$ control cases (C), $n_2=19$ patients with gestational hypertension (G), $n_3=9$ patients with mild preeclampsia (M) and $n_4=22$ patients with severe preeclampsia (S). The cases are \nwomen admitted from 2012 to 2014 to King George Hospital in Visakhapatnam, India. The healthy sample is composed by normotensive pregnant women. All women with hypertension were on antihypertensive treatment with oral Labetalol or Nifedipine. Women with severe hypertension were treated with either oral nifedipine and parenteral labetalol or a combination. For more details on the dataset, we refer to \\cite{tatapudi2017}. The prior specification is the same as in the previous section. Sections E.2 and E.3 of the Supplement contain a prior-sensitivity analysis and show rather robust results w.r.t. different prior specifications. Inference is based on 10,000 MCMC iterations with the \nfirst half used as burn-in. \n\n\\begin{table}\n\\caption{Posterior probabilities over partitions of means. Maximum a posteriori probabilities are in \\textbf{bold}.}\n\\label{tab:table2}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{lcccccccccc}\n\t\t\tpartitions&CI&CWI&LVMI&IVST&LVPW&EF&FS&EW&AW&E\/A\\\\\\hline\\hline\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.021&0.000&0.000&0.000&0.000&\\textbf{0.365}&\\textbf{0.303}&0.096&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&\\textbf{0.546}&0.001&0.083&0.016&0.078&0.190&0.021&0.036&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&0.000&0.001&0.000&0.000&0.037&0.038&0.072&0.076&0.049\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.001&0.139&0.001&0.019&0.024&0.028&0.078&0.042&0.232&0.055\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&\\textbf{0.463}&0.000&\\textbf{0.595}&0.000&0.000&0.276&0.045&\\textbf{0.498}&0.020&0.002\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.146&0.099&0.188&\\textbf{0.551}&\\textbf{0.672}&0.074&0.164&0.092&0.260&0.033\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.233&0.000&0.107&0.000&0.000&0.083&0.062&0.114&0.091&0.371\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.133&0.216&0.108&0.347&0.288&0.060&0.121&0.065&\\textbf{0.287}&\\textbf{0.491}\\\\\n\t\t\t\\hline\n\t\t\t$\\sum\\log_{15} \\left(p_i ^{-p_i}\\right)$ &0.501&0.430&0.415&0.361&0.289&0.632&0.688&0.598&0.613&0.424\n\t\\end{tabular}}\n\\end{center}\n\\end{table} \nTable \\ref{tab:table2} displays the posterior distributions for the partitions of unknown disease-specific means along with the corresponding entropy measurements, that can be used as measures of uncertainty. \nFirst note that if one takes also the ordering among distinct disease-specific locations into account, the posterior partition probabilities are, as desired, concentrated on specific orders of the associated unique values for all ten cardiac indexes. For instance, we have \n$\\mathbb{P}(\\{\\theta_{C,CI}=\\theta_{G,CI}=\\theta_{M,CI}\\}\\{\\theta_{S,CI}\\}\\mid X) =\\mathbb{P}(\\theta_{C,CI}=\\theta_{G,CI}=\\theta_{M,CI}>\\theta_{S,CI}\\mid X) = 0.463$. The ordered partitions with the highest posterior probability are displayed in Table~\\ref{tab:tableord}. \n\\begin{table}\n\\caption{Posterior probabilities over ordered partitions of means.}\n\\label{tab:tableord}\n\\begin{center}\n\t\\begin{tabular}{lcc} \n\t\t& ordered partition with& \\\\\n\t\tcardiac index& highest posterior probability&posterior prob\\\\\n\t\t\\hline\\hline\n\t\tCI&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$>$\\{\\textcolor{burgundy}{S}\\}&0.463\\\\\n\t\tCWI&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}&0.546\\\\\n\t\tLVMI&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.595\\\\\n\t\tIVST&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.548\\\\\n\t\tLVPW&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.671\\\\\n\t\tEF&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}&0.365\\\\\n\t\tFS&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}&0.303\\\\\n\t\tEW&\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$>$\\{\\textcolor{burgundy}{S}\\}&0.497\\\\\n\t\tAW&\\{\\textcolor{aoe}{C}\\}$<$\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}$<$\\{\\textcolor{burgundy}{S}\\}&0.256\\\\\n\t\tE\/A&\\{\\textcolor{aoe}{C}\\}$>$\\{\\textcolor{burntorange}{G}\\}$>$\\{\\textcolor{bostonuniversityred}{M}\\}$>$\\{\\textcolor{burgundy}{S}\\}&0.466\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/confmean0}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.45\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{figures\/confmean1}\n\t\\end{subfigure}\n\\caption{95\\% credible intervals for population-specific locations for CI and CWI}\n\\label{fig:fig5}\n\\end{figure}\n\nConsidering the posterior probabilities summarized in Table~\\ref{tab:table2} and in Table~\\ref{tab:tableord}, we find that the cardiac index (CI) is reduced in severe preeclampsia compared to all other patients, indicating reduced myocardial contractility in the presence of the most severe disorder. \nThe cardiac work index (CWI) is a good indicator to distinguish between cases and control, but not among cases. The left ventricular mass index (LVMI) is increased in severe preeclampsia patients compared to other pregnant women, indicating ventricular remodelling. While inter ventricular septal thickness (IVST) and left ventricular posterior wall thickness (LVPW) differ both between cases and controls and between severe preeclampsia and other disorders, indicating a progressive increase in the indexes with the severity of the disorder. The posterior probabilities associated to \nindexes of systolic function such as ejection fraction (EF) and fraction shortening (FS) are relatively concentrated on the partition of complete homogeneity, letting us to conclude that no differences are present among \npatients.\nAs for the parameters of the diastolic function, the posterior distribution for the E-wave indicator identifies a modified index in severe preeclampsia patients, while the mean E\/A ratio indicates a \ndecreasing diastolic function with the severity of the disorder. The posterior for the A-wave index is actually concentrated on three distinct partitions, leaving a relatively high uncertainty regarding the modifications of the index. However, considering jointly the three partitions with the highest posterior probability, differences are detected between control and cases with a total posterior probability equal to 0.779. Figure~\\ref{fig:fig5} shows point estimates and credible intervals for disorder-specific location parameters for the first two cardiac indexes. Analogous plots for all cardiac indexes can be found in Section E.1 of the Supplement.\n\nTable~\\ref{tab:table3} shows the results obtained using the prior in \\eqref{eq:eppf_dir}, instead of \\eqref{eq:prior2}. We remark that for all ten cardiac indexes, the posterior associates negligible probabilities to partitions that are in contrast with the natural order of the diagnoses. This is particularly reassuring in that the model, even without imposing such an order a priori, is able to single it out systematically across cardiac indexes.\nMoreover, we observe how the partitions identified by MAP are the same of Table~\\ref{tab:table2} for all cardiac index except AW. However, even under this alternative prior, the A-wave index is concentrated on the same three distinct partitions leading to the conclusion that there exists a difference between cases and control.\n\\begin{table}\n\t\\caption{Posterior probabilities over partitions of means. Maximum a posteriori probabilities are in \\textbf{bold}.}\n\t\\label{tab:table3}\n\t\\vspace{-0.3cm}\n\t\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\t\t\\begin{tabular}{lcccccccccc}\n\t\t\tpartitions&CI&CWI&LVMI&IVST&LVPW&EF&FS&EW&AW&E\/A\\\\\\hline\\hline\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.019&0.000&0.000&0.000&0.000&\\textbf{0.332}&\\textbf{0.247}&0.078&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&\\textbf{0.643}&0.001&0.114&0.031&0.065&0.130&0.048&0.080&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.004&0.000&0.003&0.000&0.000&0.044&0.019&0.152&0.073&0.103\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\n\t\t\t&0.004&0.000&0.000&0.000&0.000&0.037&0.105&0.013&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&0.065&0.002&0.047&0.078&0.027&0.036&0.063&\\textbf{0.424}&0.167\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&\\textbf{0.316}&0.000&\\textbf{0.527}&0.000&0.000&0.178&0.032&\\textbf{0.288}&0.002&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.023&0.000&0.000&0.000&0.000&0.019&0.103&0.006&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.173&0.089&0.124&\\textbf{0.472}&\\textbf{0.594}&0.033&0.054&0.064&0.140&0.042\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\n\t\t\t&0.002&0.000&0.001&0.003&0.000&0.044&0.031&0.017&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.018&0.000&0.000&0.000&0.000&0.061&0.067&0.016&0.000&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G},\\textcolor{burgundy}{S}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.005&0.163&0.001&0.095&0.006&0.028&0.040&0.015&0.016&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C},\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.213&0.000&0.124&0.000&0.000&0.052&0.014&0.121&0.036&0.241\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.074&0.000&0.137&0.003&0.000&0.041&0.022&0.055&0.001&0.000\\\\\n\t\t\t\t\\rowcolor{shadecolor}\\{\\textcolor{aoe}{C},\\textcolor{burgundy}{S}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\n\t\t\t&0.014&0.000&0.000&0.000&0.000&0.011&0.067&0.004&0.000&0.000\\\\\n\t\t\t\\{\\textcolor{aoe}{C}\\}\\{\\textcolor{burntorange}{G}\\}\\{\\textcolor{bostonuniversityred}{M}\\}\\{\\textcolor{burgundy}{S}\\}\n\t\t\t&0.133&0.040&0.079&0.265&0.291&0.029&0.033&0.059&0.229&\\textbf{0.448}\\\\\n\t\t\t\\hline\n\t\t\t$\\sum\\log_{15} \\left(p_i ^{-p_i}\\right)$&0.687&0.407&0.509&0.501&0.371&0.828&0.886&0.823&0.582&0.505\n\t\t\\end{tabular}}\n\t\\end{center}\n\\vspace{-0.1cm}\n\\end{table} \n\n\n\\begin{figure}[t]\n\t\\vspace{-0.1 cm}\n\t\\centering \n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[ width=\\linewidth]{figures\/EA}\n\t\t\\caption{density estimation}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip,  width=\\linewidth]{figures\/cluster_EA}\n\t\t\\caption{co-clustering}\\label{fig:fig4b}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip,  width=\\linewidth]{figures\/cluster_ord_EA}\n\t\t\\caption{co-clustering}\n\t\\end{subfigure}\n\t\\begin{subfigure}{.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[ width=\\linewidth]{figures\/LVMI}\n\t\t\\caption{density estimation}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip,  width=\\linewidth]{figures\/cluster_LVMI}\n\t\t\\caption{co-clustering}\\label{fig:fig4e}\n\t\\end{subfigure}\\hspace{0.05\\textwidth}%\n\t\\begin{subfigure}{.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={3cm 1cm 3cm 1cm}, clip, width=\\linewidth]{figures\/cluster_ord_LVMI}\n\t\t\\caption{co-clustering}\\label{fig:fig4f}\n\t\\end{subfigure}\n\\vspace{-0.1 cm}\n\\caption{ Panels (a) and (d): density estimates. Panels (b)--(c) and (e)--(f): heatmaps of the posterior probabilities of co-clustering; in (b) and (e) patients are ordered based on the diagnosis and the four black squares highlight the within-sample probabilities; in (c) and (f) patients are reordered based on co-clustering probabilities.}\n\\label{fig:fig4}\n\\vspace{-0.1 cm}\n\\end{figure} \n\nAs far as prediction and second-level clustering are concerned, Figure~\\ref{fig:fig4} displays the density estimates and the heatmap of co-clustering probabilities between pairs of patients for the E\/A ratio and LVMI. Figure~\\ref{fig:fig4b} shows that co-clustering probabilities are similar within and across diagnoses, indicating that the effect of the diseases on the distribution of the cardiac index is mostly explained through shifts between disease-specific locations. Moreover Figure~\\ref{fig:fig4b} suggests the presence of three outliers that have low probability of co-clustering with all the other subjects and that would be ignored by the model using a more traditional ANOVA structure. On the other hand, Figure~\\ref{fig:fig4e} shows a slightly different pattern for co-clustering probabilities in the fourth square, which suggests that the heterogeneity between severe preeclampsia patients and the other patients is not entirely explained by shifts in disease-specific locations. Finally, Figure~\\ref{fig:fig4f} suggests the presence of an underlying relevant factor. The corresponding figures for all ten response variables are reported in Section E.1 of the Supplement and can be used for prediction and for a graphical analysis aimed at controlling the presence of underlying relevant factors, outliers and differences across diseases distinct from shifts between disease-specific locations.\n\nOur results are coherent with almost all of the findings in \\cite{tatapudi2017}, where results were obtained through a series of independent frequentist tests. However, importantly, we are able to provide more insights thanks to the simultaneous comparison approach and the latent clustering of subjects. For instance, considering the response LVMI, \\cite{tatapudi2017} detected a significant increase in cases compared to controls and an increase in severe preeclampsia compared to gestational hypertensive and mild preeclampsia patients. Such results do not clarify whether a modification exists between the control group and gestational hypertensive patients or between the latter and mild preeclampsia patients. Moreover, in contrast to our analysis, their results do not provide any information concerning the presence of underlying common factors, outliers or distributional effects (different from shifts in locations).\n\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{s:conclusion}\nWe designed a Bayesian nonparametric model to detect clusters of hypertensive disorders over different cardiac function indexes and found modified cardiac functions in hypertensive patients compared to healthy subjects as well as progressively increased alterations with the severity of the disorder. The proposed model has application potential also beyond the \nconsidered setup when the goal is to cluster populations according to multivariate information: it borrows strength across response variables, preserves the flexibility intrinsic to nonparametric models, and correctly detects partitions of populations even in presence of small sample sizes, \nwhen alternative distribution-based clustering models tend to underestimate the number of clusters. The key component of the model is the s-HDP, a hierarchical nonparametric structure for the error terms that offers flexibility and serves as a tool to investigate the presence of unobserved factors, outliers and effects other than changes in locations.  Interesting extensions of the model include generalizations to other types of invariances in order to accommodate identifiability in generalized linear models, for instance in presence of count data and a log link function, as well as generalizations to other types of processes, beyond the Dirichlet process.\n\n\\section*{Acknowledgements}\nMost of the paper was completed while B. Franzolini was a Ph.D. student at the Bocconi University, Milan.\n A. Lijoi and I. Pr\\\"unster are partially supported by MIUR, PRIN Project 2015SNS29B.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nThe authors acknowledge K. Ishii and K. Tomiyasu for their useful remarks. The experiments at the Materials and Life Science Experimental Facility at J-PARC were performed under a user program (Proposal No. 2013A0052). M.F. is supported by a Grant-in-Aid for Scientific Research (A) (16H02125).\n\n\t\\begin{figure}[t]\n\t\\begin{center}\n\t\\includegraphics[width=75mm]{Fig3_dispersion_v1.pdf}\n\t\\caption{(Color online)~Momentum-dependence of (a) peak-position and (b) intensity in Sr$_3$Ir$_2$O$_7$. The gray lines are the results from RIXS [\\ref{Kim2012}]. The intensity is not corrected with $|{\\rm f}({\\bf Q})|^2$ and the absorption coefficient.}\n\t\\label{dispersion}\n\t\\end{center}\n\t\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction and Vocabulary }\n\\label{sec:vocabulary}\n\nOne of the important issues of contemporary physics is the\nunderstanding of strong interactions and in particular the study\nof the properties of \\textbf{strongly interacting matter} -- a system of strongly interacting particles in equilibrium.\nThe advent of the quark model of hadrons and the development of \nthe commonly accepted theory of strong interactions, \\textbf{quantum chromodynamics (QCD)},\nnaturally led to expectations that matter at very high densities\nmay exist in a state of quasi-free quarks and gluons, the \\textbf{quark-gluon\nplasma (QGP)}.\n\n\n\n\\begin{figure}[hbt]\n\n\\centering\n\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/v_qgp_hic.jpg}\n\n\\caption{\n\\textit{Left:} \n Artistic sketch of the two phases of strongly interaction matter, hadron-resonance gas and quark-gluon plasma.\n \\textit{Middle:}\nPhase diagram of QCD in temperature $T$ and baryon chemical potential $\\mu_B$, and the region covered by running or planned experiments~\\cite{Alemany:2019vsk}. The density range covered by LHC, LHC-FT and SPS experiments is indicated by the shaded areas in the figure. The lower boundary of the grey and blue shaded area follows the chemical freeze-out.\nThe upper boundary relates to the parameters at the early stage of the collisions. The potential deconfinement critical point is labelled with d-CP, the onset of deconfinement with OD. The black line at small temperatures and high densities shows the nuclear liquid-gas transition, also ending in a critical point n-CP. The density range of other experiments is indicated in the bar below the figure. This\nincludes RHIC at BNL, NICA at JINR, SIS100 at FAIR, J-PARC-HI at J-PARC, the Nuclotron at JINR (NUCL), and HIAF at HIRFL.\n \\textit{Right:} \n  Evolution of a heavy-ion collision at high energies. Successive snapshots of a central collision are shown versus time.  \n }\n \\label{fig:v_qgp}\n\\end{figure}\n\n\nDoes the QGP exist in nature?\nHow does the transition proceed from a low-density state of strongly interacting matter, \nin which quarks and gluons are confined in hadrons, to the QGP? \nIs it similar to the transition from liquid water to water vapour along\na first order transition line ending in a second order critical point and followed by a\ncross over transition, see illustration plots in Fig.~\\ref{fig:v_qgp}?\n\nThe study of high energy collisions of two atomic nuclei gives us\nthe unique possibility to address these issues in well controlled\nlaboratory experiments. \nThis is because it is observed that a system of strongly \ninteracting particles created in \\textbf{central heavy ion collisions} is close to (at least local) equilibrium.\nHow does the transition from a non-equilibrium system created in \\textbf{inelastic proton-proton interactions}\nto the equilibrium system in central heavy ion collisions look like?\n\nThese questions have motivated broad\nexperimental and theoretical efforts for about 50 years. \nSystematic measurements of particle production properties in nucleus-nucleus (A+A) collisions at different collision energies and for different masses of\ncolliding nuclei have been performed.\nBy changing collision energy and nuclear mass number \none changes macroscopic parameters of the created system -- its volume, energy, and net baryon number.\nThis allows to move across the phase diagram and look for the theoretically predicted boundaries of equilibration and matter phases, see  illustration plots in Fig.~\\ref{fig:v_qgp}.\nConsequently several physics phenomena might be \nobserved when studying experimentally nuclear collisions at high energies. \nThese are:\n\\begin{enumerate}[(i)]\n\\item \n\\textbf{onset of fireball} -- beginning of creation of large-volume ($\\gg 1$~fm$^3$) strongly interacting matter,\n\n\\item\n\\textbf{onset of deconfinement} -- beginning of QGP creation with increasing collision energy,\n\n\\item \n\\textbf{deconfinement critical point}  -- a hypothetical end point of the first order transition line to quark-gluon plasma that has properties of a second order phase transition. \n\\end{enumerate} \n\nThese phenomena are expected to lead to rapid changes of hadron production properties -- the \\textbf{critical structures} -- \nwhen changing collision energy and\/or nuclear mass number of the colliding nuclei. \n\n\n\n\n\n\\section{ Strongly interacting matter in heavy ion collisions}\n\\label{sec:strongly}\n\n\n\\textbf{Strongly interacting matter.}\nThe equation of state defines the macroscopic properties of matter in equilibrium. It  is a subject of statistical mechanics. The first step in this modelling is to clarify  the types of particle species and \ninter-particle interactions.\nOne should also choose an appropriate statistical ensemble\nwhich fixes the boundary conditions and conserves the corresponding global physical quantities, like energy and conserved charges.\nStrongly interacting matter at high energy density can be formed at the early stages of\nrelativistic \nA+A collisions.\nAs mentioned in the introduction the questions --\n\\begin{enumerate}[(i)]\n\\item\nWhat types of particles should be considered as fundamental?\n\\item\nWhat are the composite objects?\n\\item\nWhat are the fundamental forces between the matter constituents?\n\\item\nWhat are the conserved charges?\n\\end{enumerate}\n-- should be addressed.\nAnswers to these questions are changing with time as our knowledge about basic  properties of elementary particles and their interactions increases.\n\nThe first model of strongly interacting matter at high energy density was formulated in 1950 by Fermi~\\cite{fermi:1950jd}.\nIt assumes that a system created in high energy proton-proton (p+p) interactions emits  pions like black-body radiation, i.e., pions are treated as non-interacting particles,\nand the pion mass $m_\\pi\\cong 140$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace is neglected compared to the high temperature of the system.\nThe pressure $p$ and energy density $\\varepsilon$ can then be represented by the following functions of the temperature $T$ \n(the system of units with $h\/(2\\pi)=c=k_B=1$ will be used),\n\\eq{\\label{p}\np(T)~=~\\frac{\\sigma}{3}T^4~,~~~~~\\varepsilon(T)~\\equiv~T\\frac{dp}{dT}~-~p~=\\sigma T^4~,\n}\nwhere $\\sigma= \\pi^2g\/30$ is the the so-called Stephan-Boltzmann constant, with $g$ being the degeneracy factor (the number of spin and isospin states), and $g=3$ counting the three isospin states $\\pi^+,~\\pi^0,~\\pi^-$.  \n\n\\vspace{0.2cm}\n\\textbf{Hadrons and resonances.}\nThe study of particle production in high energy collisions started in the 1950s with discoveries\nof the lightest hadrons -- $\\pi$, $K$, and $\\Lambda$\n-- in cosmic-ray experiments. Soon after with the rapid advent of particle accelerators\nnew particles were discovered almost day--by--day.\nThe main feature of the strong interactions appears to be creation of new and new types of particle species -- hadrons and resonances -- when increasing the collision energy. \nA huge number (several hundreds) of different hadron and resonance species are known today. The simplest statistical model treats the hadron matter, i.e. a system of strongly interacting particles at not too high energy density, as a mixture of ideal gases of different hadron-resonance species.  \n\n\\vspace{0.2cm}\n\\textbf{Hadron-resonance gas.}\nIn the grand canonical ensemble the pressure \nfunction is then written as\n\\begin{eqnarray}\n\\label{p-id}\np^{\\rm id}(T,\\mu) \n= \\sum_i\\frac{g_i}{6\\pi^2}\\int d m\\, f_i(m)\\int_0^{\\infty} \\frac{k^4dk}{\\sqrt{k^2+m^2}}\n\\left[ \\exp\\left(\\frac{\\sqrt{k^2+m^2} - \\mu_i}{T}\\right)+\\eta_i\\right]^{-1}~,\n\\end{eqnarray}\nwhere $g_i$ is the degeneracy factor of the $i^{\\textrm{th}}$ particle and the normalized\nfunction $f_i(m)$\ntakes into account the Breit-Wigner shape of resonances with finite width\n$\\Gamma_i$ around their average mass $m_i$. For the stable hadrons,\n$f_i(m)=\\delta(m-m_i)$.\nThe sum over $i$ in Eq.~\\eqref{p-id} \nis taken over all\nnon-strange and strange hadrons listed in the Particle Data Tables.\nNote, that in the equation\n$\\eta_i =-1$ and $\\eta_i = 1$ for bosons\nand fermions, respectively, while $\\eta = 0$ corresponds to the Boltzmann approximation.\nThe chemical potential for the $i^{\\textrm{th}}$  hadron is given by\n\\begin{equation}\n\\mu_i\\ =\\ b_i\\,\\mu_B\\, +\\, s_i\\,\\mu_S\\, +\\, q_i\\,\\mu_Q\n\\label{eq:mui}\n\\end{equation}\nwith $b_i = 0,\\, \\pm 1$, $s_i = 0,\\, \\pm 1,\\, \\pm 2,\\, \\pm 3$, and\n$q_i = 0,\\, \\pm 1,\\, \\pm 2$\nbeing the corresponding baryonic number, strangeness, and electric charge of\nthe $i$th  hadron. Hadrons composed of charmed and beauty quarks are rather heavy and thus rare in the hadron-resonance gas, and their contribution to the thermodynamical functions are often neglected. \nChemical potentials are denoted as $\\mu\\equiv (\\mu_B,\\mu_S,\\mu_Q)$ and correspond to the conservation of net-baryon number, strangeness, and electric charge in the hadron-resonance gas.\nThe entropy density $s$,  net-charge densities $n_i$ (with $i=B,Q,S$), and energy density $\\varepsilon$  are calculated from the pressure function (\\ref{p-id}) according to the standard thermodynamic identities:\n\\eq{\\label{therm-id}\ns(T,\\mu)=\\left(\\frac{\\partial p}{\\partial T}\\right)_\\mu~,~~~\nn_i(T,\\mu)=\\left(\\frac{\\partial p}{\\partial \\mu_i}\\right)_T~,~~~~ \\varepsilon(T,\\mu)=Ts+\\sum_in_i\\mu_i-p~.\n}\n\nNote that only one chemical potential $\\mu_B$ is considered as independent variable in fits of the model to particle multiplicities produced in A+A reactions. Two others, $\\mu_S$ and $\\mu_Q$, should be found, at each pair of $T$ and $\\mu_B$, from the requirements that the net-strangeness density equals to zero, $n_S=0$, and the ratio of the net-electric charge density, $n_Q$, to $n_B$ equals to the ratio of the number of protons, $Z$, to the number of all nucleons, $A$ (protons and neutrons) in the colliding nuclei, $n_Q\/n_B=Z\/A$.  \nEquations (\\ref{p-id}-\\ref{therm-id}) define the ideal hadron-resonance gas  \nmodel. In spite of evident simplifications this model  rather successfully fits the rich data on mean multiplicities of hadrons measured in central A+A collisions at high energies.  \n\n\\vspace{0.2cm}\n\\textbf{Hagedorn model.} \nIs there an upper limit for the masses of mesonic and baryonic resonances? \nIn 1965 Hagedorn  formulated a statistical model assuming an exponentially increasing spectrum  of hadron-resonance states at large masses~\\cite{Hagedorn:1965st}:\n\\eq{\\label{H}\n\\rho(m)~\\cong~C\\,m^{-a}\\,\\exp\\left(\\frac{m}{T_H}\\right)~, \n}\nwhere $C$, $a$, and $T_H$ are the model parameters. \nAt that time the number of experimentally detected hadron-resonance states was much smaller than it is today. Nevertheless, Hagedorn made the brave assumption that these $m$-states interpolate the low-mass spectrum and extend to $m\\rightarrow \\infty$, and that their density at large $m$ behaves as in Eq.~(\\ref{H}).   \nThe pressure function at $\\mu=0$ then becomes\n\\eq{ \\label{pH}\np(T)~=~T\\int_0^\\infty dm\\,\\rho(m)\\,\\phi_m(T)~,\n}\nwith the function $\\phi_m(T)$ behaving at $m\/T \\gg 1$ as\n\\eq{ \\label{large-m}\n \\phi_m(T)~\\cong ~ g\\,\\left(\\frac{mT}{2\\pi}\\right)^{3\/2}\\,\\exp\\left(-\\,\\frac{m}{T}\\right)~.\n}\nThe result (\\ref{large-m}) can be found from Eq.~(\\ref{p-id}) after $k$-integration. At $m\/T\\gg1$ and $\\mu=0$\nboth quantum statistics and relativistic effects become negligible.\n\n\n There are two exponential functions in the integrand (\\ref{pH}):  $\\exp(-m\/T)$ defines the exponentially decreasing  contribution of each individual $m$-state, and $\\exp(m\/T_H)$ defines the exponentially increasing number of these $m$-states. It is clear that the $m$-integral\nin Eq.~(\\ref{pH}) exists only for $T \\le T_H$.\nTherefore, a new hypothetical\nphysical constant -- the limiting temperature\n$T_H$ -- was introduced. The numerical value of $T_H$ was estimated by Hagedorn from two sources: from the straightforward comparison of Eq.~(\\ref{H}) with the experimental mass spectrum of hadrons and resonances, $\\Delta N\/\\Delta m$, and  from the inverse slope parameter of the transverse momentum spectra of final state hadrons in p+p interactions at high energies. Both estimates gave similar values $T_H=150-160$~MeV.  \nThe hadron states with large $m$ in the Hagedorn model are named the Hagedorn fireballs. These states were defined in a democratic (bootstrap) way:\nthe Hagedorn fireball consists of an arbitrary number of non-interacting Hagedorn fireballs, each of which in turn consists of ... \n\n\nIn the 1960s it was not clear up to what masses the hadron-resonances spectrum can be extended.  \nThe answer to this question is still unclear today. The large (exponential) density of resonance states\n$\\rho(m)$ and the finite widths $\\Gamma(m)$ of these states  make their experimental observation very problematic.  \nMoreover, several conceptual problems of the Hagedorn  model were obvious from the very beginning. The lightest hadron species, e.g., pion, kaon and proton can not be considered as  composed of other (non-interacting) hadrons, and should therefore have their own (non-democratic) status. Besides, the fireballs are treated as point like non-interacting  objects. However, from nuclear physics it was already evident that at least protons and neutrons are (strongly) interacting particles: nucleons should have both attractive and repulsive interactions to be able to form stable nuclei.  Most probably, similar interactions exist between other types of baryons. Evident physical arguments suggest that the same type of repulsive and attractive interactions should exist between anti-baryon species.  \n\n\n\\vspace{0.2cm}\n\\textbf{Quark-gluon plasma.}\nThe quark model of hadron classification was proposed by \nGell--Mann~\\cite{GellMann:1964nj} and  Zweig~\\cite{Zweig:352337} in 1964. It was the alternative to the bootstrap approach. Only three types of objects -- $u$, $d$, $s$ quarks and their anti-quarks -- were needed to construct the quantum numbers of  all known hadrons and successfully predict several new ones.  A 15 years \nperiod then started in which the idea of the existence of sub-hadronic particles -- quarks and gluons -- \nwas transformed into the fundamental  theory of strong interactions,\nquantum chromodynamics (QCD). Soon after the discovery of the $J\/\\psi$-meson in 1974 three new types of quarks -- $c$, $b$, and $t$ -- were added to QCD.  \nIn parallel, an important conjecture was formulated~\\cite{Ivanenko:1965dg,Itoh:1970uw} --\nmatter at high energy density, as in super-dense star cores, may consist of quasi-free Gell-Mann-Zweig quarks instead of\ndensely packed hadrons. Some years later Shuryak investigated the properties of QCD matter and came to a qualitatively similar conclusion:  QCD matter at high temperature is best described by quark and gluon degrees of\nfreedom and the name quark-gluon plasma was coined~\\cite{Shuryak:1977ut,Shuryak:1980tp}.\n\nQuestions concerning QGP properties and properties of its transition\nto matter consisting of hadrons, have been considered since the late 1970s (see, e.g., Ref.~\\cite{Cabibbo:1975ig}). The Hagedorn model was still rather popular at that time  due to its successful phenomenological applications. For example, the temperature parameter $T_{\\rm ch}$ found from fitting the data on hadron multiplicities in p+p interactions and A+A collisions at high energies (the so-called chemical freeze-out temperature) was found to be close to the limiting Hagedorn temperature, $T_{\\rm ch} = 140-160~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace \\cong T_H $.  \nHagedorn and Rafelski~\\cite{Hagedorn:1980cv}\nas well as Gorenstein, Petrov, and Zinovjev~\\cite{Gorenstein:1981fa} \nsuggested that the \nupper limit of the hadron temperature, the Hagedorn\ntemperature $T_H$, is not the limiting temperature but the transition temperature\nto the QGP, $T_C = T_H \\approx 150$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace.  \nThe Hagedorn fireball was then interpreted\nas the quark-gluon bag formed in the early stage of the collision. It also had an exponential mass spectrum (\\ref{H}) like in the Hagedorn model,\nbut was not a point like object. The average volume of the quark-gluon bag increases linearly with its mass.\nThis causes the excluded volume effects in the system of bags and leads to the transition to the high temperature QGP phase.\nNote that the first QCD-inspired estimate of the transition temperature to the\nQGP gave $T_C \\approx 500~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace$~\\cite{Shuryak:1977ut}, the most recent QCD-based estimates obtain $T_C \\approx T_H \\approx 150~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace$~\\cite{Bazavov:2018mes}. \n\nMany physicists started to speculate that the QGP could be formed in\nA+A collisions at sufficiently high energies in which one expects that strongly interacting matter of high energy density will be created.\nTherefore the QGP might be discovered in laboratory experiments.\n\n\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/v_hic_events.jpg}\n\\caption{\nTracks produced in nucleus-nucleus collisions recorded by heavy-ion experiments\nlocated in JINR Dubna (the SKM-200 streamer chamber~\\cite{Aksinenko:1980nm}), LBL Berkeley (the LBL streamer chamber~\\cite{Sandoval:1982cn}), CERN SPS (the NA35 streamer chamber~\\cite{Stock:1987rn}) and CERN SPS (the NA49 time projection chambers~\\cite{Afanasev:1999iu}), from left to right, respectively. \n }\n\\label{fig:s_events}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{The first experiments.}\nIn parallel to the theoretical ideas and models, experimental studies of A+A\ncollisions were initiated in 1970 at the Synchrophasotron (JINR Dubna)~\\cite{Issinsky:1994it,Malakhov:2013zjq} and in 1975 at the Bevelac (LBL Berkeley)~\\cite{Alonso:1985awy}. Figure~\\ref{fig:s_events} (\\textit{first} and \\textit{second from left}) shows two examples of recorded collisions.\n\nSeveral effects were observed which could be attributed to a collective behaviour of the created system of hadrons. These are anisotropic and radial flow of particles~\\cite{Lisa:1994yr,Nagamiya:1981sd,Gustafsson:1984nd}, enhanced production of strange particles~\\cite{Anikina:1984cu} and suppressed production of pions~\\cite{Sandoval:1980bm}. They could only be\nexplained by\nassuming that  strongly interacting matter is produced in the studied collisions~\\cite{Stock:1985xe,Danielewicz:1985hn}. \nIn what follows we use the term fireball as the notation for a large volume\n($\\gg 1$~fm$^3$) system consisting of strongly interacting particles close to at least local equilibrium. They can be either hadrons and resonances or quarks and gluons.   \n\n\n\\vspace{0.2cm}\n\\textbf{Initiating the hunt for QGP.}\nThe two findings,\n\\begin{enumerate}[(i)]\n\\item\ntheoretical: QCD matter at sufficiently high temperature is in the state of a QGP;\n\\item\nexperimental: strongly interacting matter is produced in heavy ion collisions at energies of several GeV per nucleon;\n\\end{enumerate}\nled activists of the field~\\cite{Satz:2016xba} to the important decision to collide heavy ions at the maximum possible energy with the aim to discover the QGP.\nIn the 1980s the maximum possible energy for heavy ion collisions was available at CERN, Geneva.\nThis is why heavy ion physics entered the Super Proton Synchrotron (SPS) program at CERN.\nThe \\textit{Workshop on future relativistic heavy ion experiments}, GSI Darmstadt, October 7-10, 1980, organized by Bock and Stock~\\cite{Bock:1981iyr}, with an opening talk by Willis and a summary talk by Specht led to the formulation of the new program. Moreover, it initiated a series of \\textit{Quark Matter} conferences~\\cite{Satz:2016xba}.\n\n\n\\section{Evidence for the quark-gluon plasma}\n\\label{sec:q}\n\n\\textbf{Predicted QGP signals.}\nThe experimental search for a quark-gluon plasma in heavy ion collisions at the CERN SPS was shaped by several model predictions of possible QGP signals:\n\\begin{enumerate}[(i)]\n  \\item\n    suppressed production of charmonium states, in particular $J\/\\psi$ mesons~\\cite{Matsui:1986dk},\n    \\item\n    enhanced production of strange and multi-strange hadrons from the QGP~\\cite{Rafelski:1982pu},\n     \\item \n    characteristic radiation of photons and dilepton pairs from the QGP~\\cite{Shuryak:1980tp}.\n\\end{enumerate}\n\n\n\\vspace{0.2cm}\n\\textbf{Measurements at the CERN SPS.}\nThe search for the QGP at the CERN SPS was performed in two steps:\n\\begin{enumerate}[(i)]\n    \\item \n    In 1986-1987 oxygen and sulphur nuclei were accelerated to 200\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace.\n    Data on collisions with various nuclear targets were recorded by seven experiments, NA34-2, NA35, NA36, NA38, WA80, WA85 and WA94. \n    \\item\n    In 1996-2003 lead and indium beams at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace were collided with lead and indium targets. Data were recorded by nine experiments, NA44, NA45, NA49, NA50, NA52, NA57, NA60, WA97 and WA98. \n\\end{enumerate}\n\nFigure~\\ref{fig:s_events} shows a S+Au collision at 200\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace (\\textit{second from right}) and a Pb+Pb collision at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace (\\textit{right}) recorded by the NA35 streamer chamber~\\cite{Stock:1987rn} and the NA49 time projection chambers~\\cite{Afanasev:1999iu}, respectively. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\linewidth,clip=true]{plots\/q_sm1.jpg}\n        \\caption{Transverse energy distributions in S+A collisions at the top CERN SPS energy measured by HELIOS\/NA34-2~\\cite{Akesson:1988yt}.}\n    \\label{fig:detadet}\n\\end{figure}\n\nAn estimate of the energy density in A+A\ncollisions can be obtained from measurement\nof the transverse energy production and the size of the collision system. Already in \nS collisions with heavy nuclei (see Fig.~\\ref{fig:detadet}) it was found that values above 1~GeV\/fm$^3$ were\nreached (NA34~\\cite{Akesson:1987kh,Akesson:1988yt}, NA35~\\cite{Heck:1988cm}, NA49~\\cite{Margetis:1994tt}).\nMoreover the fireball showed effective temperature increasing linearly with particle mass,\na characteristic of collective radial expansion (see Fig.~\\ref{fig:exp-stat} (\\textit{left})). Also mean\nmultiplicities of produced hadrons are well reproduced by the statistical model~\\cite{Becattini:2003wp} (see Fig.~\\ref{fig:exp-stat} (\\textit{right})).\nThus conditions in collisions of heavy nuclei at the top energy of the CERN SPS are promising for the production of the QGP.\n\n\\begin{figure}\n   \\centering\n    \\includegraphics[width=0.48\\linewidth,clip=true]{plots\/q_sm2.jpg}\n    \\includegraphics[width=0.51\\linewidth,clip=true]{plots\/q_sm3.jpg}\n    \\caption{\\textit{Left:} Inverse slope parameter (effective temperature) of the transverse mass distribution versus particle mass measured by WA97, NA44 and NA49~\\cite{Antinori:2000sb}. \\textit{Right:} Mean hadron multiplicities\n    measured by NA49 compared to the statistical model fit~\\cite{Becattini:2003wp}. Pb+Pb collisions at the top CERN SPS energy. }\n   \\label{fig:exp-stat}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/q_signals1a.jpg}\n\\caption{\n\\textit{Left:} Ratio of $J\/\\psi$ meson to Drell-Yan muon pair production (data points) yields\ncompared to predictions (curves) of $J\/\\psi$ absorption by hadronic \nmatter~\\cite{Baglin:1990iv,Alessandro:2004ap} (NA38, NA50).\n\\textit{Center:} Comparison of \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace yield ratio in p+p, p+A and A+A collisions~\\cite{Bartke:1990cn,Appelshauser:1998vn}\n(NA35, NA49). \\textit{Right:} Comparison of the mid-rapidity ratios of\nof hyperon production to number of wounded nucleons in p+Be, p+Pb and Pb+Pb\ncollisions~\\cite{Antinori:2006ij} (NA57). Top CERN SPS energy.\n }\n\\label{fig:q_signalsa}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/q_sm4.jpg}\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/q_sm5.jpg}\n\\caption{\n\\textit{Left:} Direct photon signal observed in central Pb+Pb collisions~\\cite{Manko:1999tgz} (WA98).\n\\textit{Right:} Effective temperature of directly produced dimouns in In+In collisions as function of dimuon mass~\\cite{Arnaldi:2008er} (NA60). Top CERN SPS energy.\n }\n\\label{fig:q_signalsb}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{The QGP discovery.}\nResults on central collisions of medium size and heavy nuclei from the QGP search programme at the CERN SPS appeared to be consistent with the predictions for the QGP:\n\\begin{enumerate}[(i)]\n     \\item\n    The relative yield of $J\/\\psi$ mesons is significantly suppressed compared to that in p+p and p+A interactions\n    (NA38~\\cite{Baglin:1990iv}, NA50~\\cite{Alessandro:2004ap}, NA60~\\cite{Arnaldi:2007zz}) as expected for the $J\/\\psi$ melting in a QGP\n    (see Fig.~\\ref{fig:q_signalsa} (\\textit{left})).\n    \\item\n    The relative strangeness yield is consistent with the yield expected for the equilibrium QGP. Moreover, it is significantly enhanced compared to that in p+p and p+A \n    interactions (see Fig.~\\ref{fig:q_signalsa} (\\textit{center}))\n    (NA35~\\cite{Bartke:1990cn}, NA49~\\cite{Appelshauser:1998vn}). \n    Even larger enhancement is measured for the relative yield of multi-strange hyperons (see Fig.~\\ref{fig:q_signalsa} (\\textit{right})),\n    (WA97~\\cite{Andersen:1999ym}, NA57~\\cite{Antinori:2006ij}). \n    Note that QGP formation was not expected in p+p and p+A collisions.\n    \\item\n    Spectra of directly produced dimouns (virtual photons) and photons emerge above a dominant background at large mass respectively transverse momentum and show a thermal contribution with an effective temperature of about 200~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace. This is significantly larger than the expected transition temperature to QGP (see Fig.~\\ref{fig:q_signalsb})\n    (NA60~\\cite{Arnaldi:2008er} and WA98~\\cite{Manko:1999tgz}).\n    \\end{enumerate}\n\n\n\\vspace{0.2cm}\n\\textbf{Standard model of heavy-ion collisions.}\nThese and other results\nestablished the standard picture of heavy-ion collisions~\\cite{Florkowski:2010zz}:\n\\begin{enumerate}[(i)]\n    \\item \n    High density strongly interacting matter is created at the early stage of heavy-ion collisions. Starting at SPS collision energies it is in the QGP phase.\n    \\item\n    The high-density matter enters a hydrodynamic expansion, cools down and emits photons and dileptons.\n    \\item\n    At the phase transition temperature, $T_C \\approx 150~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace$, hadrons are created.\n    Statistical haronization models fit hadron yields at this stage quite well.\n    \\item \n    The hadronic matter after hadronization is still dense enough to modify the hadron composition and continue expansion.\n    \\item\n    At sufficiently low densities the hadron interaction rate drops to zero (freeze-out).\n    resonances decay and long-lived hadrons freely fly away e.g. towards particle detectors at the CERN SPS.\n\\end{enumerate}\n\n\\vspace{0.2cm}\n\\textbf{Conclusion from the QGP-search.}\nThese major achievements were compiled by the heavy-ion community~\\cite{Heinz:2000bk} and led to the CERN press release - \non February 10, 2000 the CERN Director General Luciano Maiani said:\n\\textit{\nThe combined data coming from the seven experiments on CERN's Heavy Ion programme have given a clear picture of a new state of matter. This result verifies an important prediction of the present theory of fundamental forces between quarks. It is also an important step forward in the understanding of the early evolution of the universe. We now have evidence of a new state of matter where quarks and gluons are not confined. There is still an entirely new territory to be explored concerning the physical properties of quark-gluon matter. The challenge now passes to the Relativistic Heavy Ion Collider at the Brookhaven National Laboratory and later to CERN's Large Hadron Collider.}\n\nThis was in fact the moment when the majority of heavy-ion physicists moved to\nstudy heavy-ion collisions at much higher energies at the Relativistic Heavy Ion Collider (RHIC) of Brookhaven National Laboratory (BNL).\nRich and precise results obtained during the period of 2000-2010 at RHIC provided extensive information on the properties of the QGP. There were already no doubts about QGP formation at the early stage of A+A collisions at the CERN SPS,  and all the more  at RHIC energies.  Two basic properties of the QGP were established at the RHIC BNL: jet quenching (deceleration  of high momentum partons in the hot QGP)  and a small ratio of  the shear viscosity $\\eta$ to the entropy density $s$. It was estimated that $\\eta\/s\\cong 0.1$, i.e the QGP appears to be an almost perfect liquid (see Ref.~\\cite{Adams:2005dq,Adcox:2004mh,Back:2004je,Arsene:2004fa} for details).  \n\nThe situation after the announcement of the QGP discovery in 2000 at CERN was however rather confusing. Many were pretty sure about its formation in central Pb+Pb collisions at the CERN SPS, but unambiguous evidence of the QGP state was still missing. \nNeedless to say that the Nobel prize for the QGP discovery was not yet awarded.  \nThis may be attributed to the difficulty of obtaining unique and quantitative predictions of\nthe expected QGP signals from QCD.\n\n\\vspace{0.2cm}\n\\textbf{Question marks.}\nLet us briefly discuss questions addressed to the two main signals of the QGP: the $J\/\\psi$ suppression and the strangeness enhancement.\n\n{\\bf The $J\/\\psi $ suppression.} The standard picture of $J\/\\psi$  production in collisions of hadrons and nuclei assumes\na two step process: \nthe creation of a $c\\overline{c}$ pair in hard parton collisions at the very early stage of the reaction and a subsequent formation of a bound charmonium state or two open charm hadrons. Further more it was assumed that the initial yield of $c\\overline{c}$ pairs is proportional to the yield Drell-Yan pairs. Then the $J\/\\psi $\/(Drell-Yan pairs) ratio is expected to be the same in p+p, p+A and A+A collisions providing there are no other processes which can lead to $J\/\\psi$\ndisintegration and\/or creation.\nThe measured suppression of the ratio in p+A collisions respectively to p+p interactions was interpreted as due to $J\/\\psi$ interactions with nucleons of the target nucleus and with hadronic secondaries (`co-movers'). \nIn central Pb+Pb collisions at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace the suppression was found to be significantly stronger than expected in the models including nuclear and co-mover suppression. This anomalous $J\/\\psi$  suppression was interpreted as the evidence of the QGP creation in central Pb+Pb collisions at the top CERN SPS. \nHowever, the uncertainties related to the assumption $c\\overline{c} \\sim \\textrm{Drell-Yan pairs}$ and estimates of the nuclear and co-mover suppression\nlead to uncertainty in interpretation of the anomalous $J\/\\psi$  suppression as the QGP signal.\nMoreover, models of  $J\/\\psi$  production in the later stages of the collision process have been developed:\n\\begin{enumerate}[(i)]\n\\item\nthe statistical model of $J\/\\psi$ production at the hadronization~\\cite{Gazdzicki:1999rk},\n\\item\nthe dynamical and statistical models of $J\/\\psi$ production via coalescence of $c\\overline{c}$ quarks~\\mbox{\\cite{Thews:2000rj,Levai:2000ne,BraunMunzinger:2000px,Gorenstein:2000ck}}.\n\\end{enumerate}\nClearly, in order to distinguish between different effects and verify the $J\/\\psi$ signal of the QGP creation systematic data on open charm production is needed.\n\n\\vspace{0.2cm}\n\\textbf{Strangeness enhancement.}\nA fast equilibration of $s\\overline{s}$ pairs was predicted as a QGP signature~\\cite{Rafelski:1982pu,Rafelski:2019twp}. This is mainly because of the small mass of the strange quark, $m_s\\sim 100$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace compared to the QGP temperature: $T \\ge T_C > m_s \\approx 100$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace. The estimated strangeness equilibration time was found to similar to the life time of the QGP phase in heavy-ion collisions at high energies.\nIn fact the strangeness yield measured in A+A collisions at the top SPS energy and above corresponds to the QGP equilibrium yield, for recent review see Ref.~\\cite{Rafelski:2019twp}.\nMoreover, it was estimated that the strangeness equilibration time in the confined matter is about 10 times longer than the life time of the hadronic phase in A+A collisions. This is because masses of strange hadrons, starting form the lightest one, the kaon ($m_K\\sim 500$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace) are much larger than the maximum temperature of the hadron-resonance gas $T\\le T_C \\approx 150$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace.  Thus a small yield of strangeness was expected for reactions in which the QGP was not expected, p+p and p+A interactions and A+A collisions at low energies. Consequently the enhanced production of strangeness was predicted as the next QGP signal~\\cite{Glendenning:1984ta}.\n \nThe strangeness enhancement is quantified by comparing a strange-hadron to pion ratio in A+A collisions with that in p+p interactions.\nIn particular a double ratio is calculated:\n\\eq{\\label{str-pion}\nR(\\sqrt{s_{NN}})= \\frac{\\langle K^+\\rangle_{AA}\/\\langle \\pi^+\\rangle_{AA}}{\\langle K^+\\rangle_{pp}\/\\langle \\pi^+\\rangle_{pp}}~,\n}\nwhere $\\langle\\ldots \\rangle_{AA}$ and $\\langle \\ldots \\rangle_{pp}$ denote the event averages of $K^+$ and $\\pi^+$ yields in, respectively, A+A collisions and p+p interactions at the same center of mass collision energy $\\sqrt{s_{NN}}$ of the nucleon pair.  Ratios of different strange hadrons to pions  \nwere considered, e.g.,  $\\langle K+\\overline{K}\\rangle\/\\langle \\pi\\rangle$, $\\langle \\Lambda\\rangle\/\\langle \\pi\\rangle$, $\\ldots$, $\\langle\\Omega\\rangle \/\\langle \\pi \\rangle $,  \nand then analyzed by forming the double ratios $R$, similar to the one given by Eq.~(\\ref{str-pion}).\n The confrontation of this expectation with the data was for the\nfirst time possible in 1988 when results from the SPS and the AGS\nbecame available. NA35  reported~\\cite{Bartke:1990cn} that in central S+S collisions at 200\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace\nthe strangeness to pion ratio (\\ref{str-pion}) is indeed about two times higher than in nucleon-nucleon interactions at\nthe same energy per nucleon. But an even larger enhancement ($R= 14 - 5 $) \nwas measured at the AGS at 2$A -10A$~GeV~\\cite{Pinkenburg:2001fj,Abbott:1990mm} demonstrating that strangeness enhancement {\\it increases} with {\\it decreasing} collision energy.\nMoreover, the enhancement factor (\\ref{str-pion}) should evidently go to infinity at the threshold energy of strange hadron production in nucleon-nucleon interactions. \nNote also that the strangeness neutrality introduced to statistical models using the canonical ensemble leads to a suppression of the relative yield of strange particles in systems with a low multiplicity of strangeness carriers~\\cite{Rafelski:1980gk}, e.g. p+p interactions at SPS energies.  \nIn any case, the AGS measurements indicating a strangeness enhancement larger than that at the CERN SPS show clearly difficulties in interpreting the strangeness enhancement as the QGP signal. \n\n\\vspace{0.2cm}\n\\textbf{New strategy.}\nDifficulties in interpretation of the QGP signatures forced scientists to rethink the QGP-hunt strategy. \nThe emerging new strategy was similar to the one followed by physics studying molecular liquids and gases. In these essentially simpler and familiar cases it is also sometimes  difficult to distinguish the properties of a dense gas from those of a liquid. It is much easier to identify the effects of the liquid-gas transition.  \nThus, if one believes that the QGP is formed in central Pb+Pb collisions at the top SPS energy one should observe qualitative signals of the {\\it transition} to the QGP at a lower collision energy. \nSeveral such signals were predicted within the statistical model of the early stage\n~\\cite{Gazdzicki:1998vd}.\nTheir observation would  serve as strong evidence of QGP creation in heavy-ion collisions at high enough collision energies.\n\nThis idea  motivated some of us to propose the collision energy scan at the CERN SPS with the aim to search for the {\\it onset of deconfinement}. This was the beginning of the search for the critical structures in heavy-ion collisions, for detail see the next section and Ref.~\\cite{Gazdzicki:2003fj}.\n\n\n\n\\section{ Critical structures }\n\\label{sec:cs}\n\n\n\\subsection{Evidence for the onset of deconfinement}\n\\label{sec:cs_ood}\n\n\n\\textbf{Predicted signals of the onset of deconfinement.}\nThe experimental search for the onset of deconfinement in heavy ion collisions at the CERN SPS was shaped by several model predictions of possible measurable signals:\n\\begin{enumerate}[(i)]\n    \\item \n    characteristic enhanced production of pions and suppression of the strangeness to pion ratio ~\\cite{Gazdzicki:1998vd},\n    \\item\n    softening of collective flow of hadrons~\\cite{Gorenstein:2003cu,Csernai:1999nf,Bleicher:2000sx,Bleicher:2005tb}, which should be observed in hadron distributions in transverse~\\cite{Gorenstein:2003cu} and longitudinal momenta~\\cite{Bleicher:2005tb} and azimuth angle~\\cite{Csernai:1999nf,Bleicher:2000sx}.\n\\end{enumerate}\n\n\\vspace{0.2cm}\n\\textbf{Measurements at the CERN SPS and RHIC BES.}\nThe search for the onset of deconfinement at the CERN SPS started \nin 1999 with the data taking on Pb+Pb collisions at 40\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace.\nThe data were registered by NA49, NA45, NA50 and NA57. \nIn 2000 a beam at 80\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace was delivered to NA49 and NA45.\nThe program was completed in 2002 by runs of NA49 and NA60 at 20\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace and 30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace.\nThus, together with the previously recorded data at 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, NA49 gathered data at  five collision energies. Other experiments collected data at two (NA50, NA57) or three (NA45, NA60) energies. Starting in 2010 the beam energy scan program BES was started at RHIC with the aim of covering the low\nenergy range overlapping with the CERN SPS and providing important\nconsistency checks on the measurements.\n\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/cs_ood_signals.jpg}\n\\caption{\nExamples of results illustrating the observation of the onset-of-deconfinement signals in central Pb+Pb (Au+Au) collisions~\\cite{Aduszkiewicz:2019zsv}, see text for details and more references.\n }\n\\label{fig:cs_ood_signals}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{Discovery of the onset of deconfinement.}\nResults on the collision energy dependence of hadron production in central Pb+Pb collisions from the onset-of-deconfinement search programme at the CERN SPS~\\cite{Afanasiev:2002mx,Alt:2007aa} appeared to be consistent with the predicted signals (for review see Ref.~\\cite{Gazdzicki:2010iv}):\n\\begin{enumerate}[(i)]\n\\item\nThe average number of pions per wounded nucleon, $\\langle N_\\pi \\rangle \/\\langle W \\rangle $, \nin low energy A+A collisions is smaller that this value in p+p reactions.  This relation is however changed to the opposite at  collision energies larger than $\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, the so-called {\\bf kink} structure. \n     \\item\n    The collision energy dependence of the $\\langle K^+\\rangle_{AA}\/\\langle \\pi^+\\rangle_{AA}$ ratio shows the so-called \\textbf{horn} structure. Following a fast rise the ratio passes through a maximum in the SPS range, at approximately 30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, and then decreases and settles to a plateau value at higher energies. This plateau was found to continue up to the RHIC and LHC energies.\n    \\item\n    The collision energy dependence of the inverse slope parameter of the transverse mass spectra, $T^*$, of charged kaons shows the so-called \\textbf{step} structure. Following a fast rise the $T^*$ parameter passes through a stationary region (or even a weak minimum for $K^-$), which starts at the low SPS energies, approximately 30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace, and then enters a domain of a steady increase above the top SPS energy.\n    \\end{enumerate}\n    \nFigure~\\ref{fig:cs_ood_signals} shows examples of the most recent plots~\\cite{Aduszkiewicz:2019zsv} illustrating the observation of the onset-of-deconfinement signals. As seen data from the RHIC BES~I programme\n(2010-2014) and LHC (see Ref.~\\cite{Aduszkiewicz:2019zsv} for references to original experimental papers) confirm the NA49 results and their interpretation. \n\n\nTwo comments are appropriate here.\nThe strangeness to pion ratio, e.g., \n$\\langle K^+\\rangle_{AA}\/\\langle \\pi^+\\rangle_{AA}$,\nstrongly increases with collision energy in the hadron phase. This happens because \n$m_K\/T\\gg 1$, whereas $m_\\pi\/T\\cong 1$.\nThus, a much stronger increase with increasing temperature is expected for the multiplicities of heavy strange hadrons than that of pions. The strangeness to pion ratio reaches its maximum inside the hadron phase at the onset of the deconfinement. \nThe plateau-like behavior  at high collision energies reflects the approximately constant value  of the \nstrangeness to entropy ratio in the QGP.  It equals to the ratio of the degeneracy factor of strange quarks, \n\\eq{\\label{s-quark}\ng_s=\\frac{7}{8}\\cdot 2\\cdot 2\\cdot 3 =10.5~,\n}\nto the total degeneracy factor of the quark-gluon constituents in the QGP,\n\\eq{\\label{qg}\ng= 2\\cdot 8+ \\frac{7}{8}\\cdot 2\\cdot 2\\cdot 3 \\cdot 3  = 47.5~.\n}\nThese degeneracy factors count 2 spin states of quarks and gluons, 3 flavor quark states, 8 colour states of gluons and 3 colour states of quarks, one more factor 2 appears due to antiquarks (the factor 7\/8 is due to the Fermi statistics of quarks).   \nThe strangeness to entropy ratio in the HRG at the largest  hadron temperature\n$T\\cong T_H\\cong 150$~MeV appears to be larger than this ratio in the QGP which is approximately constant at all QGP temperatures $T\\ge T_H$. Therefore, \nthe transition region from hadron matter to the QGP \nreveals itself as the {\\it suppression} of strangeness yield relative to pion yield.\n\nThe second comment concerns the inverse slope parameter $T^*$ of the transverse mass ($m_T=\\sqrt{m^2+p_T^2}$) spectrum \n\\eq{\\label{pT}\n\\frac{dN}{m_T dm_T}\\sim \\exp\\left(-\\,\\frac{m_T}{T^*}\\right)~.\n}\nThe parameter $T^*$ is sensitive to both the thermal and collective motion transverse to the collision axis and behaves as\n\\eq{\\label{T*}\nT^*\\cong T+ \\frac{1}{2}m v_T^2~,\n}\nwhere $T$ is the temperature and $v_T$ is the transverse collective (hydrodynamic) velocity of the hadronic matter at the kinetic freeze-out. The parameter $T^*$ increases strongly with collision energy \nup to the energy $\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace. This is because an increasing collision energy leads to an increase of both terms in Eq.~(\\ref{T*}) - the temperature  $T$ and velocity $v_T$ - in the hadron phase ($v_T$ increases due to the increase of the pressure). At collision energy larger than $\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}}\\xspace the parameter $T^*$ is approximately independent of the collision energy in the SPS energy range. In this region one expects the transition between confined and deconfined matter. In the transition region both values - $T$ and $v_T$ - remain approximately constant, and this leads to the plateau-like structure in the energy dependence of the $T^*$ parameter. \nAt  RHIC--LHC energies, the parameter $T^*$ again increases with collision energy. The early stage QGP pressure increases with collision energy, and thus $v_T$ in Eq.~(\\ref{T*}) increases too. \n\n\\vspace{0.1cm}\nThe workshop \\textit{Tracing the onset of deconfinement in nucleus-nucleus collisions}, ECT* Trento, April 24-29, 2004, \nsummarized the results from the energy scan programme at the CERN SPS and concluded that future measurements in the SPS energy range are needed~\\cite{Gazdzicki:2004gss}.\nThe goal is  to search for the deconfinement critical point and study system size dependence of the onset of deconfinement. Possibilities to perform  these measurements at the CERN SPS, FAIR SIS300 and RHIC were discussed. The event initiated a series of the \\textit{Critical point and onset of deconfinement} workshops.\n\n\n\\begin{figure}[hbt]\n\\centering\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_cp_hill.jpg}\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_cp_sigma.jpg}\n\\caption{\n \\textit{Left:}\n Sketch of the expected signal of the deconfinement critical point - a maximum of fluctuations in the (nuclear mass number)-(collision energy) plane.\n \\textit{Right:}\n     Results from the NA61\\slash SHINE\\xspace two dimensional scan of energy and system size for (pion multiplicity)-(transverse momentum) fluctuations in terms of the strongly intensive quantity $\\Sigma[P_T,N]$~\\cite{Gazdzicki:2017zrq}.\n }\n\\label{fig:cs_cp_signals}\n\\end{figure}\n\n\\subsection{Searching for deconfinement critical point}\n\\label{sec:cs_cp}\n\n\n\\textbf{Predicted d-CP signals.}\nThe possible existence and location of the deconfinement critical point (d-CP) is a subject of\nvivid theoretical discussion, for a recent review see Ref.~\\cite{Bzdak:2019pkr}. \nThe experimental search for the d-CP\nin A+A collisions at the CERN SPS was shaped by several model predictions (for detail see Ref.~\\cite{Bialas:1990xd}) of its potential signals:\n\\begin{enumerate}[(i)]\n    \\item \n    characteristic multiplicity fluctuations of hadrons~\\cite{Bialas:1990xd,Antoniou:2000ms,Hatta:2003wn,Antoniou:2006zb},\n    \\item\n    enhanced fluctuations of (pion multiplicity)-(transverse momentum)~\\cite{Stephanov:1999zu},\n\\end{enumerate}\n\n\nThe signals were expected to have a maximum in the parameter space of collision energy and nuclear mass number of colliding nuclei - \\textbf{the hill of fluctuations}~\\cite{Gazdzicki:2006fy}.\nThis motivated NA61\\slash SHINE\\xspace to perform a two dimensional scan at the CERN SPS~\\cite{Gazdzicki:995681} in these two parameters, which are well controlled in laboratory experiments. \n\n\n\n\\begin{figure}[hbt]\n\\centering\n\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/cs_cp_exp.jpg}\n\n\n\\caption{\n Summary of data recorded by NA61\\slash SHINE\\xspace at the CERN SPS~(\\textit{left}) and STAR at RHIC~(\\textit{right}) relevant for the search for the deconfinement-CP, see text for details. \n }\n\\label{fig:cs_cp_exp}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{Measurements at SPS and RHIC.}\nThe systematic search for the d-CP of strongly interacting matter was started in 2009 with the NA61\\slash SHINE\\xspace data taking on p+p interactions at six beam momenta in the range from 13\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace to 158\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace.\nIn the following years data on Be+Be, Ar+Sc, Xe+La and Pb+Pb collisions were\nrecorded, see Fig.~\\ref{fig:cs_cp_exp}~(\\textit{left}) for an overview.\n\nIn 2010 the beam energy scan (BES-I and BES-II) with Au+Au collisions started at the BNL RHIC~\\cite{Odyniec:2019kfh}. Search for the deconfinement critical point has been the most important goal of this programme. Above the collision energy of $\\sqrt{s_{NN}} = 7.7~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace$ ($\\approx$~30\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace) the scan was conducted in the collider mode, whereas below in the fixed target mode. The location of the recorded data in the phase diagram are shown in Fig.~\\ref{fig:cs_cp_exp}~(\\textit{right}).  \n\n\n\\vspace{0.2cm}\n\\textbf{Status of the d-CP search.}\nMany experimental results have already been obtained within the d-CP search programmes at SPS and RHIC, for a recent review see Ref.~\\cite{Czopowicz:2020twk}. Five of them were considered as possible indications of the d-CP and are presented and discussed in the following. \n\n\n\n\\begin{enumerate}[(i)]\n\\item\nA maximum of fluctuations is expected in a scan of the phase diagram (see Fig.~\\ref{fig:cs_cp_signals} (\\textit{left})). Measurements of (pion multiplicity)-(transverse momentum) fluctuations from NA61\\slash SHINE\\xspace shown in Fig.~\\ref{fig:cs_cp_signals} (\\textit{right}) do not show such a \nfeature~\\cite{Gazdzicki:2017zrq}.\n\\item\nThe energy dependence of fluctuations of conserved quantities such as the net baryon number is predicted to be sensitive to the presence of the d-CP. This holds in particular for higher moments. The scaled third and fourth moments of the net-proton multiplicity distribution in Au+Au collisions from the STAR experiment is plotted in Fig.~\\ref{fig:cs_cp_results1}~\\cite{Adam:2020unf}. The non-monotonic behaviour of the fourth moment in central collisions and its sign change around $\\sqrt{s_{NN}} \\approx 7$~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace is debated as possible indication of the d-CP.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.7\\linewidth,clip=true]{plots\/cs_cp_results1.jpg}\n\\caption{\n The energy dependence of the scaled third (\\textit{left}) and fourth (\\textit{right}) moments of the net-proton multiplicity distribution in central and peripheral Au+Au collisions from the STAR experiment~~\\cite{Adam:2020unf}.\n }\n\\label{fig:cs_cp_results1}\n\\end{figure}\n\n\\item\nAt the d-CP the correlation length diverges and leads to power-law type fluctuations of the baryon number. These were investigated by the NA61\\slash SHINE\\xspace experiment by measuring the momentum bin size dependence of the scaled second factorial moment of the proton multiplicity distribution (intermittency study) in semi-central Ar+Sc collisions at  $\\sqrt{s_{NN}} \\approx 17~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace$~\\cite{Mackowiak:2019qm}. While previous measurements by the NA49 experiment in Si+Si collisions indicated a signal the new measurements \nshown in Fig.~\\ref{fig:cs_cp_results2} do not confirm the effect.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/cs_cp_results2.jpg}\n\\caption{\n Scaled second factorial moment $\\Delta F_2$ (background subtracted) of the proton multiplicity distribution as function of the number of subdivisions M of transverse momentum space obtained in Ar+Sc collisions at $\\sqrt{s_{NN}} \\approx$ 17~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace~\\cite{Mackowiak:2019qm}.\n }\n\\label{fig:cs_cp_results2}\n\\end{figure}\n\n\\item\nThe ratio of yields of light nuclei production can be related to nucleon number fluctuations~\\cite{Liu:2019nii}. The measurements from STAR in central Au+Au collisions show strong collision energy dependence and peak at $\\sqrt{s_{NN}} \\approx 20 - 30~\\ensuremath{\\mbox{Ge\\kern-0.1em V}}\\xspace$~\\cite{Xu:2019qm}. These results are presented in Fig.~\\ref{fig:cs_cp_results3}. Such behaviour is not reproduced by model calculations without a d-CP and may thus be attributable\nto a critical point.\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/cs_cp_results3.jpg}\n\\caption{\n Energy dependence of the ratio of yields of light nuclei production in central Au+Au collisions~\\cite{Xu:2019qm} measured by the STAR experiment at RHIC. \n }\n\\label{fig:cs_cp_results3}\n\\end{figure}\n\n\\item\nThe energy and centrality dependence of short-range two-pion correlations as parameterized by source radius parameters determined from Bose-Einstein correlation analysis was used to search for indications of the d-CP~\\cite{Lacey:2014wqa,Adamczyk:2014mxp}.\nThe result for the difference $R^2_{out}$ - $R^2_{side}$ in Au+Au collisions at RHIC is shown in Fig.~\\ref{fig:cs_cp_results4}.\nA finite size scaling analysis of these results led to an estimate of the position of a d-CP at  $T \\approx$~165~MeV and \n$\\mu_B$ $\\approx$ 95~MeV.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.4\\linewidth,clip=true]{plots\/cs_cp_results4.jpg}\n\\caption{\n Centrality and energy dependence of the difference $R^2_{out}$ - $R^2_{side}$\n of radius parameters obtained from Bose-Einstein two-pion correlation analysis in Au+Au collisions from the PHENIX experiment at RHIC\n ~\\cite{Lacey:2014wqa,Adamczyk:2014mxp}.\n }\n\\label{fig:cs_cp_results4}\n\\end{figure}\n\n\\end{enumerate}\n\nThese observations, when interpreted as due to the d-CP, yield different estimates of the d-CP location on the\nphase diagram of strongly interacting matter, see Fig.~\\ref{fig:cs_cp_results_pd}~\\cite{Czopowicz:2020twk}.\nThus, as for now, the experimental results concerning the d-CP are inconclusive. New results from NA61\\slash SHINE\\xspace and STAR BES-II are expected within the coming years. \n\\vspace{0.2cm}\n\n\n\\textbf{Nuclear and deconfinement critical points.}\nThe nuclear critical point (n-CP) corresponds to the liquid-gas phase transition in the system of interacting nucleons and is located at small temperature $T_C\\approx 19$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace and large baryonic chemical potential $\\mu_B\\approx 915$~\\ensuremath{\\mbox{Me\\kern-0.1em V}}\\xspace, see Fig.~\\ref{fig:v_qgp} (\\textit{middle}) and Fig.~\\ref{fig:cs_cp_results_pd} for illustration.\n\nThe effect of the n-CP on fluctuations of conserved charges,\nbaryon number (B), electric charge (Q), and strangeness (S),\nwas studied in Refs.~\\cite{Vovchenko:2017ayq, Poberezhnyuk:2019pxs} within the \nHRG\nmodel with van der Waals interactions between baryons and between anti-baryons. \nThe second, third, and fourth order cumulants \n(susceptibilities) are\ncalculated in the grand canonical ensemble from the pressure function by taking the derivatives over the corresponding chemical potentials:\n\\eq{\n\\chi^{i}_{n}& =\\frac{\\partial ^{n}\\left( p\/T^{4}\\right) }{\\partial \\left( \\mu_{i} \n\/T\\right) ^{n}}~,\n}\nwhere $i$ stands for $B, Q, S$ and $n$ is the moment order. \n\nThe obtained results show that\nthe n-CP may significantly impact event-by-event fluctuations in A+A collisions even at high energies.  \nThus, the nuclear-CP should be taken into account in  future searches for the deconfinement-CP.\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.6\\linewidth,clip=true]{plots\/cs_cp_results_pd.jpg}\n\\caption{\nCompilation of theoretical predictions~\\cite{Stephanov:2007fk} and experimental hints~\\cite{Czopowicz:2020twk} on the location of the deconfinement critical\npoint, d-CP, in the phase diagram of strongly interacting matter. The position \nof nuclear critical point, n-CP, as suggested by theoretical and experimental results is indicated for comparision.\n }\n\\label{fig:cs_cp_results_pd}\n\\end{figure}\n\n\n\\subsection{Indication of the onset of fireball}\n\\label{sec:cs_oof}\n\n\\textbf{Predictions of reference models on system-size dependence.}\nThere are two models often used to obtain reference predictions concerning the system-size dependence of hadron production properties~\\cite{Gazdzicki:2013lda} - the Wounded Nucleon Model (WNM)~\\cite{Bialas:1976ed} and the Statistical Model (SM)~\\cite{Rafelski:1980gk}. For the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio at the CERN SPS energies they read:  \n\\begin{enumerate}[(i)]\n\\item\nThe WNM prediction: the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio is independent of the system size (number of wounded nucleons).\n\\item\nThe SM prediction: in the canonical formulation incorporating global quantum number conservation the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio increases monotonically with the system size and approaches the limit given\nby the grand canonical approximation of the model. The rate of this increase is the fastest for small systems.\n\\end{enumerate}\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.47\\linewidth,clip=true]{plots\/cs_oof_system-size.jpg}\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_oof_energy.jpg}\n\\caption{\n Measurements of the the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio in p+p, Be+Be, Ar+Sc and Pb+Pb collisions: system size dependence at 150\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace~\\cite{Podlaski:2019} (\\textit{left}) and  collision energy dependence~\\cite{Aduszkiewicz:2019zsv} (\\textit{right}). \n }\n\\label{fig:cs_oof_results}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.47\\linewidth,clip=true]{plots\/cs_oof_system-size_o.jpg}\n \\includegraphics[width=0.45\\linewidth,clip=true]{plots\/cs_oof_energy_o.jpg}\n\\caption{\n Measurements of the scaled variance $\\omega$ of the multiplicity distribution of negatively charged hadrons in inelastic p+p interactions and central Be+Be and Ar+Sc collisions~\\cite{AndreySeryakovfortheNA61\/SHINE:2017ygz}: system size dependence at 150\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace (\\textit{left})  and  collision energy dependence (\\textit{right}). \n }\n\\label{fig:cs_oof_results_o}\n\\end{figure}\n\n\\vspace{0.2cm}\n\\textbf{Unexpected result of measurements.}\nMeasurements of the system size dependence of hadron production properties at different collision energies were carried out by NA61\\slash SHINE\\xspace, for detail see Sec.~\\ref{sec:cs_ood}. \nFigures~\\ref{fig:cs_oof_results} and~\\ref{fig:cs_oof_results_o} show the unexpected result~\\cite{Podlaski:2019,AndreySeryakovfortheNA61\/SHINE:2017ygz}.\nThe \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio in Fig.~\\ref{fig:cs_oof_results} and the scaled variance of the multiplicity distribution at 150\\ensuremath{A\\,\\mbox{Ge\\kern-0.1em V}\\!\/\\!c}\\xspace in Fig.~\\ref{fig:cs_oof_results_o} are similar in inelastic p+p interactions and in central Be+Be collisions, whereas they are very different in central Ar+Sc collisions which are close to central Pb+Pb collisions. Both reference models, WNM and SM, qualitatively disagree with the data. \nThe WNM seems to work in the collisions of light nuclei (up to Be+Be) and becomes qualitatively wrong for heavy nuclei (like Pb+Pb). On the contrary, the SM is approximately valid for collisions of heavy nuclei.  However, its predictions\ndisagree with the data on p+p to Be+Be collisions.  \n\n\nThe rapid change of hadron production properties when moving from Be+Be to Ar+Sc collisions is interpreted and referred to as the onset of fireball. \nFrom Fig.~\\ref{fig:cs_oof_results}~(\\textit{right}) follows that the increase of the \\ensuremath{\\textup{K}^+}\\xspace\/\\ensuremath{\\pi^+}\\xspace ratio depends on the collision energy. On the other hand,\nthe scaled variance $\\omega$ of the multiplicity distribution shows only weak collision energy dependence \n(see Fig.~\\ref{fig:cs_oof_results_o}~(\\textit{right})).\nThe physics behind the onset of fireball is under discussion~\\cite{Motornenko:2018gdc}.\nWhereas one does not observe the formation of fireball in the collisions of light nuclei at the SPS energies, the large size system of strongly interacting matter is probably formed at LHC energies even in p+p high multiplicity events. \n\n\n\\vspace{0.3cm}\n\\textbf{In summary}, the scans in collision energy and nuclear mass number of colliding nuclei\nperformed at SPS and RHIC indicate four domains\nof hadron production separated by two thresholds: the onset of deconfinement and the\nonset of fireball. The sketch presented in Fig.~\\ref{fig:future}~(\\textit{left}) illustrates this conclusion.\n\n\n\\section{Plans for future measurements}\n\\label{sec:futures}\n\n\\begin{figure}[hbt]\n\n\\centering\n \\includegraphics[width=0.99\\linewidth,clip=true]{plots\/future_phases.jpg}\n \\caption{\\textit{Left:} \nTwo-dimensional scan conducted by NA61\\slash SHINE\\xspace varying collision energy and nuclear mass number of colliding nuclei indicates four domains of hadron\nproduction separated by two thresholds: the onset of deconfinement and the onset\nof fireball. The onset of deconfinement is well established in central Pb+Pb~(Au+Au)\ncollisions, its presence in collisions of low mass nuclei, in particular, inelastic p+p\ninteractions is questionable.\n\\textit{Right:} \nRegions in the phase diagram of strongly interacting matter studied by present (red) and future (green) heavy ion facilities.\n }\n\\label{fig:future}\n\\end{figure}\n\nLet us close by discussing possible future measurements which are suggested by this review and which should be considered as priorities: \n\\begin{enumerate}[(i)]\n\\item\nA collision energy scan in the onset of deconfinement region to measure open and hidden charm production in Pb+Pb collisions and establish the impact of the onset on the heavy quark sector. \nThis requires high statistics data collected with detectors optimized for open and hidden charm measurements. Detailed physics arguments and possible experimental set-ups are presented in Refs.~\\cite{Aduszkiewicz:2309890,Agnello:2018evr}.\n\\item\nA detailed study of the onset of fireball and its collision energy dependence in the onset of deconfinement region. The goal is to understand the underlying physics of this phenomenon, for details see Ref.~\\cite{Aduszkiewicz:2309890}. This requires a two dimensional scan in the nuclear mass number of the colliding nuclei and in collision energy performed with small steps in nuclear mass number.\n\\end{enumerate}\nConclusive results from the data recorded by NA61\\slash SHINE\\xspace and RHIC BES-II are needed to\nplan future measurements for the deconfinement-CP search.\n\nFigure~\\ref{fig:future}~(\\textit{right}) presents a compilation of present and future facilities and their region of coverage in the phase diagram of strongly interacting matter. \nCharm measurements are planned by NA61\\slash SHINE\\xspace~\\cite{Aduszkiewicz:2309890}, NA60+~\\cite{Agnello:2018evr} at the CERN SPS, they are considered by MPD~\\cite{Kekelidze:2018nyo} at NICA and J-PARC-HI~\\cite{Sako:2019hzh} at J-PARC.\nA detailed two dimensional scan is considered by NA61\\slash SHINE\\xspace at the CERN SPS~\\cite{Aduszkiewicz:2309890}.\n\n\n\\begin{acknowledgments} \nThe work was supported by the the National Science Centre Poland grant 2018\\slash 30\\slash A\\slash ST2\\slash 00226 and the German Research Foundation grant GA1480\\slash 8-1. The work of M.I.G. is supported by the Program of Fundamental Research of the Department of Physics and Astronomy of National Academy of Sciences of Ukraine.\n\\end{acknowledgments}\n\n\\bibliographystyle{ieeetr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}