commit 5d41a3cc313ca78d6d675f21a818687d468c8dcd Author: Ian Jauslin Date: Sun Nov 2 10:06:28 2025 -0500 Initial commit diff --git a/He_Jauslin_Lebowitz_Peled_2025.tex b/He_Jauslin_Lebowitz_Peled_2025.tex new file mode 100644 index 0000000..18a125c --- /dev/null +++ b/He_Jauslin_Lebowitz_Peled_2025.tex @@ -0,0 +1,2008 @@ +\documentclass[12pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage[normalem]{ulem} + +% \linespread{2} + +\usepackage{amsmath} +\usepackage{amsthm,physics} +\usepackage{array} +\usepackage{authblk} +\usepackage{amssymb} +\usepackage{xcolor} +\usepackage[numbers,sort&compress]{natbib} +\usepackage{hyperref} +\usepackage{mathtools} +\usepackage{xargs}[2008/03/08] +\usepackage[shortlabels]{enumitem} +\usepackage{comment} +\usepackage{subcaption} +\usepackage{mathrsfs} + +\theoremstyle{plain} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} + 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+\long\def\obs#1{{\color{lightgray}#1}} + +\usepackage{dsfont} +\newcommand{\indicator}[1]{\mathds{1}_{#1}} +\newcommand{\dist}{\operatorname{dist}} +\renewcommand{\cp}{\mathrm{cp}} +\newcommand{\set}[1]{\left\{#1\right\}} +\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} +\renewcommand{\max}{\mathrm{max}} + +\global\long\def\per{\mathrm{per}} +\global\long\def\eps{\epsilon} +\global\long\def\E{\mathbb{E}} +\global\long\def\Z{\mathbb{Z}} +\global\long\def\P{\mathbb{P}} +\def\T{\mathbb{T}} +\def\R{\mathbb{R}} +\def\N{\mathbb{N}} +\DeclareMathOperator\diam{diam} +\DeclareMathOperator\CE{CE} + +\begin{document} + +\title{Liquid-vapor transition in a model of a continuum particle system with finite-range modified Kac pair potential} + +\author[1]{Qidong He} +\author[2]{Ian Jauslin} +\author[3]{Joel Lebowitz} +\author[4]{Ron Peled} +\affil[1,2,3]{Department of Mathematics, Rutgers University} +\affil[3]{Department of Physics, Rutgers University} +\affil[4]{Department of Mathematics, University of Maryland, College Park} + +\date{} + +\maketitle + +\abstract + +We prove the existence of a phase transition in dimension $d>1$ in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range $\gamma^{-1}$, with $\gamma>0$. +This transition is ``close'', for small positive $\gamma$, to the one proved previously by Lebowitz and Penrose in the van der Waals limit $\gamma\downarrow0$. +It is of the type of the liquid-vapor transition observed when a fluid, like water, heated at constant pressure, boils at a given temperature. +Previous results on phase transitions in continuum systems with stable potentials required the use of unphysical four-body interactions or special symmetries between the liquid and vapor. + +The pair interaction we consider is obtained by partitioning space into cubes of volume $\gamma^{-d}$, and letting the Kac part of the pair potential be uniform in each cube and act only between adjacent cubes. +The ``short-range'' part of the pair potential is quite general (in particular, it may or may not include a hard core), but restricted to act only between particles in the same cube. + +Our setup, the ``boxed particle model'', is a special case of a general ``spin'' system, for which we establish a first-order phase transition using reflection positivity and the Dobrushin--Shlosman criterion. + +\tableofcontents + + + +\section{Introduction} + +In 1998, Lebowitz, Mazel, and Presutti~\cite{lebowitz1998rigorous} wrote: +``An outstanding problem in equilibrium statistical mechanics is to derive rigorously the existence of a liquid-vapor phase transition (LVT) in a continuous system of particles interacting with any kind of reasonable potential, say Lennard--Jones or hard core plus attractive square well.'' +This situation has remained largely unchanged over the past quarter century. +This is so despite the fact that the LVT is ubiquitous. +It is observed every time we boil a pot of water and is displayed prominently in textbooks as a paradigm of phase transitions in physical systems; see Figure \ref{fig:phase-diagram}. +The LVT is also observed in all computer simulations of systems with the above type of pair potentials \cite{hansen1969phase,mcdonald1972equation}. +These simulations show that such pair potentials are, in fact, adequate for describing the commonly observed LVT, so why can we not prove it mathematically? +In fact, the LVT is qualitatively described by approximate theories with mean-field-type interactions (where all particles interact with the same strength), dating back to the nineteenth century; see below and \cite{van73,Maxwell75}. +However, for pair potentials without any symmetry, it has been proven rigorously only in the case of an attractive Kac type pair potential of the form $\gamma^d\varphi(\gamma r)$, where $d$ is the spatial dimension, in the infinite-range limit $\gamma\downarrow 0$, thus the need for a rigorous proof of the existence of the LVT in a continuum particle system with finite range (or rapidly decaying) pair interactions. +We do this here for a simplified model for small $\gamma>0$. +We give a brief historical background of the LVT in Section \ref{sec:history}. + +Mathematically speaking, we are interested in proving, for a continuous system of particles with stable pair interactions (see Appendix \ref{app:ruelle}) and no special symmetries, the existence of more than one infinite-volume Gibbs measure having different densities for some ranges of inverse temperature $\beta=1/T$ (setting Boltzmann's constant equal to $1$) and chemical potential $\lambda$ \cite{ruelle1971existence}. +For a system of particles in a region $\Lambda\subset\R^d$ with pair interactions $u(x-x')$, $x,x'\in\R^d$, the probability of having $N$ particles in a configuration $X_{\Lambda}=(x_1,\dots,x_N)\in\Lambda^N$ given a specified configuration in $\Lambda^c$, $Y_{\Lambda^c}$ (boundary condition), is given, in the grand-canonical Gibbs measure, by +\begin{equation} +\label{eqn:grand-canonical finite volume Gibbs measure} + \P^{\Lambda}_{\beta,\lambda}(\dd{X}_{\Lambda}\mid Y_{\Lambda^c})=\frac{1}{\Xi^{\Lambda}_{\beta,\lambda}} + \frac{1}{N!}e^{-\beta[-\lambda N+U(X_\Lambda\mid Y_{\Lambda^c})]}\prod_{i=1}^N\dd{x}_i, +\end{equation} +where +\begin{equation} +\label{eqn:particle model Hamiltonian} + U(X_\Lambda\mid Y_{\Lambda^c})=\sum_{1\le i0$, a good qualitative description of the LVT observed in real systems \cite{Bernal33,Millot92}. + +The idea behind this equation of state (EOS) is that the force between atoms is strongly repulsive (hard-core like) at short distances and weakly attractive at large distances, which respectively give rise to the first and second terms on the RHS of \eqref{eqn:introduction_van-der-Waals}. +In fact, the second term can be derived by assuming a ``mean field'' attractive interaction between the particles, i.e., every pair of particles interact with a potential independent of the distance between them, whose strength is inversely proportional to the size of the system. +The first term in \eqref{eqn:introduction_van-der-Waals} can be written as $T/(\rho^{-1}-b)$, where the denominator represents the effective volume available to each particle. +It is, in fact, the exact pressure of a one-dimensional system of hard rods of diameter $b$ and can be considered an approximation for a strong short-range repulsion in higher-dimensional systems. + +The canonical free energy density $f(T,\rho)$ is defined as the thermodynamic limit +\begin{equation} + f(T,\rho):=-\lim_{\substack{N\to\infty,\Lambda\uparrow\R^d\\N/\abs{\Lambda}\to\rho}}\frac{T}{\abs{\Lambda}}\log Z(T,N,\Lambda), +\end{equation} +where +\begin{equation} + Z(T,N,\Lambda):=\frac{1}{N!}\int_{\Lambda^N}\dd{x}_1\dots\dd{x}_N e^{-\frac{1}{T} U(X_\Lambda)} +\end{equation} +is the canonical partition function, $\abs{\Lambda}$ is the volume of $\Lambda$, and $U(X_\Lambda)$ is the interaction potential of the $N$ particles in $\Lambda$, i.e., the first sum in \eqref{eqn:particle model Hamiltonian}. +Using the fact (see e.g. \cite{Ruelle69}) that the pressure is given by +\begin{equation} +\label{eqn:introduction_pressure} + p(T,\rho)=\rho^2\pdv{\rho}\left[\frac{1}{\rho}f(T,\rho)\right], +\end{equation} +\eqref{eqn:introduction_van-der-Waals} is equivalent to the following expression for $f(T,\rho)$: +\begin{equation} +\label{eqn:introduction_free-energy} + f(T,\rho)=-T\rho\log\frac{1-b\rho}{\rho}-\frac{1}{2}a\rho^2, +\end{equation} +up to a term independent of $\rho$. +The Maxwell construction thus corresponds to the Gibbs double tangent construction for $f(T,\rho)$; see Figure \ref{fig:constructions}. + +The question then, as now, is how to derive the liquid-vapor phase transition from a ``realistic'' pair interaction between the particles. + +\begin{figure} + \centering + \begin{subfigure}[t]{0.46\textwidth} + \centering + \includegraphics[width=0.9\columnwidth]{maxwell_construction.tikz.pdf} + \caption{The solid line represents isotherms as given by \eqref{eqn:introduction_van-der-Waals} while the dotted line is the Maxwell construction giving equal areas to the solid color regions. + This gives the coexistence of liquid and vapor phases at the same $T$ and $p$.} + \label{fig:maxwell-construction} + \end{subfigure} + \hspace{12pt} + \begin{subfigure}[t]{0.46\textwidth} + \centering + \includegraphics[width=0.9\columnwidth]{double_tangent.tikz.pdf} + \caption{The Gibbs double tangent construction for $T0$ \cite{presutti2008scaling} but that the transition line should be, for small $\gamma>0$, close to its limiting value as $\gamma\downarrow0$. +This has been established by Presutti \cite{presutti2008scaling} for lattice gases on $\Z^{d}$ ($d\ge 2$) when $v(r)$ is just the single-site hard-core exclusion \cite{cassandro1996phase,bodineau1997phase,bovier1997low}. +However, for general lattice systems, without the particle-hole symmetry present in this case, there is no proof of the existence of an LVT at $\gamma>0$ close to that for $\gamma\downarrow 0$. +The situation is similar for continuum systems with stable pair potentials: the only proof of the existence of an LVT for continuum systems with finite-range potentials and no symmetries is given by Lebowitz, Mazel, and Presutti \cite{lebowitz1998rigorous,lebowitz1999liquid}, who had to resort to \emph{unphysical}, \emph{long-range four-body} repulsion to take the place of the short-range repulsive pair interactions to ensure stability of the system against collapse by the attractive Kac potential. +Their result, therefore, did not alleviate the need for proving the LVT for realistic pair potentials. +We also note here that Ruelle \cite{ruelle1971existence} proved the existence of a phase transition for the continuum two-species symmetric Widom-Rowlinson model and that Johansson \cite{johansson1995separation} proved a phase transition for a 1D continuum model whose pair potential decays like $r^{-\alpha}$, $\alpha\in(1,2)$. + + + +\paragraph{The present work} + +To obtain insight into the LVT for pair potentials of the form \eqref{eqn:introduction_pair-potential} at small $\gamma>0$ and its relation to the mean-field limit $\gamma\downarrow 0$, we study here a simplified model which we call the ``box model'' and which will be described in detail below. +In this model, we prove the existence of an LVT for small $\gamma>0$ given some properties of the free energy $f_{0}(T,\rho)$ of the reference system, the one in which the Kac potential is absent. +These properties are essentially the same as those used by LP to prove the LVT in the limit $\gamma\downarrow 0$, some of which are direct consequences of the existence of the thermodynamic limit for particle systems with superstable and tempered interactions \cite[\S3.1]{Ruelle69}; see Appendix \ref{app:ruelle}. +We show that, in the box model framework, an LVT occurs \emph{close} to the $\gamma\downarrow 0$ limit, in a precise sense, for sufficiently small $\gamma>0$. +Our result also applies to soft-core potentials, which LP did not consider. + +The main benefit of the ``box model'' is that it is reflection positive. We remark that the idea of using reflection positivity to justify mean-field predictions has appeared before, in a different context, in the works~\cite{biskup2003rigorous,biskup2006mean} (see also~\cite[Section 4]{biskup2009reflection}). + +\subsection{The boxed particle model and the box model}\label{sec:the box model} + +In this paper, we introduce and study the \emph{box model}, which is a generalization of a particle model called the \emph{boxed particle model} obtained by modifying the pair potential in \eqref{eqn:introduction_pair-potential} that was studied by Lebowitz and Penrose in the mean-field limit $\gamma\downarrow 0$. +Let us first define the boxed particle model, which we will generalize to the box model in Section \ref{sec:box_model_def}. + +The boxed particle model is defined by modifying \eqref{eqn:introduction_pair-potential} (primarily in the Kac potential part $\varphi$, but the short-range part will also be modified). +To do this, we partition $\R^d$ into a lattice of \emph{mesoscopic} cubes of side length $\gamma^{-1}$, and replace the long-range Kac interaction of LP with one that is constant for particles inside each of the cubes, and has a (possibly different) constant value for particles in adjacent cubes. +We will be quite general about the other interaction between particles given by $v(r)$ in \eqref{eqn:introduction_pair-potential}, and merely assume that it is superstable and tempered \cite{Ruelle69}. +However, we will neglect the interaction between different cubes due to $v(r)$. +A possible choice for $v(r)$ is a hard-core interaction, but we will be more general than that and allow interactions like the Lennard--Jones or Morse pair potentials (see Appendix \ref{app:ruelle} for more details). + +More formally, we define the boxed particle model as a model for a system of particles interacting via the following pair interaction: +\begin{equation} +\label{eqn:boxed pair interaction} + u_\gamma(x,y):= + \begin{cases} + v(x-y)-J_1\gamma^d & \mbox{if }\operatorname{Box}_\gamma(x)=\operatorname{Box}_\gamma(y)\\ + -J_2\gamma^d & \mbox{if }\operatorname{Box}_\gamma(x)\sim\operatorname{Box}_\gamma(y)\\ + 0 & \mbox{otherwise} + \end{cases} +\end{equation} +where $J_1>-2dJ_2$ and $J_2>0$ are constants, $\operatorname{Box}_\gamma(x):=\gamma^{-1}\floor{\gamma x}+[0,\gamma^{-1})^d$ denotes the unique cube in the mesoscopic lattice containing $x\in\R^d$, and the symbol $\sim$ means that two cubes are nearest neighbors, i.e., they share a $(d-1)$-dimensional face, and $v$ is a superstable and tempered potential (see Appendix \ref{app:ruelle} and \cite{Ruelle69}). +We fix the chemical potential in the boxed particle model to $\lambda+\frac12J_1\gamma^d$ (we add $\frac12J_1\gamma^d$ to simplify the notation in the ``spin'' model introduced in the next paragraph). + +It is straightforward to check that the box model is equivalent to a ``spin'' model on the lattice $\mathbb Z^d$ with nearest neighbor interactions, where each point $v$ corresponds to a mesoscopic lattice cube as above, and the ``spin'' at $v$ is given by the density $\eta_v:=N_v/\gamma^{-d}$ of particles inside the cube corresponding to $v$. +Indeed, by integrating over the positions of the $N_v$ particles in each cube $v$, we find that the boxed particle model is equivalent to the following effective Hamiltonian on configurations of densities in the cubes: +\begin{equation} +\begin{split} +\label{eq:Hamiltonian_box} +H_{\lambda,\gamma}(\eta) +:= &\gamma^{-d}\left[-\lambda\sum_v\eta_v- \frac{1}{2}J_{1}\sum_v \eta_v^{2} - J_{2}\sum_{v\sim w}\eta_v\eta_w + \sum_v f_{\gamma}(\eta_v)\right]\\ +=&\gamma^{-d}\left[-\lambda\sum_v\eta_v- \frac{1}{2}(J_1+2d J_2)\sum_v \eta_v^{2} + \frac{1}{2}J_{2}\sum_{v\sim w}|\eta_v-\eta_w|^2 + \sum_v f_{\gamma}(\eta_v)\right], +\end{split} +\end{equation} +and +\begin{equation} + f_\gamma(\eta_v):=-\frac{1}{\beta\gamma^{-d}}\log\frac1{(\eta_v\gamma^{-d})!}\int_{([0,\gamma^{-1})^{d})^{\eta_v\gamma^{-d}}}\dd{x}_1\cdots \dd{x}_{\eta_v\gamma^{-d}}\prod_{i0$ (the coupling strengths) and $\beta>0$ (the inverse temperature)---these parameters will be held fixed throughout our arguments and will be omitted from the notation. Let $\lambda\in\R$ (the chemical potential). The formal Hamiltonian of the box model is given by +\begin{equation}\label{eq:Hamiltonian_box_rewritten} +H_{\lambda,\gamma}(\eta) +\coloneqq \gamma^{-d}\left(\sum_{v}\left[-\lambda\eta_v- \frac{1}{2}\alpha\eta_v^{2} + f_{\gamma}(\eta_v)\right]+\frac{1}{2}J_{2}\sum_{v\sim w}\abs{\eta_{v}-\eta_{w}}^{2}\right), +\end{equation} +where $f_{\gamma}:S_\gamma\to\R\cup\{+\infty\}$ is a measurable function for each $\gamma$, satisfying the convergence assumption in Section \ref{sec:convergence_assumptions} (in particular, these conditions allow for $f_\gamma$ to be the free energy for a system of particles interacting via a superstable, tempered pair interaction; see Section \ref{sec:convergence_assumptions} for details, and so the boxed particle model introduced above is a special case of the box model). +We note that $f_{\gamma}$ may depend on the fixed parameters $\alpha$, $J_2$, and $\beta$, but we will not make this dependence explicit. + +Given an integer $L\ge 2$, the Gibbs measure of the box model on the discrete torus $\Lambda_L:=\mathbb Z^d/(L \mathbb Z)^d$ is +\begin{equation}\label{eq:box model finite-volume Gibbs measure periodic} + \frac{1}{\Xi^{L, \per}_{\lambda,\gamma}}e^{-\beta H_{\lambda,\gamma}^{L,\per}(\eta)}\prod_{v\in\Lambda_L}\dd{\nu}_\gamma(\eta_v) +\end{equation} +where $\nu_\gamma$ is Lebesgue measure on $S_\gamma=[0,\infty)$ in the continuous case and $\nu_\gamma$ is the normalized counting measure $\gamma^d\sum_{\rho\in S_\gamma}\delta_\rho$ in the discrete case, where $H_{\lambda,\gamma}^{L,\per}$ is given by the expression~\eqref{eq:Hamiltonian_box_rewritten} for $H_{\lambda,\gamma}$, changing the sums to run over $v\in\Lambda_L$ and $\{v,w\}\in E(\Lambda_L)$ (the edge set of the discrete torus graph), and where +\begin{equation} +\label{eq:box model finite-volume partition function periodic} + \Xi^{L,\per}_{\lambda,\gamma}:=\int e^{-\beta H_{\lambda,\gamma}^\Lambda(\eta)}\prod_{v\in\Lambda}\dd{\nu}_\gamma(\eta_v) +\end{equation} +is the partition function (normalizing constant). One may similarly define Gibbs measures with free or prescribed boundary conditions. + + + +\subsubsection{Convergence assumptions}\label{sec:convergence_assumptions} + +We assume that the functions $(f_\gamma)_{0<\gamma\le 1}$ satisfy one of the following two conditions: (in the discrete case, for convenience in stating the assumption, we extend $f_{\gamma}$ to $[0,\infty)$ by a linear interpolation) +\begin{enumerate} + \item Hard-core case: There is $\rho_{\cp}\in(0,\infty)$ and a continuous $f:[0,\rho_{\cp})\to\R$ such that: +\begin{enumerate} + \item For every $\rho_0\in[0,\rho_{\cp})$, + \begin{equation}\label{eq:uniform convergence below eta cp} + \lim_{\gamma\downarrow0}f_{\gamma}(\rho) = f(\rho)\quad\text{uniformly in $\rho\in[0,\rho_0]$.} + \end{equation} + \item + \begin{equation}\label{eq:convergence at eta cp} + \liminf_{\substack{\gamma\downarrow0\\\rho\to\rho_{\cp}}} f_{\gamma}(\rho) \ge \lim_{\rho\uparrow\rho_{\cp}} f(\rho)\in(-\infty,\infty] + \end{equation} + (the existence of the limit $\lim_{\rho\uparrow\rho_{\cp}} f(\rho)$ in $(-\infty,\infty]$ is part of the assumption). + \item There is $\rho_{\max}\in(\rho_{\cp},\infty)$ such that $f_{\gamma}(\rho)=\infty$ for all $\rho>\rho_{\max}$ and $0<\gamma\le 1$. In addition, for every $\rho_1>\rho_{\cp}$, + \begin{equation}\label{eq:uniform convergence above eta cp} + \lim_{\gamma\downarrow0}\inf_{\rho\ge \rho_1}f_{\gamma}(\rho) = \infty. + \end{equation} +\end{enumerate} +\item Soft-core case: There is a continuous $f:[0,\infty)\to\R$ such that: +\begin{enumerate} + \item For every $\rho_0\in[0,\infty)$, + \begin{equation}\label{eq:uniform convergence below infinity} + \lim_{\gamma\downarrow0}f_{\gamma}(\rho) = f(\rho)\quad\text{uniformly in $\rho\in[0,\rho_0]$.} + \end{equation} + \item + \begin{equation}\label{eq:growth at infinity} + \alpha_{\max}:=\liminf_{\substack{\gamma\downarrow0\\\rho\to\infty}}\frac{f_{\gamma}(\rho)}{\frac{1}{2}\rho^2}\in(0,\infty]. + \end{equation} +\end{enumerate} +\end{enumerate} +To unify some of our later statements, we set $\alpha_{\max}\coloneqq\infty$ in the hard-core case. +We note that the above assumptions imply that the box model is well-defined for sufficiently small $\gamma$ and $\alpha<\alpha_\max$, in the sense made precise in Lemma~\ref{lem:model is well defined}. + + +\begin{remark} +\begin{enumerate} +\item In the hard-core case, it follows from~\eqref{eq:uniform convergence below eta cp} that +$\liminf_{\substack{\gamma\downarrow0\\\rho\to\rho_{\cp}}} f_{\gamma}(\rho) \le \liminf_{\rho\uparrow\rho_{\cp}} f(\rho)$. This complements the first inequality in~\eqref{eq:convergence at eta cp}, making it an equality. + +\item +In the soft-core case, we note that~\eqref{eq:uniform convergence below infinity} and~\eqref{eq:growth at infinity} imply that +\begin{equation}\label{eq:growth of f at infinity} + \liminf_{\rho\to\infty}\frac{f(\rho)}{\frac{1}{2}\rho^2}\ge\alpha_\max. +\end{equation} + +\item +The above assumptions are satisfied for the boxed particle model, for which we recall that $f_{\gamma}(\rho)$ (see (\ref{eq:box model finite-volume partition function periodic})) is the free energy for a system of particles that take positions in the hypercubic volume $\gamma^{-d}$ and interact via a superstable and tempered pair potential. +This is a consequence of well-known results by Ruelle \cite{Ruelle69, ruelle1963classical}. +For more detailed references and a proof that these conditions are satisfied for superstable and tempered potentials, see Appendix \ref{app:ruelle} and Proposition \ref{prop:convergence_assumptions}. +In particular, the assumptions are verified when the interaction is a hard core, or the Lennard--Jones potential, or the Morse potential; see Proposition \ref{prop:superstable_examples}. +\end{enumerate} +\end{remark} + + + +\subsubsection{Main results} +\label{sec:main results} + +Our first result identifies the limiting grand-canonical pressure of the box model, showing that it coincides with the expression obtained by Gates--Penrose in the limit $\gamma\downarrow 0$ \cite{gates1969vani}. +Define the {\it canonical mean-field free energy density} (following the nomenclature of \cite[(10.2.1.3)]{presutti2008scaling}) +\begin{equation}\label{eq:mean-field free energy density} + \phi_{\lambda}(\rho) +:= -\lambda\rho-\frac{1}{2}\alpha\rho^{2}+ f(\rho). +\end{equation} + +\begin{theorem} +\label{thm:comparison_with_GP} +In every dimension $d\ge 1$, +\begin{equation} +\lim_{\substack{L\to\infty\\\gamma\downarrow0}}\frac{1}{\beta\gamma^{-d}|\Lambda_L|}\log\Xi^{L, \per}_{\lambda,\gamma} = -\inf_{\rho}\phi_{\lambda}(\rho), +\end{equation} +with the infimum over $\rho\in[0,\rho_\cp)$ in the hard-core case and over $\rho\in[0,\infty)$ in the soft-core case. +\end{theorem} +The same result (with the same proof) also holds for free boundary conditions, or for prescribed boundary conditions which are uniformly bounded as $L\to\infty, \gamma\downarrow 0$. + +Lebowitz--Penrose~\cite{lebowitz1966rigorous} (canonical ensemble), followed by Gates--Penrose~\cite{gates1969vani} (grand-canonical ensemble), proved the existence of a liquid-vapor phase transition in the mean-field limit $\gamma\downarrow 0$ whenever the function $\phi_\lambda$ is non-convex. Our main result establishes the liquid-vapor phase transition at positive $\gamma$ (i.e., before taking the mean-field limit) in the box model. + +\begin{theorem}\label{thm:main} + Suppose the dimension $d\ge 2$. Suppose $\beta>0$ and $0<\alpha<\alpha_{\max}$ are such that $\phi_\lambda$ is non-convex (this property does not depend on $\lambda$). Then there exists $\gamma_0 > 0$ such that for all $0<\gamma\le\gamma_0$ there exists $\lambda(\gamma)$ for which the box model admits two distinct translation-invariant Gibbs measures that differ from each other in their value for the average density. +\end{theorem} + +Moreover, as we formulate next, we show that the critical chemical potential and the densities of the liquid and vapor phases tend to their mean-field values as $\gamma\downarrow 0$. + +In the hard-core case, the function $f$ is defined on the interval $[0,\rho_{\cp})$. We extend its domain to $[0,\rho_\cp]$ by setting +\begin{equation}\label{eq:f extension to rho cp} + f(\rho_{\cp}):=\lim_{\rho\uparrow\rho_\cp} f(\rho), +\end{equation} +noting that $f(\rho_\cp)\in(-\infty,\infty]$ by~\eqref{eq:convergence at eta cp}. This also extends the domain of $\phi_\lambda$ to $[0,\rho_\cp]$, for all $\lambda$, via~\eqref{eq:mean-field free energy density}. + +Suppose $\beta>0$ and $0<\alpha<\alpha_{\max}$ are such that $\phi_\lambda$ is non-convex. Let $\lambda_*\in\R$ be such that $\phi_{\lambda_*}$ attains its global minimum at (at least) two points $\rho_{*,-}<\rho_{*,+}$ with $\phi_{\lambda_\ast}$ non-constant on the interval $[\rho_{*,-},\rho_{*,+}]$; such a $\lambda_*$ necessarily exists by the non-convexity of $\phi_{\lambda}$ and \eqref{eq:convergence at eta cp}, in the hard-core case, or \eqref{eq:growth of f at infinity}, in the soft-core case. +Let +\begin{equation} + \mathcal{M}:=\set{\rho\mid \phi_{\lambda_*}(\rho)=\inf_{\rho'}\phi_{\lambda_*}(\rho')} +\end{equation} +be the points where the global minimum is attained. Fix $\rho_{*,0}$ with $\rho_{*,-}<\rho_{*,0}<\rho_{*,+}$ and $\rho_{*,0}\notin\mathcal{M}$. + +\begin{theorem} +\label{thm:detail} + Suppose the dimension $d\ge 2$ and proceed in the above setup. + Then there exists $\gamma_0 > 0$ such that, for all $0<\gamma\le\gamma_0$, there exists $\lambda_c(\gamma)$ for which the box model with $\lambda=\lambda_c(\gamma)$ admits two translation-invariant Gibbs measures $\P^\pm_{\lambda_c(\gamma), \gamma}$ and we have + \begin{equation}\label{eq:convergence of critical chemical potential} + \lim_{\gamma\downarrow0}\lambda_c(\gamma) = \lambda_* + \end{equation} + and, for every open set $U\subset\R$ containing $\mathcal{M}$, + \begin{equation}\label{eq:density concentration} + \lim_{\gamma\downarrow0}\P^-_{\lambda_c(\gamma), \gamma}(\eta_0\in U\cap[0,\rho_{*,0}))=\lim_{\gamma\downarrow0}\P^+_{\lambda_c(\gamma), \gamma}(\eta_0\in U\cap(\rho_{*,0},\infty))=1. + \end{equation} +\end{theorem} + +\begin{remark}\label{remark:remark after main theorem} + In the (generic) case when $\mathcal{M}$ consists solely of the two points $\rho_{*,\pm}$, it follows from~\eqref{eq:density concentration} that the distribution of $\eta_0$ under $\P^\pm_{\lambda_c(\gamma), \gamma}$ converges in distribution to a delta measure at $\rho_{*,\pm}$. +\end{remark} + +\subsection{The $(J,\omega)$-spin model and a condition for first-order phase transition}\label{J omega model section} + +The main idea of the proof of Theorem \ref{thm:main} is to use the reflection positivity of the box model to derive a chessboard estimate, which allows us to use a result by Dobrushin and Shlosman \cite{shlosman1986method} guaranteeing the existence of the phase transition. +For the sake of completeness, we state and prove a version of the Dobrushin--Shlosman criterion that is suited to our result in Section \ref{sec:dobrushin_shlosman}. + +Our analysis applies to a wider class of models. In this section, we introduce this class and state a condition that ensures a first-order phase transition therein. + + + +\subsubsection{The $(J,\omega)$-spin model} +Given $J\ge0$ and a Borel measure $\omega$ on $\R^n$ with finite, positive total measure (i.e., $\omega(\R^n)\in(0,\infty)$), we give the name $(J,\omega)$-spin model to the $n$-component Gaussian free field with coupling constant $J$ and single-site measure $\omega$. This is the model on configurations $\eta:\Z^d\to\R^n$ whose formal Hamiltonian is +\begin{equation}\label{eq:(J,omega)-spin model Hamiltonian} +H(\eta):=J\sum_{v\sim w}\norm{\eta_{v}-\eta_{w}}^{2}. +\end{equation} + +To apply the chessboard estimate, we consider the model on the discrete torus $\Lambda_L=\mathbb Z^d/(L \mathbb Z)^d$: Its finite-volume Hamiltonian is +\begin{equation}\label{Jw_ham} +H^L_J(\eta):=J\sum_{\{v,w\}\in E(\Lambda_L)}\norm{\eta_{v}-\eta_{w}}^{2} +\end{equation} +on configurations $\eta:\Lambda_L\to\R^n$, where $E(\Lambda_L)$ denotes the usual edge set of $\Lambda_L$, thought of as a graph, and $\|\cdot\|$ is the Euclidean norm (when convenient, we also regard such configurations as periodic functions on the entire $\Z^d$). +The corresponding finite-volume Gibbs measure is +\begin{equation} + \P^{L,\per}_{J,\omega}(\dd{\eta})=\frac1{\Xi^{L,\per}_{J,\omega}} e^{-H^L_J(\eta)}\prod_{v\in\Lambda_L} \omega(\dd{\eta}_v) + \label{prob_Jw} +\end{equation} +where $\Xi^{L,\per}_{J,\omega_\lambda}$ is the partition function: +\begin{equation} + \Xi^{L,\per}_{J,\omega}:=\int e^{-H^L_J(\eta)}\prod_{v\in\Lambda_L}\omega(\dd{\eta}_v) + . + \label{partition_function} +\end{equation} + +We will need (a bound on) the grand-canonical pressure +\begin{equation} +\label{eqn:spin-model_free-energy} + \psi_{J,\omega}:=\liminf_{L\to\infty}\frac{1}{L^d}\log\Xi_{J,\omega}^{L,\per}. +\end{equation} +For later applications, we note the simple inequality +\begin{equation} + \label{eqn:f-upper-bound} + e^{\psi_{J,\omega}} \ge \sup_{S} e^{-dJ\diam(S)^{2}}\omega(S) +\end{equation} +where the supremum is over all measurable $S\subset\R^n$ and $\diam(S):=\sup_{x,y\in S}\|x-y\|$. Indeed, for each such $S$ we have +\begin{equation} +\label{eqn:partition-function-lower-bound} + \Xi_{J,\omega}^{L,\per} + \ge \int_{S^{\Lambda_L}} e^{-H^L_J(\eta)}\prod_v \omega(\dd{\eta}_v) \ge e^{-J\diam(S)^2|E(\Lambda_L)|}\omega(S)^{|\Lambda_L|}. +\end{equation} + + + +\subsubsection{A continuous family of $(J,\omega)$-spin models} +\label{sec:continuous family of spin models} + +We consider a \emph{continuous family of $(J,\omega)$-spin models}, indexed by $\lambda$ in an interval $[\lambda_-,\lambda_+]$. +Precisely, we consider a continuous function $\lambda\mapsto J_\lambda\ge 0$, and a function $\lambda\mapsto \omega_\lambda$ from $[\lambda_-,\lambda_+]$ to the set of Borel measures on $\R^n$ with finite, positive total measure that is continuous in distribution, in the sense that, for any converging sequence $\lambda_k\in[\lambda_-,\lambda_+]$ with $\lambda_k\to \lambda$ and any bounded continuous $g:\mathbb R^n\to \mathbb R$, + \begin{equation} + \label{eqn:good regions_continuity} + \lim_{k\to\infty}\frac{1}{\omega_{\lambda_k}(\R^n)}\int g \dd{\omega}_{\lambda_k}=\frac{1}{\omega_{\lambda}(\R^n)}\int g \dd{\omega}_\lambda + . + \end{equation} + + +The box model defined in (\ref{eq:Hamiltonian_box}) corresponds to choosing $n=1$, letting +\begin{equation} + J_{\lambda,\gamma}=\frac12\beta J_2\gamma^{-d} +\end{equation} +and letting +\begin{equation} + \omega_{\lambda,\gamma}(\dd{\rho})=e^{\beta\gamma^{-d}(-\lambda\rho-\frac12\alpha\rho^2+f_{\gamma}(\rho))}\nu_\gamma(\dd\rho), +\end{equation} +where $\nu_\gamma$ was defined after~\eqref{eq:box model finite-volume Gibbs measure periodic}. +These choices define a continuous family of $(J,\omega)$-spin models; see Section \ref{sec:deduction of main theorem}. + + + + +\subsubsection{First-order phase transition} + +We now introduce further conditions that we will use for showing the existence of a first-order phase transition. +In these conditions, it is convenient to normalize the single-site measures using the pressures of~\eqref{eqn:spin-model_free-energy}: define, for each $\lambda\in[\lambda_-,\lambda_+]$, +\begin{equation} + \tilde \omega_\lambda:=e^{-\psi_{J_\lambda,\omega_\lambda}}\omega_\lambda + . + \label{normalized_omega} +\end{equation} + + +\begin{assumption}\label{as:good regions} + There exist closed sets $G_-,G_+\subset\R^n$ and $\theta_1,\theta_2,\theta_3\ge 0$ such that + \begin{enumerate} + \item The complement of $G_-\cup G_+$ has small measure: for all $\lambda\in[\lambda_-,\lambda_+]$, + \begin{equation} + \label{eqn:good regions_double-well} + \tilde\omega_{\lambda}((G_-\cup G_+)^{c})\le\theta_1 + . + \end{equation} + \item $G_-$ and $G_+$ are separated: for all $\lambda\in[\lambda_-,\lambda_+]$, + \begin{equation} + \label{eqn:good regions_separation} + e^{-\frac{1}{2}J_\lambda\dist(G_-, G_+)^2}(\tilde\omega_{\lambda}(G_-)\tilde\omega_{\lambda}(G_+))^{1/2} \le\theta_2 + \end{equation} + with $\mathrm{dist}(G_-,G_+):=\inf_{x_-\in G_-,x_+\in G_+}\norm{x_--x_+}$. + + \item The region $G_\#$ is dominant at $\lambda_\#$: + \begin{equation} + \label{eqn:good regions_endpoints} + \tilde\omega_{\lambda_\#}(G_\#^{c})\le\theta_3, + \quad\#\in\set{-,+}. + \end{equation} + \end{enumerate} +\end{assumption} + +\begin{theorem} +\label{thm:phase-transition} + Suppose the dimension $d\ge 2$. + For each $\epsilon\in(0,1/2)$ there exists $c(\eps, d)>0$ depending only on $\eps$ and the dimension $d$ such that the following holds. + If Assumption~\ref{as:good regions} holds with $\max\{\theta_1,\theta_2,\theta_3\}\le c(\eps, d)$, then there exists $\lambda_{\mathrm c}\in(\lambda_-,\lambda_+)$ such that the $(J_{\lambda_c},\omega_{\lambda_c})$-spin model has two distinct translation-invariant Gibbs measures $\P_{-},\P_{+}$. + Moreover, + \begin{equation}\label{eq:P plus minus properties} + \P_{\pm}(\eta_{0}\in G_{\pm})\ge 1-\epsilon + . + \end{equation} +\end{theorem} + +\begin{remark} + An explicit expression for the constant $c(\eps,d)$ can be obtained by examining the proofs in Section \ref{sec:Dobrushin-Shlosman_estimates}. +\end{remark} + +\section{The Dobrushin--Shlosman criterion} +\label{sec:dobrushin_shlosman} + +The main tool in the proof of Theorem~\ref{thm:phase-transition} is the Dobrushin--Shlosman criterion \cite{dobrushin1981phases},~\cite[Section 4]{shlosman1986method} for the existence of a first-order phase transition. +We present below a version of the criterion, adapted to our setting with the $(J,\omega)$-spin model. + +\begin{theorem}[Dobrushin--Shlosman criterion]\label{thm:DS theorem} +Let $(J_\lambda,\omega_\lambda)_{\lambda\in[\lambda_-,\lambda_+]}$ be a continuous family of $(J,\omega)$-spin models. +For each $\lambda\in[\lambda_-,\lambda_+]$, let $\P_\lambda$ be a translation-invariant Gibbs measure of the $(J_\lambda,\omega_\lambda)$-spin model, such that the family of probability measures $(\P_\lambda)_{\lambda\in[\lambda_-,\lambda_+]}$ is tight. +Let $0<\eps<\frac{1}{2}$. +Suppose that there exist disjoint closed subsets $G_-,G_+\subset \mathbb \R^n$ and $\delta_1,\delta_2>0$ such that +\begin{enumerate} + \item $\delta_1+\delta_2 \leqslant 1-\frac \epsilon 2-\sqrt{1-\epsilon}$; \label{itm:DS theorem_constants} + \item \label{itm:all lambda} for all $\lambda\in[\lambda_-,\lambda_+]$ and $v,w\in \mathbb Z^d$, +\begin{align} +&\mathbb P_\lambda(\eta_0\notin G_-\cup G_+)\le \delta_1,\label{eta_out}\\ +&\mathbb P_\lambda(\eta_v\in G_-, \eta_w\in G_+)\le \delta_2;\label{eta_+-} +\end{align} +\item $\P_{\lambda_\#}(\eta_0\in G_\#)\ge 1-\eps$ for $\#\in\{-,+\}$. +\label{itm:DS theorem_endpoints} +\end{enumerate} +Then, there exists $\lambda_c\in[\lambda_-,\lambda_+]$ such that the $(J_{\lambda_c},\omega_{\lambda_c})$-spin model admits two distinct translation-invariant Gibbs measures $\P_{\lambda_c,-}$ and $\P_{\lambda_c,+}$. +Moreover, +\begin{equation} +\P_{\lambda_c,\#}(\eta_0\in G_\#)\ge 1-\eps +,\quad +\#\in\set{-,+}. +\end{equation} +\end{theorem} + +We give a proof of the theorem for completeness, starting with the following lemma. + +\begin{lemma} +\label{claim:DS-ergodic} +Define $\Delta_L:=\set{-L,-L+1,\dots,L-1,L}^d$ for each $L\ge 2$. +Let $\lambda\in[\lambda_-,\lambda_+]$. Let $\eta$ be sampled from $\P_\lambda$. +Define + \begin{equation}\label{eq:Pi def} + \Pi_\#(\eta):=\lim_{L\rightarrow\infty}\frac{1}{\abs{\Delta_L}}\sum_{v\in\Delta_L}\indicator{G_\#}(\eta_v),\quad \#\in\{-,+\}, + \end{equation} + which exist almost surely by the Birkhoff-Khinchin ergodic theorem. + Under the assumptions of Theorem \ref{thm:DS theorem}, for all $\delta_3>0$, + \begin{equation}\label{eq:large limiting density} + \P_\lambda\left(\max\set{\Pi_-,\Pi_+}\ge 1-\delta_3\right)\ge 1-\frac{2(\delta_1+\delta_2)}{\delta_3}. + \end{equation} +\end{lemma} + +\begin{proof} +For each $L\ge 2$ and configuration $\eta$, define +\begin{align} + \Pi_{\#,L}(\eta){}&:=\frac{1}{\abs{\Delta_L}}\sum_{v\in\Delta_L}\indicator{G_\#}(\eta_v),\quad\#\in\set{-,+},\\ + \Psi_L(\eta){}&:=\frac{1}{\abs{\Delta_L}^2}\sum_{v,w\in\Delta_L}\indicator{(G_-^2\cup G_+^2)^c}(\eta_v,\eta_w). +\end{align} +Let $L\ge 2$. +By \eqref{eta_out} and \eqref{eta_+-}, +\begin{equation} +\begin{multlined} + \E_\lambda[\Psi_L]\le\frac{1}{\abs{\Delta_L}^2}\sum_{v,w\in\Delta_L}\left[ + \P_\lambda(\eta_v\not\in G_-\cup G_+) + +\P_\lambda(\eta_w\not\in G_-\cup G_+)\right. + \\ + \left. + +\P_\lambda(\eta_v\in G_-,\eta_w\in G_+) + +\P_\lambda(\eta_w\in G_-,\eta_v\in G_+)\right] + \le 2 (\delta_1+\delta_2). +\end{multlined} +\end{equation} +Hence, by Markov's inequality, +\begin{equation} +\label{eqn:Dobrushin-Shlosman_Markov-inequality} + \P_\lambda(\Psi_L<\delta_3)\ge 1-\frac{2(\delta_1+\delta_2)}{\delta_3}. +\end{equation} +Now, since $G_-$ and $G_+$ are disjoint, $\Pi_{-,L}+\Pi_{+,L}\le 1$, so +\begin{equation} +\label{eqn:Dobrushin-Shlosman_manipulation} + \max\set{\Pi_{-,L},\Pi_{+,L}} + \ge + \max\set{\Pi_{-,L},\Pi_{+,L}}(\Pi_{-,L}+\Pi_{+,L}) + \ge + \Pi_{-,L}^2+\Pi_{+,L}^2=1-\Psi_L. +\end{equation} +Combining \eqref{eqn:Dobrushin-Shlosman_Markov-inequality} and \eqref{eqn:Dobrushin-Shlosman_manipulation}, we get that +\begin{equation} + \P_\lambda(\max\{\Pi_{-,L},\Pi_{+,L}\}\ge 1-\delta_3)\ge 1-\frac{2(\delta_1+\delta_2)}{\delta_3}. +\end{equation} +We conclude the proof using that $\Pi_{\#,L}\rightarrow\Pi_{\#}$ almost surely as $L\rightarrow\infty$. +\end{proof} + +We are now ready to prove Theorem~\ref{thm:DS theorem}. + +\begin{proof}[Proof of Theorem~\ref{thm:DS theorem}] +Define +\begin{equation} + T_\#:=\left\{\lambda\in[\lambda_-,\lambda_+] \mid \P_\lambda(\eta_0\in G_\#)\ge 1-\eps\right\},\quad \#\in\{-,+\}. +\end{equation} +We consider two cases. + +First, suppose that $T_-\cup T_+\neq[\lambda_-,\lambda_+]$. +In other words, there exists $\lambda_c\in[\lambda_-,\lambda_+]$ such that +\begin{equation}\label{eq:P_lambda choice} + \P_{\lambda_c}(\eta_0\in G_\#)< 1-\eps,\quad\#\in\{-,+\}. +\end{equation} +By considering the ergodic decomposition of $\P_{\lambda_c}$ (see, e.g., \cite[Theorem 14.17]{georgii2011gibbs}), we have that (recalling~\eqref{eq:Pi def}) +\begin{equation}\label{eq:probaility as expectation} + \P_{\lambda_c}(\eta_0\in G_\#) = \E_{\lambda_c}[\Pi_\#],\quad \#\in\{-,+\}. +\end{equation} +Using Item \ref{itm:DS theorem_constants} of the assumptions of the theorem, it is straightforward to check that there exists $\delta_3>0$ such that $(1-\delta_3)(1-2(\delta_1+\delta_2)/\delta_3)=1-\epsilon$ and $\delta_3\le\epsilon$. + +\begin{claim}\label{claim:phase-transition_nonzero-prob} +\begin{equation} +\label{eqn:phase-transition_nonzero-prob} + \P_{\lambda_c}(\Pi_\#>1-\delta_3)>0,\quad\#\in\set{-,+}. +\end{equation} +\end{claim} + +\begin{proof} +We treat only the $\#=-$ case, as the other case is similar. +Suppose by contradiction that $\P_{\lambda_c}(\Pi_->1-\delta_3)=0$. +By Markov's inequality, Lemma~\ref{claim:DS-ergodic}, and the choice of $\delta_3$, we have that +\begin{equation} +\label{eqn:phase-transition_observable-lower-bound} + \E_{\lambda_c}[\Pi_+] + \ge\P_{\lambda_c}(\Pi_+\ge 1-\delta_3)(1-\delta_3) + \ge\left(1-\frac{2(\delta_1+\delta_2)}{\delta_3}\right)(1-\delta_3)=1-\epsilon. +\end{equation} +On the other hand, from~\eqref{eq:P_lambda choice} and~\eqref{eq:probaility as expectation}, it follows that $\E_{\lambda_c}[\Pi_+]<1-\epsilon$, which contradicts \eqref{eqn:phase-transition_observable-lower-bound}. +\end{proof} + +We now deduce that $\P_{\lambda_c}$ is not ergodic. +Indeed, suppose by contradiction that $\P_{\lambda_c}$ is ergodic. +As $\set{\Pi_\#>1-\delta_3}$ is a translation-invariant event, Claim \ref{claim:phase-transition_nonzero-prob} implies that $\mathbb P_{\lambda_c}(\Pi_\#>1-\delta_3)=1$, $\#\in\set{-,+}$. +Thus, there exists a configuration $\eta$ such that $\Pi_\#(\eta)>1-\delta_3$, $\#\in\set{-,+}$, so $1\ge \Pi_-(\eta)+\Pi_+(\eta)>2(1-\delta_3)\ge 2(1-\epsilon)>1$, a contradiction. +Therefore, $\mathbb P_{\lambda_c}$ is not ergodic, so its ergodic decomposition~\cite[Theorem 14.17]{georgii2011gibbs} contains two translation-invariant Gibbs measures $\P_{\lambda_c,-},\P_{\lambda_c,+}$ of the $(J_{\lambda_c},\omega_{\lambda_c})$-spin model +such that +\begin{equation} + \P_{\lambda_c,\#}(\eta_0\in G_\#) + =\E_{\lambda_c,\#}[\Pi_\#] + \ge 1-\delta_3\ge 1-\epsilon,\quad\#\in\{-,+\}, +\end{equation} +establishing Theorem~\ref{thm:DS theorem} in this case. + +Second, suppose that $T_-\cup T_+=[\lambda_-,\lambda_+]$. By Item \ref{itm:DS theorem_endpoints} of the assumptions, $\lambda_-\in T_-$ and $\lambda_+\in T_+$, so $T_-$ and $T_+$ are both non-empty. +As $[\lambda_-,\lambda_+]$ is connected, it follows that there exists $\lambda_c\in\overline{T}_-\cap\overline{T}_+$ (where $\overline{A}$ denotes the closure of a set $A$). +Let $(\lambda_{n,-})_{n\ge 1}\subset T_-$ satisfy $\lambda_{n,-}\to\lambda_c$. +As the family $(\P_\lambda)_{\lambda\in[\lambda_-,\lambda_+]}$ is tight, it follows that $\P_{\lambda_{n_k,-}}\to \P_{\lambda_c,-}$ in distribution for some subsequence $(n_k)_{k\ge 1}$. The limit measure is a Gibbs measure of the $(J_{\lambda_c},\omega_{\lambda_c})$-spin model (see Proposition \ref{prop:convergence-of-Gibbs-measures}) and is clearly translation-invariant. +Moreover, since $G_-$ is closed, +\begin{equation} + \P_{\lambda_c,-}(\eta_0\in G_-)\ge \limsup_{k\rightarrow\infty} \P_{\lambda_{n_k,-}}(\eta_0\in G_-)\ge 1-\eps. +\end{equation} +By a symmetric argument, we obtain a translation-invariant Gibbs measure $\P_{\lambda_c,+}$ of the $(J_{\lambda_c},\omega_{\lambda_c})$-spin model satisfying $\P_{\lambda_c,+}(\eta_0\in G_+)\ge 1-\eps$. +This completes the proof of the theorem. +\end{proof} + + + +\section{Phase co-existence in the $(J,\omega)$-spin model} + +In this section, we prove Theorem \ref{thm:phase-transition}. +Our proof is based on a criterion for the existence of first-order phase transitions due to Dobrushin and Shlosman \cite{shlosman1986method}; see Theorem \ref{thm:DS theorem}. +We re-prove this Theorem in Section \ref{sec:dobrushin_shlosman} in greater detail than the original reference. +In particular, our proof of this result applies to models with unbounded spin values. + + +\subsection{Reflection positivity and chessboard estimate} +\label{sec:chessboard_estimate} + +We rely on the {\it chessboard estimate}, which follows from the reflection positivity of the $(J,\omega)$-spin model. +It is convenient to use \emph{reflections through hyperplanes intersecting edges}. + +\subsubsection{Notation} +We continue to work on the discrete torus $\Lambda_L=\mathbb Z^d/(L \mathbb Z)^d$, restricting to even values of $L$. +For $1\le i\le d$ and $p\in \Z+\frac{1}{2}$, define the reflection of $\Lambda_L$ through the hyperplane orthogonal to direction $i$ at coordinate $p$, +\begin{equation}\label{eq:reflection transformation} + (\tau_{i,p}(v))_j:=\begin{cases} 2p-v_j\ (\mathrm{mod}\ L)&\text{if } j=i\\ + v_j&\text{otherwise}\end{cases}. +\end{equation} + +Now $\tau_{i,p}$ naturally acts on spin configurations $\eta$, and on functions of spin configurations. With a slight abuse of notation, we denote these actions by $\tau_{i,p}$ as well: +\begin{equation} + (\tau_{i,p}\eta)_v:=\eta_{\tau_{i,p}(v)} + ,\quad + \tau_{i,p} f(\eta):=f(\tau_{i,p}(\eta)) + . +\end{equation} + +Given $x\in\Z^{d}$ and $\vec{\ell}\in\Z_{\ge 0}^{d}$ such that $2(\vec\ell_i+1)$ divides $L$ for all $i$, define $R=R_{\vec{\ell},x}:=\prod_{i=1}^{d}[x_i,x_i+\vec{\ell}_i]\cap\Z^{d}$ as the box with corner $x$ and side lengths $\vec{\ell}$. +Let $T^R_{L}$ be the group of isomorphisms of $\Lambda_L$ that is generated by the reflections +\begin{equation} + \bigcup_{i=1}^d \set{\tau_{i,p}\mid p\in x_i-\frac{1}{2}+(\vec{\ell}_i+1)\Z} \label{eq:P^R vert} + . +\end{equation} +Note that +\begin{equation}\label{eq:size of T^R_L} + \abs{T^R_L}=\prod_{i=1}^d\frac{L}{\vec{\ell}_i+1}. +\end{equation} +We also let $T^R$ be the group of isomorphisms of $\Z^d$ generated by the reflections in~\eqref{eq:P^R vert} (so that $T^R$ is infinite). + +Recall that configurations of the $(J,\omega)$-spin model are functions $\eta:\Z^d\to\R^n$. An \emph{observable}, i.e., a measurable function $f:(\R^n)^{\Z^d}\to\R$, is called \emph{$R$-local} if $f(\eta)$ depends only on the restriction of $\eta$ to $R$. + +\subsubsection{The chessboard estimate} +We are now ready to state the chessboard estimate. +Its proof is standard (see e.g. \cite[Theorem 5.8]{biskup2009reflection}, \cite[Theorem 10.11 and Remark 10.15]{friedli2017statistical}, or \cite[Section 2.7.1]{peled2019lectures}), and follows from the reflection positivity of the $(J,\omega)$-spin model, which is also standard (see e.g. \cite[Section 10.3.2]{friedli2017statistical}). +We will not reproduce either proof here. + +\begin{lemma}[Chessboard estimate] +\label{lem:chessboard} +Let $J>0$ and $\omega$ be a Borel measure on~$\R^n$ with finite, positive total measure. +Let $L\in\Z_{\ge 0}$ and $\vec{\ell}\in\Z_{\ge 0}^{d}$ satisfy that $2(\vec\ell_i+1)$ divides $L$ for all $i$. Let $x\in\Z^d$ and $R:=R_{\vec\ell,x}$. Let $A\subset T^{R}_L$ and $(f_{\tau})_{\tau\in A}$ be bounded $R$-local observables. +Then +\begin{equation} +\mathbb P_{J,\omega}^{L,\per}\left(\prod_{\tau\in A}\tau f_{\tau}\right)\le\prod_{\tau\in A}\|f_{\tau}\|_{J,\omega}^{R|L} +\end{equation} +where $\|\cdot\|_{J,\omega}^{R|L}$ is the chessboard seminorm of $f$, defined by +\begin{equation} +\label{eq:vertex chessboard norm def} +\norm{f}_{J,\omega}^{R\mid L} :=\Bigg[\mathbb P_{J,\omega}^{L,\per}\Bigg(\prod_{\tau\in T^{R}_{L}}(\tau f)\Bigg)\Bigg]^{1/\abs{T^{R}_{L}}}, +\end{equation} +noting that $\abs{T^{R}_{L}}$ is given by~\eqref{eq:size of T^R_L}. +\end{lemma} + +In our use of the chessboard estimate, it is convenient to pass to infinite volume. We first define the notion of a torus-limit Gibbs measure, taking care to take the limit on tori with highly divisible side lengths, to satisfy the assumptions of the chessboard estimate for arbitrary $\vec{\ell}\in\Z_{\ge 0}^d$. + +\begin{definition} +\label{def:torus-limit} +We call a Gibbs measure $\P_{J,\omega}$ a \emph{torus-limit Gibbs measure of the $(J,\omega)$-spin model} if it is obtained as a limit in distribution along a subsequence of the (finite-volume) torus Gibbs measures $(\P^{k!,\per}_{J,\omega})_k$. +\end{definition} + +\begin{corollary}[The chessboard estimate in the limit] +\label{cor:infinite-volume-chessboard-estimate} +Let $J\ge 0$ and $\omega$ be a Borel measure on~$\R^n$ with finite, positive total measure. Let $x\in\Z^d$, $\vec{\ell}\in\Z_{\ge 0}^{d}$ and $R:= R_{\vec\ell,x}$. Let $A\subset T^{R}$ be finite and $(f_{\tau})_{\tau\in A}$ bounded, $R$-local, lower semi-continuous observables. +Let $\P_{J,\omega}$ be a torus-limit Gibbs measure of the $(J,\omega)$-spin model. +Then, +\begin{equation}\label{eq:chessboard estimate infinite volume} +\P_{J,\omega}\left(\prod_{\tau\in A}\tau f_{\tau}\right) +\le \prod_{\tau\in A}\norm{f_{\tau}}_{J,\omega}^{R} +\end{equation} +where we define +\begin{equation} + \norm{f_{\tau}}_{J,\omega}^{R}:=\limsup_{k\rightarrow\infty}\norm{f_{\tau}}_{J,\omega}^{R\mid k!}. +\end{equation} +\end{corollary} + +\begin{proof} +By an immediate application of the Portmanteau theorem and the finite-volume chessboard estimate (Lemma \ref{lem:chessboard}). +\end{proof} + +Though not needed for our results, we remark that it is desirable to extend~\eqref{eq:chessboard estimate infinite volume} to general periodic Gibbs measures, rather than just torus-limit Gibbs measures. Such an extension was demonstrated in~\cite{hadas2025columnar} for a different model. However, while the proof of~\cite{hadas2025columnar} applies in some generality, it does not cover the $(J,\omega)$-spin model for $\omega$ with non-compact support due to the fact that its interaction energy $\|\eta_v-\eta_w\|^2$ is unbounded. + + + +\subsection{Proof of Theorem \ref{thm:phase-transition}} + +We now prove Theorem \ref{thm:phase-transition} by verifying the assumptions of the Dobrushin--Shlosman criterion (Theorem \ref{thm:DS theorem}), where we take $\P_{\lambda}$ to be a torus-limit Gibbs measure (see Remark \ref{rem:existence-torus-limit}) of the $(J_\lambda,\omega_\lambda)$-spin model, $\lambda\in[\lambda_-,\lambda_+]$. +This requires verifying the tightness of this family of measures as well as three probabilistic estimates. +In Section \ref{sec:torus-limit-measures-tight}, we resolve the first issue by taking advantage of the continuity of the mappings $\lambda\mapsto J_\lambda$ and $\lambda\mapsto\omega_\lambda$. +In Section \ref{sec:Dobrushin-Shlosman_estimates}, we use Assumption \ref{as:good regions} to tackle the second issue. + + + +\subsubsection{Tightness of torus-limit Gibbs measures} +\label{sec:torus-limit-measures-tight} +It is convenient to use the following notation in this section: Given a Borel measure $\omega$ on $\R^n$ with finite, positive total measure, we denote its normalized probability measure by +\begin{equation} +\label{eqn:normalized probability measure} + \bar{\omega}:=\frac{\omega}{\omega(\R^n)}. +\end{equation} + +We formulate the tightness property of the torus-limit Gibbs measures in slightly greater generality than needed. + +\begin{proposition}[Tightness of torus-limit Gibbs measures] +\label{prop:torus-limit-measures-tight} +Let $I$ be an arbitrary index set. +Suppose that $\set{J_{i}\mid i\in I}$ is a bounded subset of $\R_{\ge 0}$ and $\set{\omega_{i}\mid i\in I}$ a family of Borel measures on $\R^n$ with finite, positive total measure, such that the family $\set{\bar{\omega}_{i}\mid i\in I}$ is tight. +For each $i\in I$, let $\P_i$ be a torus-limit Gibbs measure of the $(J_i,\omega_i)$-spin model. +Then, the family of measures $\set{\P_i\mid i\in I}$ is tight. +\end{proposition} + +Proposition \ref{prop:torus-limit-measures-tight} is sufficient for our purpose. +Indeed, recall that we are considering a continuous family $(J_\lambda,\omega_\lambda)_{\lambda\in[\lambda_-,\lambda_+]}$ of $(J,\omega)$-spin models. +The continuity of the mappings $\lambda\mapsto J_\lambda$ and $\lambda\mapsto\omega_\lambda$ imply that the set $\set{J_\lambda\mid\lambda\in[\lambda_-,\lambda_+]}$ is bounded and that the family of probability measures $\set{\bar{\omega}_\lambda\mid\lambda\in[\lambda_-,\lambda_+]}$, as the continuous image of a compact set, is compact, thus tight. + +\smallskip +We begin the proof of Proposition~\ref{prop:torus-limit-measures-tight} by establishing the tightness of the family of finite-volume torus Gibbs measures of the $(J_i,\omega_i)$-spin models, $i\in I$. + +\begin{lemma} +\label{lem:torus-measures-tight-family} +Let $I$ be an arbitrary index set. +Suppose that $\set{J_{i}\mid i\in I}$ is a bounded subset of $\R_{\ge 0}$ and $\set{\omega_{i}\mid i\in I}$ a family of Borel measures on $\R^n$ with finite, positive total measure, such that the family $\set{\bar{\omega}_{i}\mid i\in I}$ is tight. +Then, the set of (finite-volume) torus Gibbs measures +\begin{equation} +\set{\P_{J_i,\omega_i}^{L,\per}\mid L\ge 2 \text{ even, }i\in I} +\end{equation} +is tight. +\end{lemma} + +\begin{proof} +Let $\epsilon>0$. +We will construct a family $(K_{v,\epsilon})_{v\in\Z^d}$ of compact subsets of $\R^n$ such that +\begin{equation} +\P^{L,\per}_{J_{i},\omega_{i}}(\eta_v \in K_{v,\epsilon} \text{ for all }v\in\Z^d) \ge 1 -\epsilon,\quad\text{for all } L\ge2\text{ even and }i\in I. +\end{equation} +This completes the proof of the lemma, since the product $\prod_{v\in\Z^{d}}K_{v,\epsilon}$ of these compact sets is a compact subset of $(\R^n)^{\Z^d}$ by Tychonoff's theorem. + +We proceed with the construction. +By a union bound, it suffices to choose the sets $(K_{v,\epsilon})_{v\in\Z^d}$ such that +\begin{equation} +\label{eqn:tightness-union-bound} +\begin{multlined} +\sum_{v\in\Z^{d}} \P^{L,\per}_{J_{i},\omega_{i}}(\eta_v \notin K_{v,\epsilon}) +\le\epsilon, +\quad\text{for all } L\ge2\text{ even and }i\in I. +\end{multlined} +\end{equation} +Fix $i\in I$ and an even $L\ge 2$. +By the chessboard estimate (Lemma \ref{lem:chessboard}), for any $v\in\Z^{d}$ and compact subset $K\subset\R^{n}$, we have that +\begin{equation} +\P^{L,\per}_{J_{i},\omega_{i}}(\eta_v \notin K) \le \P^{L,\per}_{J_{i},\omega_{i}}(\eta_w \notin K\text{ for all }w\in\Lambda_{L})^{1/L^d}, +\end{equation} +where +\begin{equation} +\P^{L,\per}_{J_{i},\omega_{i}}(\eta_w \notin K\text{ for all }w\in\Lambda_{L}) +=\frac{1}{\Xi_{J_{i},\omega_{i}}^{L,\per}} +\int_{(K^{c})^{\Lambda_{L}}} e^{-H^{L}_{J_{i}}(\eta)} \prod_{w\in\Lambda_{L}} \omega_{i}(\dd{\eta}_w). +\end{equation} +Bounding $H^{L}_{J_{i}}(\eta)\ge0$ and using the lower bound \eqref{eqn:partition-function-lower-bound} on $\Xi_{J_{i},\omega_{i}}^{L,\per}$, we conclude that +\begin{equation} +\label{eqn:torus-measures-tight-family_probability-bound} +\P^{L,\per}_{J_{i},\omega_{i}}(\eta_v \notin K) +\le \frac{\omega_{i}(K^{c})}{e^{-dJ_{i}\diam(S)^{2}}\omega_{i}(S)} +=\frac{\bar{\omega}_{i}(K^{c})}{e^{-dJ_{i}\diam(S)^{2}}\bar{\omega}_{i}(S)} +\end{equation} +for all measurable $S\subset\R^{n}$. +By the tightness of the family of probability measures $\set{\bar{\omega}_i\mid i\in I}$, we may choose $S$ to be a compact subset of $\R^n$ such that $\inf_{i\in I}\bar{\omega}_i(S)\ge 1/2$. +Next, fix a function $\delta:\Z^{d}\rightarrow\R_{>0}$ such that $\sum_{v\in\Z^{d}}\delta(v)\le 1$. +Using tightness again and recalling that the set $\set{J_i\mid i\in I}$ is bounded, we choose, for each $v\in\Z^d$, a compact set $K_{v,\epsilon}\subset\R^{n}$ such that +\begin{equation} +\label{eqn:torus-measures-tight-family_choice} +\bar{\omega}_{i}(K_{v,\epsilon}^c) +\le \epsilon\delta(v) +e^{-d\diam(S)^{2}\sup_{i'\in I}J_{i'}} +\inf_{i'\in I}\bar{\omega}_{i'}(S),\quad\text{for all }i\in I. +\end{equation} +It remains to check that the family $(K_{v,\epsilon})_{v\in\Z^d}$ thus chosen satisfies \eqref{eqn:tightness-union-bound}, but this is an immediate consequence of \eqref{eqn:torus-measures-tight-family_probability-bound}, \eqref{eqn:torus-measures-tight-family_choice}, and the fact that $\sum_{v\in\Z^d}\delta(v)\le 1$. +\end{proof} + +\begin{remark} +\label{rem:existence-torus-limit} + It follows immediately from Lemma \ref{lem:torus-measures-tight-family} that every $(J,\omega)$-spin model admits at least one torus-limit Gibbs measure. + In turn, this justifies our choice of $\P_\lambda$ as a torus-limit Gibbs measure of the $(J_\lambda,\omega_\lambda)$-spin model for each $\lambda\in[\lambda_-,\lambda_+]$, as made at the beginning of this subsection. +\end{remark} + +We now deduce Proposition \ref{prop:torus-limit-measures-tight}. + +\begin{proof}[Proof of Proposition \ref{prop:torus-limit-measures-tight}] +By Lemma \ref{lem:torus-measures-tight-family}, the set of torus Gibbs measures +\begin{equation} +\mathcal{G}=\set{\P^{L,\per}_{J_{i},\omega_{i}}\mid L\ge 2\text{ even},i\in I} +\end{equation} +is tight, namely (using Prokhorov's theorem \cite[Theorem 5.1]{billingsley2013convergence}), its closure $\overline{\mathcal{G}}$ is compact. +Further, for each $i\in I$, $\P_{i}$ is the limit in distribution along a subsequence of the torus Gibbs measures $(\P^{k!,\per}_{J_{i},\omega_{i}})_{k\ge 1}$ by definition. +Thus, the set of torus-limit Gibbs measures $\set{\P_{i}\mid i\in I}$ is a subset of $\overline{\mathcal{G}}$. +As $\overline{\mathcal{G}}$ is compact, $\set{\P_{i}\mid i\in I}$ has compact closure, hence a tight family of measures (again using Prokhorov's theorem \cite[Theorem 5.2]{billingsley2013convergence}). +\end{proof} + + + +\subsubsection{Verification of key estimates} +\label{sec:Dobrushin-Shlosman_estimates} + +We now verify the three estimates in the Dobrushin--Shlosman criterion (Theorem \ref{thm:DS theorem}) with the help of Assumption \ref{as:good regions}. + +We start with \eqref{eta_out} of Item \ref{itm:all lambda}. + +\begin{proposition} +\label{prop:Dobrushin-Shlosman_double-well} +Under Assumption \ref{as:good regions}, for all $\lambda\in[\lambda_{-},\lambda_{+}]$, if $\P_\lambda$ is a torus-limit Gibbs measure of the $(J_\lambda,\omega_\lambda)$-spin model, then +\begin{equation} +\P_{\lambda}(\eta_{0}\not\in G_{-}\cup G_{+})\le\theta_1. +\end{equation} +\end{proposition} + +\begin{proof} +Let $R:=\set{0}^{d}$ denote the origin of $\Z^{d}$. +Consider the $R$-local event $E:=\set{\eta\in\Omega\mid \eta_{0}\in (G_{-}\cup G_{+})^{c}}$. +By Corollary \ref{cor:infinite-volume-chessboard-estimate}, +\begin{equation} +\label{eqn:Dobrushin-Shlosman_double-well_chessboard-estimate} +\P_{\lambda}(E) +\le\limsup_{k\rightarrow\infty}\norm{E}^{R\mid k!}_{J_\lambda,\omega_\lambda}, +\end{equation} +where the chessboard seminorm +\begin{equation} +\label{eqn:Dobrushin-Shlosman_double-well_chessboard-seminorm} +\norm{E}^{R\mid k!}_{J_\lambda,\omega_\lambda} +=\P^{k!,\per}_{J_{\lambda},\omega_{\lambda}}\Bigg(\bigcap_{\tau\in T^{R}_{k!}}\tau(E)\Bigg)^{1/(k!)^{d}} +\end{equation} +is well-defined for all $k\ge 2$. +To estimate the probability on the RHS of \eqref{eqn:Dobrushin-Shlosman_double-well_chessboard-seminorm}, we use the following simple bound on the partition function of the $(J_{\lambda},\omega_{\lambda})$-spin model in $\Lambda_{k!}$. +Let $\nu>0$. +By \eqref{eqn:spin-model_free-energy}, there exists $k_\nu\in\N$ such that, for all $k\ge k_\nu$, +\begin{equation} +\label{eqn:Dobrushin-Shlosman_double-well_partition-function-lower-bound} + \frac{1}{(k!)^{d}}\log\Xi_{J_\lambda,\omega_\lambda}^{k!,\per} + \ge\psi_{J_\lambda,\omega_\lambda}-\nu. +\end{equation} +On the other hand, the event $\bigcap_{\tau\in T^{R}_{k!}}\tau(E)$ consists of configurations with all spins in $(G_{-}\cup G_{+})^{c}$. +Thus, +\begin{equation} +\label{eqn:Dobrushin-Shlosman_double-well_final} +\P^{k!,\per}_{J_{\lambda},\omega_{\lambda}}\Bigg(\bigcap_{\tau\in T^{R}_{k!}}\tau(E)\Bigg) +\le\frac{\omega_{\lambda}((G_{-}\cup G_{+})^{c})^{ +(k!)^{d}}}{e^{(\psi_{J_\lambda,\omega_\lambda}-\nu)(k!)^{d}}} +\le (e^\nu \theta_1)^{(k!)^d}, +\end{equation} +where we used \eqref{eqn:good regions_double-well} of Assumption \ref{as:good regions} in the last inequality. +Inserting \eqref{eqn:Dobrushin-Shlosman_double-well_final} into \eqref{eqn:Dobrushin-Shlosman_double-well_chessboard-seminorm}, recalling \eqref{eqn:Dobrushin-Shlosman_double-well_chessboard-estimate}, and taking $\nu\rightarrow0$ complete the proof. +\end{proof} + +Next, we verify Item \ref{itm:DS theorem_endpoints}. + +\begin{proposition} +\label{prop:endpoint} +Under Assumption \ref{as:good regions}, for $\#\in\set{-,+}$, if $\P_{\lambda_\#}$ is a torus-limit Gibbs measure of the $(J_{\lambda_\#},\omega_{\lambda_\#})$-spin model, then +\begin{equation} +\label{eqn:endpoint} +\P_{\lambda_{\#}}(\eta_{0}\in G_{\#})\ge1-\theta_3. +\end{equation} +\end{proposition} + +\begin{proof} +The proof is analogous to that of Proposition \ref{prop:Dobrushin-Shlosman_double-well}: replacing $(G_{-}\cup G_{+})^{c}$ by $G_{\#}^{c}$ and using \eqref{eqn:good regions_endpoints} of Assumption \ref{as:good regions}, we get that +\begin{equation} +\P_{\lambda_{\#}}(\eta_{0}\in G_{\#}^{c})\le\theta_3, +\end{equation} +which immediately implies \eqref{eqn:endpoint}. +\end{proof} + +Finally, we verify \eqref{eta_+-} of Item \ref{itm:all lambda}. + +\begin{proposition} +\label{prop:Dobrushin-Shlosman_interface} +For all $\delta_2>0$, there exist constants $\theta_1(\delta_2),\theta_2(\delta_2)>0$ such that, if Assumption \ref{as:good regions} holds with $0<\theta_i\le\theta_i(\delta_2)$, $i=1,2$, then, for all $\lambda\in[\lambda_{-},\lambda_{+}]$, torus-limit Gibbs measure $\P_\lambda$ of the $(J_\lambda,\omega_\lambda)$-spin model, and $v,w\in\Z^{d}$, +\begin{equation} +\P_{\lambda}(\eta_{v}\in G_{-},\eta_{w}\in G_{+})\le\delta_2. +\end{equation} +\end{proposition} + +\begin{remark} + It is possible to extract quantitative estimates for the constants $\theta_i(\delta_2)$, $i=1,2$, as promised by the proposition from a fully explicit chessboard-Peierls argument, in terms of the smallness of certain series. + We do not attempt this here. +\end{remark} + +\begin{proof} +The proof is by a standard chessboard-Peierls argument, which we do not belabor here and refer the reader to the literature \cite{frohlich1978phase} for detailed implementations. +Intuitively, the conditions $\eta_v\in G_-$, $\eta_w\in G_+$ imply the existence of geometric interfaces in $\Z^d$, consisting of edges connecting vertices which are costly neighbors due to their spin values. +Of particular relevance to the chessboard-Peierls argument are the set of boundary edges of the connected component of $\set{u\in\Z^d\mid\eta_u\in G_-}$ containing $v$, where each edge connects a spin in $G_-$ and another in $G_-^c=G_+\cup(G_-\cup G_+)^c$, and an analogous set of edges for $w$. +The probability of observing these costly edge events simultaneously is then controlled using the chessboard estimate. + +In our case, we consider the following edge events. +Let $R:=\set{v_1,v_2}$ be an edge of $\Z^d$. +Define +\begin{align} +\label{eqn:Dobrushin-Shlosman_interface_edge-event} +E_{v_1,v_2}^{-,+}&:=\set{\eta\in\Omega\mid\eta_{v_1}\in G_{-},\eta_{v_2}\in G_{+}},\\ +E_{v_1,v_2}^{-,0}&:=\set{\eta\in\Omega\mid\eta_{v_1}\in G_{-},\eta_{v_2}\in (G_{-}\cup G_{+})^{c}}, +\end{align} +and similarly $E_{v_1,v_2}^{+,-}$ and $E_{v_1,v_2}^{+,0}$. +In view of the above outline of the chessboard-Peierls argument, to prove the proposition, it suffices to show that the chessboard seminorm of each of the four edge events can be made arbitrarily small by taking $\delta$ correspondingly small. +We establish this for $E_{v_1,v_2}^{-,+}$ and $E_{v_1,v_2}^{-,0}$ in individual claims below, and note that symmetric arguments yield identical bounds for $E_{v_1,v_2}^{+,-}$ and $E_{v_1,v_2}^{+,0}$. + +We start with $E_{v_1,v_2}^{-,0}$ which is simpler to deal with. + +\begin{claim} +Under Assumption \ref{as:good regions}, for all $\lambda\in[\lambda_-,\lambda_+]$, +\begin{equation} +\label{eqn:Dobrushin-Shlosman_interface_edge-event_-,0 probability} +\norm{E^{-,0}_{v_1,v_2}}^R_{J_\lambda,\omega_\lambda}\le\theta_1. +\end{equation} +\end{claim} + +\begin{proof} +Observe that the edge event $E^{-,0}_{v_1,v_2}$ is contained in the single-vertex event that $\eta_{v_2}\in(G_-\cup G_+)^c$. +Recall from the proof of Proposition \ref{prop:Dobrushin-Shlosman_double-well} that the chessboard seminorm of the latter is bounded by $\delta$ (the proof there is written for when $v_2$ is the origin, but this is inconsequential). +The monotonicity of the chessboard seminorm then implies \eqref{eqn:Dobrushin-Shlosman_interface_edge-event_-,0 probability}. +\end{proof} + +The treatment of $E_{v_1,v_2}^{-,+}$ is complicated by the fact that it is not an open event, i.e., an open subset of $\Omega$, so its indicator function does not fulfill the lower semi-continuity requirement of the chessboard estimate in the limit (Corollary \ref{cor:infinite-volume-chessboard-estimate}). +To overcome this technical nuisance, we construct below a sequence of open events containing $E_{v_1,v_2}^{-,+}$ satisfying suitable properties. + +\begin{claim} +\label{clm:Dobrushin-Shlosman_interface_edge-event_-,+} +There exists a decreasing sequence of open events $(E_{v_1,v_2;j}^{-,+})_{j\ge 1}$ such that +\begin{enumerate} + \item $\bigcap_{j=1}^\infty E_{v_1,v_2;j}^{-,+}=E_{v_1,v_2}^{-,+}$; \label{itm:Dobrushin-Shlosman_interface_edge-event_-,+_intersection} + \item under Assumption \ref{as:good regions}, for all $\lambda\in[\lambda_-,\lambda_+]$, + \begin{equation} + \limsup_{j\rightarrow\infty}\norm{E_{v_1,v_2;j}^{-,+}}^R_{J_\lambda,\omega_\lambda}\le\theta_2^2. + \end{equation} + \label{itm:Dobrushin-Shlosman_interface_edge-event_-,+_seminorm} +\end{enumerate} +\end{claim} + +\begin{proof} +For each $j\ge 1$, define the open $2^{-j}$-extensions of a set $A\subset\R^n$ by +\begin{equation} + O_j(A):=A+(-2^{-j},2^{-j})^n, +\end{equation} +where $+$ denotes the Minkowski sum. +Observe that if $A$ is closed, then +\begin{equation}\label{eq:intersection of extensions} + A=\bigcap_{j=1}^\infty O_{j}(A). +\end{equation} +Using the above notation, define, for each $j\ge 1$, +\begin{equation} +\label{eqn:-+bound1} + E_{v_1,v_2;j}^{-,+}:=\set{\eta\in\Omega\mid\eta_{v_1}\in O_j(G_{-}),\eta_{v_2}\in O_{j}(G_{+})}, +\end{equation} +which forms a decreasing sequence of open events. +Property \ref{itm:Dobrushin-Shlosman_interface_edge-event_-,+_intersection} immediately follows from \eqref{eq:intersection of extensions}. +To prove Property \ref{itm:Dobrushin-Shlosman_interface_edge-event_-,+_seminorm}, fix $j\ge 1$ and consider the chessboard seminorm +\begin{equation}\label{eq:chessboard norm bound on kappa events} + \norm{E_{v_1,v_2;j}^{-,+}}^R_{J_\lambda,\omega_\lambda} + =\limsup_{k\rightarrow\infty}\norm{E_{v_1,v_2;j}^{-,+}}^{R\mid k!}_{J_\lambda,\omega_\lambda} + =\limsup_{k\rightarrow\infty}\P^{k!,\per}_{J_{\lambda},\omega_{\lambda}}\Bigg(\bigcap_{\tau\in T^{R}_{k!}}\tau(E^{-,+}_{v_1,v_2;j})\Bigg)^{2/(k!)^{d}} +\end{equation} +Note that the probability on the RHS of \eqref{eq:chessboard norm bound on kappa events} is well-defined for all $k\ge 4$. +To bound this probability, let $\nu>0$ be arbitrary and recall the lower bound \eqref{eqn:Dobrushin-Shlosman_double-well_partition-function-lower-bound} on the partition function of the $(J_\lambda,\omega_\lambda)$-spin model in $\Lambda_{k!}$, which holds for all $k\ge k_\nu$ for some $k_\nu\ge 4$. +Taking into account that the constraint $\bigcap_{\tau\in T^{R}_{k!}}\tau(E^{-,+}_{v_1,v_2;j})$ forces exactly $1/2d$ of the edges in $\Lambda_{k!}$ to connect a spin in $O_j(G_{-})$ to another in $O_j(G_{+})$, we get that +\begin{equation} + \P^{k!,\per}_{J_{\lambda},\omega_{\lambda}}\Bigg(\bigcap_{\tau\in T^{R}_{k!}}\tau(E^{-,+}_{v_1,v_2;j})\Bigg)^{2/(k!)^{d}} + \le + \frac{\omega_{\lambda}(O_j(G_{-}))\omega_{\lambda}(O_j(G_{+}))e^{-J_{\lambda}\dist(O_{j}(G_{-}),O_{j}(G_{+}))^{2}}} + {e^{2(\psi_{J_\lambda,\omega_\lambda}-\nu)}} +\end{equation} +for all $k\ge k_\nu$. +Taking the limit superior $k\to\infty$ and then $\nu\to0$, we conclude that +\begin{equation} + \norm{E_{v_1,v_2;j}^{-,+}}^R_{J_\lambda,\omega_\lambda} + \le + \frac{\omega_{\lambda}(O_j(G_{-}))\omega_{\lambda}(O_j(G_{+}))e^{-J_{\lambda}\dist(O_{j}(G_{-}),O_{j}(G_{+}))^{2}}} + {e^{2\psi_{J_\lambda,\omega_\lambda}}}. +\end{equation} +Finally, using again \eqref{eq:intersection of extensions} for the closed sets $G_-,G_+$ and using our assumption~\eqref{eqn:good regions_separation}, we deduce Property \ref{itm:Dobrushin-Shlosman_interface_edge-event_-,+_seminorm}. +\end{proof} +\end{proof} + + + +\section{Phase co-existence in the box model}\label{sec:phase co-existence in box model} + +In this section, we derive Theorems~\ref{thm:main} and \ref{thm:detail} as consequences of Theorem \ref{thm:phase-transition}. + +Our convergence assumption describes four possible behaviors for $f_{\gamma}$ and the limiting $f$ (discrete vs.\ continuous and hard-core vs.\ soft-core). Our proofs will be streamlined to apply uniformly to all these cases, as much as possible. In the discrete case, for convenience in using the convergence assumption, we extend $f_{\gamma}$ from $S_\gamma=\gamma^d\Z\cap[0,\infty)$ to $[0,\infty)$ by a linear interpolation. + +Throughout the section we fix $\beta>0$ and $0<\alpha<\alpha_{\max}$ such that the mean-field free energy $\phi_{\lambda}$ (see~\eqref{eq:mean-field free energy density}) is non-convex, as in the assumption of Theorem~\ref{thm:main}. The coupling constant $J_2>0$ and the dimension $d\ge 2$ are also held fixed. + + + +\subsection{Deduction of Theorems~\ref{thm:main} and \ref{thm:detail}} +\label{sec:deduction of main theorem} + +\paragraph{Translating to the $(J,\omega)$-spin model language} + +Let $0<\gamma\le 1$. +To apply Theorem~\ref{thm:phase-transition}, we recall from Section \ref{sec:continuous family of spin models} the expression of the box model as a continuous family $(J_{\lambda,\gamma},\omega_{\lambda,\gamma})$ of $(J,\omega)$-spin models, indexed by the chemical potential $\lambda\in\R$: +\begin{align} + J_{\lambda,\gamma}{}&=\frac{1}{2}J_2\beta\gamma^{-d}, + \\ + \dd{\omega}_{\lambda,\gamma}(\rho){}&=e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma(\rho), +\end{align} +where the reference measure $\nu_\gamma$ was defined after~\eqref{eq:box model finite-volume Gibbs measure periodic} and we introduced the shorthand +\begin{equation} + \phi_{\lambda,\gamma}(\rho):=-\lambda\rho-\frac{1}{2}\alpha\rho^2+f_{\gamma}(\rho). +\end{equation} + +Our assumptions imply that $\omega_{\lambda,\gamma}(\R)\in(0,\infty)$ for sufficiently small $\gamma$ as required for $(J_{\lambda,\gamma},\omega_{\lambda,\gamma})$ to be a continuous family of $(J,\omega)$-spin models with respect to $\lambda$. +The next lemma makes this precise, and also, in the soft-core case, clarifies that $f_{\gamma}$ must increase quadratically at infinity. +\begin{lemma}\label{lem:model is well defined} + There exists $\gamma_0>0$ such that $\omega_{\lambda,\gamma}(\R)\in(0,\infty)$ for all $0<\gamma\le \gamma_0$ and $\lambda\in\R$. + In addition, for each $\alpha_0\in (0,\alpha_\max)$ there exists $\gamma_{\alpha_0}>0$ and $\rho_{\alpha_0}\in[0,\infty)$ such that $f_\gamma(\rho)\ge \frac{1}{2}\alpha_0\rho^2$ for $\rho\ge\rho_{\alpha_0}$ and $0<\gamma\le \gamma_{\alpha_0}$. +\end{lemma} +\begin{proof} + The uniform convergence statements~\eqref{eq:uniform convergence below eta cp} (hard-core case) and~\eqref{eq:uniform convergence below infinity} (soft-core case) imply that if $\gamma$ is sufficiently small, then $f_\gamma<\infty$ on a subset of $S_\gamma$ of positive $\nu_\gamma$-measure, so that $\omega_{\lambda,\gamma}(\R)>0$. + + In the hard-core case, $f_\gamma(\rho)=\infty$ for $\rho>\rho_\max$ so that the quadratic lower bound on $f_\gamma$ holds trivially. In the soft-core case, for each $\alpha_0\in (0,\alpha_\max)$, for the quadratic lower bound to fail there need to exist sequences $\gamma_n\downarrow 0$ and $\rho_n\uparrow\infty$ on which $f_{\gamma_n}(\rho_n)<\frac{1}{2}\alpha_0\rho_n^2$. However, as $\alpha_0<\alpha_\max$, this contradicts~\eqref{eq:growth at infinity}. + + The quadratic growth at infinity implies that $\int_{[\rho_1,\infty)} e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma<\infty$ for some $\rho_1<\infty$ and all sufficiently small $\gamma$. To conclude that $\omega_{\lambda,\gamma}(\R)<\infty$ for small $\gamma$ and $\lambda\in\R$, it then suffices that $\inf_{\rho\in[0,\rho_1]}f_\gamma(\rho)>-\infty$ for small $\gamma$. In the soft-core case, this follows directly from~\eqref{eq:uniform convergence below infinity}. In the hard-core case, it follows from~\eqref{eq:uniform convergence below eta cp},~\eqref{eq:convergence at eta cp} and~\eqref{eq:uniform convergence above eta cp}. +\end{proof} + +The family $(J_{\lambda,\gamma}, \omega_{\lambda,\gamma})$ is a continuous family of $(J,\omega)$-spin models with respect to $\lambda\in\R$, for any fixed values of $\beta,\gamma$, in the sense of Section \ref{sec:continuous family of spin models}. This follows from the continuity of $\phi_{\lambda,\gamma}$ in $\lambda$ and the dominated convergence theorem (using Lemma~\ref{lem:model is well defined}). + + + +\paragraph{Applying Theorem~\ref{thm:phase-transition}} + +In the hard-core case, set, in addition to \eqref{eq:f extension to rho cp}, +\begin{equation}\label{eq:infinite value in extension} + f(\rho):=\infty\quad\text{for $\rho>\rho_\cp$}. + \end{equation} +With this extension, the domain of the function $\phi_\lambda$ also extends to $[0,\infty)$, for all $\lambda$, via its definition~\eqref{eq:mean-field free energy density}. We point out for later use that +\begin{equation}\label{eq:continuity of phi lambda} + \text{$\phi_\lambda:[0,\infty)\to(-\infty,\infty]$ is continuous except possibly at $\rho_\cp$}. +\end{equation} +We proceed with the notation $\lambda_*, \rho_{*,-},\rho_{*,0},\rho_{*,+}$ as introduced before Theorem~\ref{thm:detail}. Define +\begin{equation} m_*:=\min_{\rho\in[0,\infty)}\phi_{\lambda_*}(\rho), +\end{equation} +noting that it is necessarily finite. + +\begin{remark} + By Theorem \ref{thm:comparison_with_GP}, $-m_*$ is equal to the Gates--Penrose mean-field pressure. +\end{remark} + +The following proposition verifies the assumption needed to apply Theorem~\ref{thm:phase-transition}. +\begin{proposition} +\label{prop:phase-transition} + There exist $\kappa,\delta_0>0$ and $0<\gamma_0\le 1$ such that the following holds. Let $\delta\in(0,\delta_0)$. + Define $\lambda_\pm(\delta) := \lambda_*\pm \kappa \delta$ and the closed sets +\begin{equation}\label{eq:G+- def} +\begin{split} + G_-(\delta) &:= \phi_{\lambda_*}^{-1}([m_*, m_*+\delta])\cap [0,\rho_{*,0}),\\ + G_+(\delta) &:=\left(\phi_{\lambda_*}^{-1}([m_*, m_*+\delta])\cap (\rho_{*,0},+\infty)\right)\cup I(\delta), +\end{split} +\end{equation} +where $I(\delta):=\emptyset$ in either the soft-core case, or in the hard-core case when $\phi_{\lambda_*}(\rho_\cp)>m_*$, and where $I(\delta):=[\rho_\cp,\rho_\cp+\delta]$ in the hard-core case when $\phi_{\lambda_*}(\rho_\cp)=m_*$. + +Then, for each $0<\gamma\le\gamma_0$, Assumption~\ref{as:good regions} is satisfied for the continuous family $(J_{\lambda,\gamma}, \omega_{\lambda,\gamma})_{\lambda\in[\lambda_-(\delta),\lambda_+(\delta)]}$ of $(J,\omega)$-spin models and the sets $G_\pm(\delta)$, with parameters $\theta_1(\delta,\gamma),\theta_2(\delta,\gamma),\theta_3(\delta,\gamma)$ which tend to zero as $\gamma\downarrow 0$. + +\end{proposition} +\begin{remark} +The interval $I(\delta)$, added in the hard-core case when $\phi_{\lambda_*}(\rho_\cp)=m_*$, is required for the proposition to hold since it is possible, when $\phi_{\lambda_*}(\rho_\cp)=m_*$, that the measure $\tilde\omega_{\lambda,\gamma}$ concentrates its mass at densities $\rho>\rho_\cp$, which, in the absence of $I(\delta)$, would violate~\eqref{eqn:good regions_double-well} of Assumption~\ref{as:good regions}. We note, however, that the precise choice of $I(\delta)$ is unimportant and the proposition remains true (with the same proof) with $I(\delta)=[\rho_\cp, \rho_\cp+f(\delta)]$ for any $f(\delta)>0$. +\end{remark} + +We now deduce Theorem~\ref{thm:detail} from Proposition \ref{prop:phase-transition}, noting that Theorem~\ref{thm:detail} immediately implies Theorem~\ref{thm:main}. + +\begin{proof}[Proof of Theorem~\ref{thm:detail}] + Let $\kappa,\delta_0>0$ and $0<\gamma_0\le 1$ be as in Proposition~\ref{prop:phase-transition}, and recall the notation $c(\epsilon,d)$ from Theorem~\ref{thm:phase-transition}. + Fix two strictly decreasing sequences $(\delta_n)_{n\ge 1}\subset(0,\delta_0)$ and $(\epsilon_n)_{n\ge 1}\subset(0,1/2)$ such that $\lim_{n\to\infty}\delta_n=\lim_{n\to\infty}\epsilon_n=0$. + By Proposition~\ref{prop:phase-transition}, for each $n\ge 1$, there exists $0<\gamma_n\le\gamma_0$ such that + \begin{equation} + \label{eqn:detail-consolidation} + \sup_{0<\gamma\le\gamma_n}\operatornamewithlimits{max}_{1\le i\le 3}\theta_i(\delta_n,\gamma)\le c(\epsilon_n,d). + \end{equation} + Without loss of generality, we may take $(\gamma_n)_{n\ge 1}$ to be strictly decreasing to $0$. + + We now specify a function $\lambda_c:(0,\gamma_1)\rightarrow\R$ satisfying the requirements of Theorem~\ref{thm:detail}, as follows. + Since $(\gamma_n)_{n\ge 1}$ is strictly decreasing to $0$, for each $0<\gamma<\gamma_1$, there exists a unique $n(\gamma)\ge 1$ such that $\gamma\in[\gamma_{n(\gamma)+1},\gamma_{n(\gamma)})$. + Using \eqref{eqn:detail-consolidation}, we let $\lambda_c(\gamma)$ be given by Theorem~\ref{thm:phase-transition}, i.e., such that $\lambda_c(\gamma)\in(\lambda_\ast-\kappa\delta_{n(\gamma)},\lambda_\ast+\kappa\delta_{n(\gamma)})$ and the $(J_{\lambda_c(\gamma),\gamma},\omega_{\lambda_c(\gamma),\gamma})$-spin model admits two distinct translation-invariant Gibbs measures $\P^{\pm}_{\lambda_c(\gamma),\gamma}$ satisfying + \begin{equation} + \label{eqn:detail-consolidation-density concentration} + \P^{\pm}_{\lambda_c(\gamma),\gamma}(\eta_{0}\in G_{\pm}(\delta_{n(\gamma)}))\ge 1-\epsilon_{n(\gamma)}. + \end{equation} + + We now verify that the function $\lambda_c$ chosen above satisfies the requirements of Theorem~\ref{thm:detail}. + Property \eqref{eq:convergence of critical chemical potential} follows from the construction $\lambda_c(\gamma)\in(\lambda_\ast-\kappa\delta_{n(\gamma)},\lambda_\ast+\kappa\delta_{n(\gamma)})$, the monotone convergence of $(\delta_n)_{n\ge 1}$ to $0$, and the monotonicity of $n(\gamma)$ in $\gamma$. + For Property \eqref{eq:density concentration}, let $U\subset\R$ be an open set containing $\mathcal{M}$. + By the continuity of $\phi_{\lambda_\ast}$, there exists $\delta_U>0$ such that, for all $0<\delta\le\delta_U$, $G_-(\delta)\subset U\cap[0,\rho_{\ast,0})$ and $G_+(\delta)\subset U\cap(\rho_{\ast,0},\infty)$. + By \eqref{eqn:detail-consolidation-density concentration}, for all small enough $\gamma>0$, + \begin{equation} + \label{eqn:detail-consolidation-density concentration-minus measure} + \P^{-}_{\lambda_c(\gamma),\gamma}(\eta_{0}\in U\cap[0,\rho_{\ast,0})) + \ge\P^{-}_{\lambda_c(\gamma),\gamma}(\eta_{0}\in G_{-}(\delta_{n(\gamma)})) + \ge 1-\epsilon_{n(\gamma)}, + \end{equation} + \begin{equation} + \label{eqn:detail-consolidation-density concentration-plus measure} + \P^{+}_{\lambda_c(\gamma),\gamma}(\eta_{0}\in U\cap(\rho_{\ast,0},\infty)) + \ge\P^{+}_{\lambda_c(\gamma),\gamma}(\eta_{0}\in G_{+}(\delta_{n(\gamma)})) + \ge 1-\epsilon_{n(\gamma)}. + \end{equation} + Now, \eqref{eq:density concentration} follows by taking $\gamma\downarrow 0$ in \eqref{eqn:detail-consolidation-density concentration-minus measure} and \eqref{eqn:detail-consolidation-density concentration-plus measure}, and using the monotone convergence of $(\epsilon_n)_{n\ge 1}$ to $0$ and again the monotonicity of $n(\gamma)$ in $\gamma$. +\end{proof} + + + +\subsection{Verifying Assumption~\ref{as:good regions}} + +In this section, we deduce Proposition~\ref{prop:phase-transition} from the next proposition, which will itself be established in Section \ref{sec:proof of normalized measure estimate}. +The normalized measures $\tilde{\omega}_{\lambda,\gamma}$ are defined as in~\eqref{normalized_omega}. + +\begin{proposition} +\label{prop:normalized-measure} +For all non-empty Borel $B\subseteq[0,\infty)$ such that $\inf_{\rho\in B} f(\rho)<\infty$ and all compact $K\subset\R$, +\begin{equation} +\label{eqn:normalized-measure} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d\log\tilde{\omega}_{\lambda,\gamma}(B)+\beta\left[\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)-\inf_{\rho\in[0,\infty)}\phi_{\lambda}(\rho)\right]\right\}\le 0, +\end{equation} +with $\overline{B}$ denoting the closure of $B$. +\end{proposition} + +\begin{lemma}\label{lem:terminal rho} +There exists $\rho_{\mathrm{T}}<\infty$ such that +\begin{equation} + \inf_{\substack{\lambda\in[\lambda_*-1,\lambda_*+1]\\\rho\in[\rho_{\mathrm{T}},\infty)}} \phi_{\lambda}(\rho) \ge m_*+1. +\end{equation} +\end{lemma} + +\begin{proof} + In the hard-core case, $f(\rho)=\infty$ when $\rho>\rho_\cp$ by~\eqref{eq:infinite value in extension}, so we may take any $\rho_{\mathrm{T}}\in(\rho_\cp,\infty)$. + In the soft-core case, the claim follows from the quadratic lower bound~\eqref{eq:growth of f at infinity}, the fact that $\alpha<\alpha_\max$ and the definition~\eqref{eq:mean-field free energy density} of $\phi_{\lambda}$. +\end{proof} + +\begin{proof}[Deduction of Proposition \ref{prop:phase-transition}] +Fix +\begin{equation} + 0<\delta_0<\begin{cases}\min\{\phi_{\lambda_*}(\rho_{*,0})-m_*, 1\}&\substack{\text{in the soft-core case,}\\\text{or the hard-core case with $\phi_{\lambda_*}(\rho_\cp)=m_*$}}\\ + \min\{\phi_{\lambda_*}(\rho_{*,0})-m_*, 1, \phi_{\lambda_*}(\rho_\cp)-m_*\}&\text{hard-core case with $\phi_{\lambda_*}(\rho_\cp)>m_*$} + \end{cases} +\end{equation} +arbitrarily. +Fix the $0<\gamma_0\le 1$ of Lemma~\ref{lem:model is well defined}. Fix $0<\kappa<\min\{1, \frac{1}{\rho_{*,+}+\rho_{\mathrm{T}}}\}$ for the $\rho_{\mathrm{T}}$ of Lemma~\ref{lem:terminal rho}. + +Let $\delta\in(0,\delta_0)$. We first note that the sets $G_\pm(\delta)$ are closed. Recall that $\phi_{\lambda_*}$ is continuous in the soft-core case, and is continuous on $[0,\rho_\cp]$ (allowing it to be infinite at $\rho_\cp$) and is infinite on $(\rho_\cp,\infty)$ in the hard-core case. +Thus, as $m_*+\delta<\phi_{\lambda_*}(\rho_{*,0})$, we have that $G_-(\delta)= \phi_{\lambda_*}^{-1}([m_*, m_*+\delta])\cap [0,\rho_{*,0}-\eps]$ and $G_+(\delta) = (\phi_{\lambda_*}^{-1}([m_*, m_*+\delta])\cap [\rho_{*,0}+\eps,\infty))\cup I(\delta)$ for some $\eps>0$, whence $G_\pm(\delta)$ are closed (noting that $I(\delta)$ is closed). + +Fix $K:=[\lambda_*-\kappa \delta, \lambda_*+\kappa\delta]$. Proposition~\ref{prop:normalized-measure} implies that for any set $B$ as in the proposition, there exists a function $\eps_{B}:(0,\gamma_0]\to[0,\infty)$ satisfying $\lim_{\gamma\downarrow0}\eps_{B}(\gamma)=0$ such that +\begin{equation} +\label{eqn:normalized measure with error} + \tilde{\omega}_{\lambda,\gamma}(B)\le \exp{\gamma^{-d}\left(\eps_B(\gamma)-\beta\left[\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)-\inf_{\rho\in[0,\infty)}\phi_{\lambda}(\rho)\right]\right)}\quad\text{for}\quad\substack{\lambda\in K,\\0<\gamma\le\gamma_0}. +\end{equation} + +We start with~\eqref{eqn:good regions_separation} of Assumption~\ref{as:good regions}. We apply~\eqref{eqn:normalized measure with error} with $B=G_\pm(\delta)$, noting that $\inf_{\rho\in G_\pm(\delta)} f(\rho)<\infty$ by the definition of $G_\pm(\delta)$ and $m_*$. We obtain, for each $\lambda\in K$ and $0<\gamma\le\gamma_0$, +\begin{equation} +\begin{split} + {}&e^{-\frac{1}{2}J_{\lambda,\gamma}\dist(G_-(\delta), G_+(\delta))^2}(\tilde\omega_{\lambda,\gamma}(G_-(\delta))\tilde\omega_{\lambda,\gamma}(G_+(\delta)))^{1/2} + \\ + \le{}& + \begin{multlined}[t] + \exp\left\{-\frac{1}{2}J_{\lambda,\gamma}\dist(G_-(\delta), G_+(\delta))^2+\frac{1}{2}\gamma^{-d}\left(\epsilon_{G_-(\delta)}(\gamma)+\epsilon_{G_+(\delta)}(\gamma) + \vphantom{\frac{1}{2}}\right.\right. + \\ + \left.\left.-\beta\left[\inf_{\rho\in G_-(\delta)}\phi_{\lambda}(\rho)+\inf_{\rho\in G_+(\delta)}\phi_{\lambda}(\rho)-2\inf_{\rho\in [0,\infty)}\phi_{\lambda}(\rho))\right]\right)\right\} + \end{multlined} + \\ + \le{}&\exp{-\frac{1}{2}\gamma^{-d}\left[\frac{1}{2}J_2\beta\dist(G_-(\delta), G_+(\delta))^2-\epsilon_{G_-(\delta)}(\gamma)-\epsilon_{G_+(\delta)}(\gamma)\right]}. +\end{split} +\end{equation} +Since $G_-(\delta)$ and $G_+(\delta)$ are closed and disjoint we have that $\dist(G_-(\delta), G_+(\delta))>0$. Therefore, +\begin{equation}\label{eq:verifying the second property} + \lim_{\gamma\downarrow 0}\sup_{\lambda\in K} e^{-\frac{1}{2}J_{\lambda,\gamma}\dist(G_-(\delta), G_+(\delta))^2}(\tilde\omega_{\lambda,\gamma}(G_-(\delta))\tilde\omega_{\lambda,\gamma}(G_+(\delta)))^{1/2} = 0. +\end{equation} + +We continue with~\eqref{eqn:good regions_double-well} of Assumption~\ref{as:good regions}. We apply~\eqref{eqn:normalized measure with error} with $B=(G_-(\delta)\cup G_+(\delta))^{c}$, noting that $\inf_{\rho\in B} f(\rho)<\infty$ since $\rho_{*,0}\in B$ and, in the hard-core case, $\rho_{*,0}<\rho_{*,+}\le\rho_\cp$. We obtain, for each $\lambda\in K$ and $0<\gamma\le\gamma_0$, +\begin{equation}\label{eq:towards the first property} +\begin{multlined} + \tilde\omega_{\lambda,\gamma}((G_-(\delta)\cup G_+(\delta))^{c}) + \\ + \le \exp{\gamma^{-d}\left(\eps_{(G_-(\delta)\cup G_+(\delta))^{c}}(\gamma)-\beta\left[\inf_{\rho\in \overline{(G_-(\delta)\cup G_+(\delta))^{c}}}\phi_{\lambda}(\rho)-\inf_{\rho\in[0,\infty)}\phi_{\lambda}(\rho)\right]\right)}. +\end{multlined} +\end{equation} +Now, for each $\lambda\in K$, on the one hand, +\begin{equation} + \inf_{\rho\in[0,\infty)}\phi_{\lambda}(\rho)\le \phi_{\lambda}(\rho_{*,+})\le m_* + \kappa\delta\rho_{*,+}. +\end{equation} +On the other hand, recalling the definition of $G_\pm(\delta)$ from~\eqref{eq:G+- def} and applying Lemma~\ref{lem:terminal rho} using the fact that $K\subset[\lambda_*-1,\lambda_*+1]$ (since $\kappa,\delta<1$), +\begin{equation}\label{eq:infimum away from Gs} + \inf_{\rho\in (G_-(\delta)\cup G_+(\delta))^{c}}\phi_{\lambda}(\rho)\ge m_* + \delta - \kappa\delta\rho_{\mathrm{T}} = m_* + (1 - \kappa\rho_{\mathrm{T}})\delta. +\end{equation} +Moreover, we claim that +\begin{equation}\label{eq:closure does not change inf} + \inf_{\rho\in \overline{(G_-(\delta)\cup G_+(\delta))^{c}}}\phi_{\lambda}(\rho) = \inf_{\rho\in (G_-(\delta)\cup G_+(\delta))^{c}}\phi_{\lambda}(\rho). +\end{equation} +This is clear in the soft-core case since $\phi_\lambda$ is continuous. +It follows in the hard-core case when $\phi_{\lambda_*}(\rho_\cp)>m_*$, since in this case $\rho_\cp$ belongs to the open set $(G_-(\delta)\cup G_+(\delta))^{c}$ by the definition of $G_\pm(\delta)$ from~\eqref{eq:G+- def} and our choice of $\delta$, whence the boundary $\partial (G_-(\delta)\cup G_+(\delta))$ consists only of continuity points of $\phi_\lambda$ by~\eqref{eq:continuity of phi lambda}. +It also follows in the hard-core case when $\phi_{\lambda_*}(\rho_\cp)=m_*$ since a neighborhood of $\rho_\cp$ is contained in $G_+(\delta)$ by the definition~\eqref{eq:G+- def} (making use of $I(\delta)$), so again the boundary $\partial (G_-(\delta)\cup G_+(\delta))$ consists only of continuity points of $\phi_\lambda$ by~\eqref{eq:continuity of phi lambda}. +Therefore, since $\kappa<\frac{1}{\rho_{*,+}+\rho_{\mathrm{T}}}$, +\begin{equation} + \inf_{\lambda\in K}\left(\inf_{\rho\in \overline{(G_-(\delta)\cup G_+(\delta))^{c}}}\phi_{\lambda}(\rho)-\inf_{\rho\in[0,\infty)}\phi_{\lambda}(\rho)\right)>0. +\end{equation} +Together with~\eqref{eq:towards the first property}, this implies +\begin{equation}\label{eq:verifying the first property} + \lim_{\gamma\downarrow0}\sup_{\lambda\in K}\tilde\omega_{\lambda,\gamma}((G_-(\delta)\cup G_+(\delta))^{c}) = 0. +\end{equation} + +Lastly, we check~\eqref{eqn:good regions_endpoints} of Assumption~\ref{as:good regions}. +Let $\#\in\set{-,+}$. +We apply~\eqref{eqn:normalized measure with error} with $B=G_\#(\delta)^{c}$, noting that $\inf_{\rho\in B} f(\rho)<\infty$ as, again, $\rho_{*,0}\in B$. We obtain, for each $0<\gamma\le\gamma_0$, +\begin{equation}\label{eq:towards the third property} + \tilde\omega_{\lambda_\#,\gamma}(G_\#(\delta)^{c})\le \exp{\gamma^{-d}\left(\eps_{G_\#(\delta)^{c}}(\gamma)-\beta\left[\inf_{\rho\in \overline{G_\#(\delta)^{c}}}\phi_{\lambda_\#}(\rho)-\inf_{\rho\in[0,\infty)}\phi_{\lambda_\#}(\rho)\right]\right)} +\end{equation} +where we recall that $\lambda_\pm(\delta) = \lambda_*\pm \kappa \delta$. +Now, on the one hand, +\begin{equation} + \inf_{\rho\in G_{\mp}(\delta)}\phi_{\lambda_\pm}(\rho) + \ge \inf_{\rho\in G_{\mp}(\delta)}\phi_{\lambda_\ast}(\rho) + +\inf_{\rho\in G_{\mp}(\delta)}(-(\lambda_\pm-\lambda_\ast)\rho) + \ge m_*-(\lambda_\pm-\lambda_*)\rho_{*,0}, +\end{equation} +which, using that +\begin{equation} + \overline{G_\pm(\delta)^c} + =G_\pm(\delta)^c\cup\partial G_\pm(\delta) + = G_\mp(\delta)\cup \overline{(G_-(\delta)\cup G_+(\delta))^{c}} +\end{equation} +and using~\eqref{eq:closure does not change inf} and~\eqref{eq:infimum away from Gs}, implies that +\begin{equation} +\begin{multlined} + \inf_{\rho\in \overline{G_\#(\delta)^{c}}}\phi_{\lambda_\#}(\rho)\ge m_*+\min\{-(\lambda_\#-\lambda_*)\rho_{*,0}, (1 - \kappa\rho_{\mathrm{T}})\delta\} + \\ + =\begin{cases}m_*+\min\{\kappa\rho_{*,0},1 - \kappa\rho_{\mathrm{T}}\}\delta&\#=-\\ + m_*+\min\{-\kappa\rho_{*,0},1 - \kappa\rho_{\mathrm{T}}\}\delta&\#=+ + \end{cases}. +\end{multlined} +\end{equation} +On the other hand, +\begin{equation} + \inf_{\rho\in[0,\infty)}\phi_{\lambda_\#}(\rho)\le \phi_{\lambda_\#}(\rho_{*,\#}) = m_* - (\lambda_\#-\lambda_*)\rho_{*,\#} = \begin{cases}m_*+\kappa\delta\rho_{*,-}&\#=-\\ + m_*-\kappa\delta\rho_{*,+}&\#=+ + \end{cases}. +\end{equation} +Therefore, using that $\rho_{*,-}<\rho_{*,0}<\rho_{*,+}$ and using again that $\kappa<\frac{1}{\rho_{*,+}+\rho_{\mathrm{T}}}$, +\begin{equation} + \inf_{\rho\in \overline{G_\#(\delta)^{c}}}\phi_{\lambda_\#}(\rho)-\inf_{\rho\in[0,\infty)}\phi_{\lambda_\#}(\rho)>0. +\end{equation} +Together with~\eqref{eq:towards the third property}, this implies +\begin{equation}\label{eq:verifying the third property} + \lim_{\gamma\downarrow0}\tilde\omega_{\lambda_\#,\gamma}(G_\#(\delta)^{c}) = 0. +\end{equation} + +The proposition follows from~\eqref{eq:verifying the second property},~\eqref{eq:verifying the first property} and~\eqref{eq:verifying the third property}. +\end{proof} + + + +\subsection{Technical lemmas} + +We will deduce Proposition \ref{prop:normalized-measure} and Theorem \ref{thm:comparison_with_GP} from the two technical lemmas introduced in this section. +Recall that, in the discrete case, we extended $f_{\gamma}$ to $[0,\infty)$ by a linear interpolation. + +\begin{lemma} +\label{lem:normalized-measure_measure} + For any compact $K\subset\R$ and non-empty, Borel $B\subseteq[0,\infty)$ such that $\inf_{\rho\in B} f(\rho)<\infty$, + \begin{equation} + \label{eqn:normalized-measure_measure} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d\log\omega_{\lambda,\gamma}(B)+\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\}\le 0 + \end{equation} +\end{lemma} + +\begin{claim} +\label{claim:box-model_hard-core_lower-bound} +For all non-empty, bounded Borel $B\subseteq[0,\infty)$ such that $\inf_{\rho\in B} f(\rho)<\infty$ and all compact $K\subset\R$, +\begin{equation} + \liminf_{\gamma\downarrow0}\inf_{\lambda\in K}\left\{\inf_{\rho\in B}\phi_{\lambda,\gamma}(\rho)-\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\}\ge 0. +\end{equation} +\end{claim} + +\begin{proof} +Suppose, to obtain a contradiction, that the claim does not hold. Therefore, there exist $\epsilon>0$ and sequences $(\gamma_j)\subset(0,1]$, $(\lambda_j)\subset K$, and $(\rho_j)\subset B$, with $\gamma_j\downarrow 0$, such that $\phi_{\lambda_j,\gamma_j}(\rho_j)<\infty$ and +\begin{equation} + \phi_{\lambda_j,\gamma_j}(\rho_j)\le \inf_{\rho\in \overline{B}}\phi_{\lambda_j}(\rho)-\epsilon +\end{equation} +for all $j$. +By compactness, we may further assume that $\lambda_j\rightarrow\lambda_\infty\in K$ and $\rho_j\rightarrow\rho_\infty\in \overline{B}$. +To obtain a contradiction, it suffices to show that +\begin{equation}\label{eq:limiting phi value} + \liminf_{j\rightarrow\infty}\phi_{\lambda_j,\gamma_j}(\rho_j)\ge\phi_{\lambda_\infty}(\rho_\infty), +\end{equation} +using that $\lim_{j\rightarrow\infty}\inf_{\rho\in \overline{B}}\phi_{\lambda_j}(\rho)=\inf_{\rho\in \overline{B}}\phi_{\lambda_\infty}(\rho)$ by the boundedness of $B$ and the continuity of $\phi_\lambda$. +In the soft-core case,~\eqref{eq:limiting phi value} follows from~\eqref{eq:uniform convergence below infinity}. In the hard-core case,~\eqref{eq:limiting phi value} follows from~\eqref{eq:uniform convergence below eta cp} if $\rho_\infty<\rho_\cp$, follows from~\eqref{eq:convergence at eta cp} (and~\eqref{eq:f extension to rho cp}) if $\rho_\infty=\rho_\cp$, and follows from~\eqref{eq:uniform convergence above eta cp} if $\rho_\infty>\rho_\cp$. +\end{proof} + +\begin{claim} +\label{clm:tail bound} +For any compact $K\subset\R$, +\begin{equation} + \lim_{\rho_1\to\infty}\limsup_{\gamma\downarrow0}\sup_{\lambda\in K}\left\{\gamma^d\log\omega_{\lambda,\gamma}([\rho_1,\infty))\right\}=-\infty. +\end{equation} +\end{claim} +\begin{proof} + Fix $\alpha_0\in(\alpha,\alpha_\max)$. + Lemma~\ref{lem:model is well defined} shows that there exist $\gamma_{\alpha_0}>0$ and $\rho_{\alpha_0}\in[0,\infty)$ such that $f_\gamma(\rho)\ge \frac{1}{2}\alpha_0\rho^2$ for $\rho\ge\rho_{\alpha_0}$ and $0<\gamma\le \gamma_{\alpha_0}$. + Therefore, for all $\lambda\in K$, and taking $0<\gamma\le \gamma_{\alpha_0}$ sufficiently small and $\rho_1\ge\rho_{\alpha_0}$ sufficiently large, it holds that + \begin{multline} + \int_{[\rho_1,\infty)}e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma(\rho) \le \int_{[\rho_1,\infty)}e^{-\beta\gamma^{-d}(-\rho\min K + \frac{1}{2}(\alpha_0-\alpha)\rho^2)}\dd{\nu}_\gamma(\rho)\\ + \le\int_{[\rho_1,\infty)}e^{-\beta\gamma^{-d}(\rho+1)}\dd{\nu}_\gamma(\rho) \le e^{-\beta\gamma^{-d}\rho_1} \nu_\gamma([\rho_1,\rho_1+1))\sum_{k=1}^\infty e^{-\beta\gamma^{-d}k} \le e^{-\beta\gamma^{-d}\rho_1}, + \end{multline} + where the second inequality uses that $-\rho\min K + \frac{1}{2}(\alpha_0-\alpha)\rho^2\ge \rho+1$ for sufficiently large $\rho$, the third inequality uses the monotonicity of the integrand and the $1$-periodicity of $\nu_\gamma$, and the final inequality uses that $\gamma$ is sufficiently small and the definition of $\nu_\gamma$. + The claim follows. +\end{proof} + +\begin{proof}[Proof of Lemma \ref{lem:normalized-measure_measure}] +Let $K\subset\R$ be compact and $B\subseteq[0,\infty)$ be non-empty and Borel, satisfying that $\inf_{\rho\in B} f(\rho)<\infty$. +To prove \eqref{eqn:normalized-measure_measure}, we first use Claim \ref{clm:tail bound} to find $\rho_1>0$ such that +\begin{equation} +\label{eqn:normalized-measure_rho1-second-property} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d\log\int_{[\rho_1,\infty)}e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma(\rho)\right\} + \le -\beta\sup_{\lambda\in K}\inf_{\rho\in\overline{B}}\phi_\lambda(\rho). +\end{equation} +Let $B_1:=B\cap[0,\rho_1]$ and $B_2:=B\cap(\rho_1,\infty)$. +Splitting $\omega_{\lambda,\gamma}(B)=\omega_{\lambda,\gamma}(B_1)+\omega_{\lambda,\gamma}(B_2)$ and using the elementary inequality $\log(a+b)\le\log2+\max\set{\log a,\log b}$, we bound the LHS of \eqref{eqn:normalized-measure_measure} by +\begin{equation} +\label{eqn:normalized-measure_measure_max-bound} +\begin{multlined} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d\log2+\max\set{ + \log\omega_{\lambda,\gamma}(B_1), + \log\omega_{\lambda,\gamma}(B_2) + }+\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\} + \\ + =\max + \left\{\limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d + \log\omega_{\lambda,\gamma}(B_1) + +\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\}, + \right. + \\ + \left.\limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d + \log\omega_{\lambda,\gamma}(B_2) + +\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\} + \right\}. +\end{multlined} +\end{equation} +On the one hand, using Claim \ref{claim:box-model_hard-core_lower-bound} and that $\inf_{\rho\in B_1}\phi_{\lambda,\gamma}(\rho)\ge \inf_{\rho\in B}\phi_{\lambda,\gamma}(\rho)$, +\begin{equation} +\label{eqn:normalized-measure_measure_main-part} +\begin{multlined} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d + \log\omega_{\lambda,\gamma}(B_1) + +\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\} + \\ + \le \limsup_{\gamma\downarrow 0}\gamma^d\log\nu_\gamma(B_1) + -\beta\liminf_{\gamma\downarrow 0}\inf_{\lambda\in K}\left\{\inf_{\rho\in B}\phi_{\lambda,\gamma}(\rho)-\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\} + \le 0. +\end{multlined} +\end{equation} +On the other hand, using \eqref{eqn:normalized-measure_rho1-second-property}, +\begin{equation} +\label{eqn:normalized-measure_measure_tail} +\begin{multlined} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d + \log\omega_{\lambda,\gamma}(B_2) + +\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\} + \\ + \le\limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d\log\int_{[\rho_1,\infty)} e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma(\rho) + \right\}+\beta\sup_{\lambda\in K}\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho) + \le 0. +\end{multlined} +\end{equation} +Combining \eqref{eqn:normalized-measure_measure_max-bound}, \eqref{eqn:normalized-measure_measure_main-part}, and \eqref{eqn:normalized-measure_measure_tail}, we get \eqref{eqn:normalized-measure_measure}. +\end{proof} + +\begin{lemma} + \label{lem:normalized-measure_normalization} + For any compact $K\subset\R$, + \begin{equation} + \label{eqn:normalized-measure_normalization} + \liminf_{\gamma\downarrow 0}\inf_{\lambda\in K}\inf_{L\ge 1}\left\{\frac{1}{\gamma^{-d}\abs{\Lambda_L}}\log\Xi^{L,\per}_{\lambda,\gamma}+\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}\ge 0, + \end{equation} + where we introduced the shorthand $\psi_{\lambda,\gamma}:=\psi_{J_{\lambda,\gamma},\omega_{\lambda,\gamma}}$. +\end{lemma} + +\begin{claim} +\label{claim:box-model_pressure-localization} +For any compact $K\subset\R$, +\begin{equation} + \lim_{\xi\downarrow0}\limsup_{\gamma\downarrow0}\sup_{\lambda\in K}\inf_{\rho_0\in[0,\infty)}\left\{\sup_{\rho\in[\rho_0,\rho_0+\xi]}\phi_{\lambda,\gamma}(\rho)-\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}\le 0. +\end{equation} +\end{claim} + +\begin{proof} + Let $\epsilon>0$. In the soft-core case, choose $\rho_1<\infty$ such that + \begin{equation} + \inf_{\rho\in[0,\rho_1]}\phi_\lambda(\rho)=\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\quad\text{for all $\lambda\in K$.} + \end{equation} + This is possible since $K$ is bounded and using the quadratic growth~\eqref{eq:growth of f at infinity} of $f$, together with our choice $\alpha<\alpha_\max$ (and the definition~\eqref{eq:mean-field free energy density} of $\phi_\lambda$). In the hard-core case, choose $\rho_1<\rho_\cp$ such that + \begin{equation} + \sup_{\lambda\in K}\left\{\inf_{\rho\in[0,\rho_1]}\phi_\lambda(\rho)-\inf_{\rho\in[0,\rho_\cp]}\phi_\lambda(\rho)\right\}\le\epsilon. + \end{equation} + This is possible since $K$ is bounded and as $f$ is continuous on $[0,\rho_\cp]$ (at $\rho_\cp$, we mean this in the generalized sense~\eqref{eq:f extension to rho cp} if $f(\rho_\cp)=\infty$). + + As $\phi_\lambda$ is continuous, for each $\lambda\in K$, there exists $\rho_0(\lambda)\in[0,\rho_1]$ such that $\phi_\lambda(\rho_0(\lambda))=\inf_{\rho\in[0,\rho_1]}\phi_\lambda(\rho)$. + Then, for all small enough $\xi,\gamma>0$, + \begin{equation} + \begin{split} + {}&\sup_{\lambda\in K}\inf_{\rho_0\in[0,\infty)}\left\{\sup_{\rho\in[\rho_0,\rho_0+\xi]}\phi_{\lambda,\gamma}(\rho)-\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\} + \\ + \le{}&\sup_{\lambda\in K}\left\{\sup_{\rho\in[\rho_0(\lambda),\rho_0(\lambda)+\xi]}\phi_{\lambda,\gamma}(\rho)-\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\} + \\ + \le{}&\sup_{\lambda\in K}\left\{\sup_{\rho\in[\rho_0(\lambda),\rho_0(\lambda)+\xi]}\phi_{\lambda}(\rho)-\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}+\epsilon + \\ + \le{}&\sup_{\lambda\in K}\left\{\phi_{\lambda}(\rho_0(\lambda))-\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}+2\epsilon + \\ + \le{}&3\epsilon, + \end{split} + \end{equation} + where we used the uniform convergence assumption~\eqref{eq:uniform convergence below eta cp} (hard-core case) or~\eqref{eq:uniform convergence below infinity} (soft-core case) in the second inequality, the uniform continuity of $(\lambda,\rho)\mapsto\phi_\lambda(\rho)$ on $K\times[0,\rho_1+\xi]$ in the third, and the definition of $\rho_0(\lambda)$ and $\rho_1$ in the last. + The proof is complete after taking $\gamma\downarrow 0$, $\xi\downarrow 0$, and $\epsilon\downarrow 0$. +\end{proof} + +\begin{proof}[Proof of Lemma \ref{lem:normalized-measure_normalization}] +To prove~\eqref{eqn:normalized-measure_normalization}, we bound, using \eqref{eqn:partition-function-lower-bound}, +\begin{equation} + \inf_{L\ge 1}\frac{1}{\gamma^{-d}\abs{\Lambda_L}}\log\Xi^{L,\per}_{\lambda,\gamma} + \ge \sup_{S}\left\{-dJ_{\lambda,\gamma}\diam(S)^2+\log\omega_{\lambda,\gamma}(S)\right\}. +\end{equation} +Let $\xi>0$. +By restricting to sets $S$ of the form $[\rho_0,\rho_0+\xi]$, where $\rho_0\in [0,\infty)$, we bound the LHS of \eqref{eqn:normalized-measure_normalization} below by +\begin{equation} +\begin{multlined} + \liminf_{\gamma\downarrow 0}\inf_{\lambda\in K}\sup_{\rho_0\in[0,\infty)}\left\{-\gamma^d dJ_{\lambda,\gamma}\xi^2+\gamma^d\log\omega_{\lambda,\gamma}(S)+\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\} + \\ + =-\frac{1}{2}\beta d J_2\xi^2 + +\liminf_{\gamma\downarrow 0}\inf_{\lambda\in K}\sup_{\rho_0\in[0,\infty)}\left\{\gamma^d\log\omega_{\lambda,\gamma}(S)+\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}, +\end{multlined} +\end{equation} +where $S$ is the shorthand for $[\rho_0,\rho_0+\xi]$. +Thus, it suffices to show that +\begin{equation} + \lim_{\xi\downarrow 0}\liminf_{\gamma\downarrow 0}\inf_{\lambda\in K}\sup_{\rho_0\in[0,\infty)}\left\{\gamma^d\log\int_{[\rho_0,\rho_0+\xi]}e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma(\rho)+\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}\ge 0. +\end{equation} +We bound +\begin{equation} +\begin{split} + {}&\sup_{\rho_0\in[0,\infty)}\left\{\gamma^d\log\int_{[\rho_0,\rho_0+\xi]}e^{-\beta\gamma^{-d}\phi_{\lambda,\gamma}(\rho)}\dd{\nu}_\gamma(\rho)+\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\} + \\ + \ge{}&\sup_{\rho_0\in[0,\infty)}\left\{\gamma^d\log\nu_\gamma([\rho_0,\rho_0+\xi])-\beta\sup_{\rho\in[\rho_0,\rho_0+\xi]}\phi_{\lambda,\gamma}(\rho)+\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\} + \\ + \ge{}&\inf_{\rho_0\in[0,\infty)}\left\{\gamma^d\log\nu_\gamma([\rho_0,\rho_0+\xi])\right\} + -\beta\inf_{\rho_0\in[0,\infty)}\left\{\sup_{\rho\in[\rho_0,\rho_0+\xi]}\phi_{\lambda,\gamma}(\rho)-\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}. +\end{split} +\end{equation} +Since +\begin{equation} + \liminf_{\gamma\downarrow 0}\inf_{\rho_0\in[0,\infty)}\left\{\gamma^d\log\nu_\gamma([\rho_0,\rho_0+\xi])\right\}=0, +\end{equation} +we deduce~\eqref{eqn:normalized-measure_normalization} using Claim~\ref{claim:box-model_pressure-localization}. +\end{proof} + + + +\subsection{Deduction of Proposition~\ref{prop:normalized-measure} from Lemma~\ref{lem:normalized-measure_measure} and Lemma~\ref{lem:normalized-measure_normalization}} +\label{sec:proof of normalized measure estimate} + +Let $K\subset\R$ be compact and $B\subseteq[0,\infty)$ be non-empty and Borel, satisfying that $\inf_{\rho\in B} f(\rho)<\infty$. +By \eqref{normalized_omega}, +\begin{equation} + \gamma^d\log\tilde{\omega}_{\lambda,\gamma}(B) + =\gamma^d\log\omega_{\lambda,\gamma}(B) + -\gamma^d\psi_{\lambda,\gamma}, +\end{equation} +so the LHS of \eqref{eqn:normalized-measure} is bounded above by +\begin{equation} +\begin{multlined} + \limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{\gamma^d\log\omega_{\lambda,\gamma}(B)+\beta\inf_{\rho\in \overline{B}}\phi_{\lambda}(\rho)\right\} + \\ + +\limsup_{\gamma\downarrow 0}\sup_{\lambda\in K}\left\{-\gamma^d\psi_{\lambda,\gamma}-\beta\inf_{\rho\in[0,\infty)}\phi_\lambda(\rho)\right\}. +\end{multlined} +\end{equation} +The proposition now follows from Lemmas \ref{lem:normalized-measure_measure} and \ref{lem:normalized-measure_normalization} and the definition \eqref{eqn:spin-model_free-energy} of $\psi_{\lambda,\gamma}$. + + + +\subsection{Proof of Theorem~\ref{thm:comparison_with_GP}} + +Let $\lambda\in\R$. +By Lemma~\ref{lem:normalized-measure_normalization}, +\begin{equation} +\label{eqn:comparison_with_GP-lower_bound} + \liminf_{\substack{L\to\infty \\ \gamma\downarrow 0}}\left\{\frac{1}{\gamma^{-d}|\Lambda_L|}\log\Xi^{L, \per}_{\lambda,\gamma}+\beta\inf_{\rho}\phi_\lambda(\rho)\right\}\ge 0. +\end{equation} +For an upper bound, we use the trivial bound ${H^L_{J_{\lambda,\gamma}}(\eta)}\ge 0$ in \eqref{partition_function} to obtain +\begin{equation} + \sup_{L\ge 1}\left\{\frac{1}{\gamma^{-d}|\Lambda_L|}\log\Xi^{L, \per}_{\lambda,\gamma}\right\}\le\log\omega_{\lambda,\gamma}([0,\infty)). +\end{equation} +Applying Lemma~\ref{lem:normalized-measure_measure} with $B=[0,\infty)$, we get that +\begin{equation} +\label{eqn:comparison_with_GP-upper_bound} + \limsup_{\substack{L\to\infty\\\gamma\downarrow 0}}\left\{\frac{1}{\gamma^{-d}|\Lambda_L|}\log\Xi^{L, \per}_{\lambda,\gamma}+\beta\inf_{\rho}\phi_{\lambda}(\rho)\right\} + \le\limsup_{\gamma\downarrow 0}\left\{\gamma^d\log\omega_{\lambda,\gamma}(B)+\beta\inf_{\rho}\phi_{\lambda}(\rho)\right\} + \le 0. +\end{equation} +The theorem follows from \eqref{eqn:comparison_with_GP-lower_bound} and \eqref{eqn:comparison_with_GP-upper_bound}. + + + +\subsection*{Acknowledgements} +JLL thanks Roman Koteck\'y for emphasizing the importance of the problem of rigorously establishing the liquid-vapor phase transition in a 2022 IHES meeting. + +QH is supported by an SAS fellowship at Rutgers University. +The research of RP is partially supported by the Israel Science Foundation grants 1971/19 and 2340/23, by the European Research Council Consolidator grant 101002733 (Transitions) and by the National Science Foundation grant DMS-2451133. +Part of this work was completed while RP was a visiting fellow at the Mathematics Department of Princeton University, a visitor of the Institute for Advanced Study and a consultant at Rutgers University. RP is grateful for their support. +IJ gratefully acknowledges support through NSF Grant DMS-2349077, and the Simons Foundation, Grant Number 825876. + + +\bibliographystyle{plain} +\bibliography{bibliography} + + +\appendix + +\section{Convergence assumptions for particle systems}\label{app:ruelle} + +The convergence assumptions in Section \ref{sec:convergence_assumptions} are satisfied for systems of particles interacting via {\it well-behaved} pair potentials. +These results were proved by Ruelle \cite{ruelle1963classical, Ruelle69}, and are recalled in this appendix. + +Consider a continuum particle system in the box $[0,\gamma^{-1}]^d$, interacting via the Hamiltonian +\begin{equation} + H(x_1,\cdots,x_N)=\sum_{id$ and $A,R_0>0$. +Let $f_\gamma(\rho)$ denote the canonical free energy +\begin{equation} + f_\gamma(N \gamma^{d}):=-\frac1{\beta \gamma^{-d}}\log \frac1{N!}\int \dd{x}_1\cdots \dd{x}_Ne^{-\beta H(x_1,\cdots,x_N)} + \label{fparticle} +\end{equation} +($\beta$ is the inverse temperature, which we consider fixed, and so do not make it explicit in the notation.) + +\begin{theorem}{\rm(\cite[Theorem 3.4.4]{Ruelle69})}:\label{theo:particle} + If the potential is stable and tempered, then there exists $\rho_{\mathrm{cp}}\in[0,\infty]$ and a convex (and thus continuous) function $f:[0,\rho_{\mathrm{cp}})\to \mathbb R$ such that, for all $\rho \geqslant 0$ and any $\rho_\gamma$ such that $\rho_\gamma\to\rho$, the following hold. + If $\rho<\rho_{\mathrm{cp}}$ + \begin{equation} + \lim_{\gamma\downarrow 0}f_\gamma(\rho_\gamma)=f(\rho) + \label{ruelle_lim1} + \end{equation} + if $\rho=\rho_{\mathrm{cp}}$ + \begin{equation} + \liminf_{\gamma\downarrow 0}f_\gamma(\rho_\gamma)\geqslant \lim_{\rho\uparrow\rho_{\mathrm{cp}}}f(\rho) + \label{ruelle_lim2} + \end{equation} + and if $\rho>\rho_{\mathrm{cp}}$ then + \begin{equation} + \lim_{\gamma\downarrow0}f_\gamma(\rho_\gamma)=\infty + . + \label{ruelle_lim3} + \end{equation} +\end{theorem} + +The limit in (\ref{ruelle_lim1}) is actually uniform, as mentioned (in a slightly different context) in \cite[Remark 3.3.13]{Ruelle69}. +We give the argument here for the sake of completeness. + +\begin{corollary}\label{cor:uniform} + The limit in (\ref{ruelle_lim1}) is uniform. +\end{corollary} + +\begin{proof} + Suppose the limit were not uniform, then there would be $\epsilon>0$ and a sequence $\gamma_i\to0$ such that $|f_{\gamma_i}(n_{\gamma_i})-f(\rho)| \geqslant \epsilon$. + This contradicts Theorem \ref{theo:particle} since it applies to {\it any} $\rho_\gamma$ and in particular to $n_{\gamma_i}$. +\end{proof} + +As we will see below, these results allow us to prove assumptions 1(a), 1(b), 1(c), and 2(a) of Section \ref{sec:convergence_assumptions}. +To obtain assumption 2(b), we will need to impose a stronger stability condition. +The potential is said to be {\it superstable} if \cite[Section 3.2.9]{Ruelle69} +\begin{equation} + H(x_1,\cdots,x_N) \geqslant N(C N \gamma^{d}-D) + \label{superstability} +\end{equation} +for some $C>0$ and $D \geqslant 0$. +(Note that superstability trivially implies stability.) + +\begin{lemma}\label{lemma:superstable} + If the potential is superstable, then + \begin{equation} + f_\gamma(\rho) + \geqslant \rho\left(\rho C-D + +\frac 1{\beta}(\log\rho-1) + \right) + . + \end{equation} +\end{lemma} + +\begin{proof} + Plugging (\ref{superstability}) into (\ref{fparticle}), we find + \begin{equation} + f_\gamma(N \gamma^{d}) + \geqslant -\frac1{\beta \gamma^{-d}}\log\frac{\gamma^{-Nd}}{N!}e^{-\beta N^2\gamma^{d}C+\beta ND} + \end{equation} + and we conclude using $N! \geqslant N^N e^{-N}$. +\end{proof} + +Superstability will thus allow us to ensure assumption 2(b) of Section \ref{sec:convergence_assumptions}. +However, the condition (\ref{superstability}) is not very explicit. +Ruelle derived \cite{ruelle1963classical} a more elementary condition on the potential that implies superstability. +Since in \cite{ruelle1963classical}, this condition is formulated only in three dimensions, and is not separated clearly from the rest of the discussion, we state Ruelle's result here and give a proof, following Ruelle's original. + +Ruelle proved \cite{ruelle1963classical} that, if $\phi$ is superstable, then $f_\gamma$ grows at least quadratically at infinity. +This result is not written explicitly in \cite{ruelle1963classical} as a theorem, so we reproduce its proof here. + + +\begin{lemma}{\rm(\cite{ruelle1963classical})}\label{lemma:super} + If $\phi(x) \geqslant \phi_0(x)$ where $\phi_0$ is continuous, Lebesgue-integrable, $\int \phi_0(x)\dd{x}>0$ and the Fourier transform of $\phi_0$ is non-negative, then $\phi$ is superstable for sufficiently small $\gamma$. +\end{lemma} + +\begin{proof} + We bound + \begin{equation} + H(x_1,\cdots,x_N) \geqslant + \sum_{i\displaystyle l} + \sum_{i,j}e^{ip(x_i-x_j)u} + \geqslant\sum_{i,j}\left(1-\sum_{n=1}^\infty \frac{p^{2(2n-1)}}{(2(2n-1))!}(\gamma^{-1}\sqrt d)^{2(2n-1)}\right) + =\\[0.5cm]\hfill + =N^2\left(1-\frac{\cosh(\gamma^{-1}p\sqrt d)-\cos(\gamma^{-1}p\sqrt d)}2\right) + . + \end{array} + \end{equation} + Defining + \begin{equation} + f(p):=\max\left\{0,\left(1-\frac{\cosh(p\sqrt d)-\cos(p\sqrt d)}2\right)\right\} + \end{equation} + we thus have + \begin{equation} + \sum_{i,j}e^{ip(x_i-x_j)u} + \geqslant + N^2f(\gamma^{-1}p) + . + \end{equation} + Plugging this into (\ref{phi2_fourier}) we have, recalling that $\hat\phi_0 \geqslant 0$, + \begin{equation} + \sum_{i,j}\phi_0(x_i-x_j)\geqslant + \frac{N^2}{(2\pi)^{\frac d2}}\int \dd{k}\hat\phi_0(k)f(\gamma^{-1}|k|) + = + \frac{N^2}{\gamma^d(2\pi)^{\frac d2}}\int \dd{k}\hat\phi_0(\gamma k)f(|k|) + . + \end{equation} + Note that $f$ has compact support and $\hat\phi_0$ is bounded (since $\phi_0$ is integrable) so, by the dominated convergence theorem, + \begin{equation} + \lim_{\gamma\to0}\int \dd{k}\hat\phi_0(\gamma k)f(|k|) + =\hat\phi_0(0)\int \dd{k}f(|k|) + \end{equation} + and so, for any $A <\hat\phi_0(0)\int \dd{k}f(|k|)$, there exists $\gamma_0$ such that, if $\gamma \geqslant \gamma_0$, then + \begin{equation} + \int \dd{k}\hat\phi_0(\gamma k)f(|k|) \geqslant A + . + \end{equation} + Putting all this together, we find that + \begin{equation} + H(x_1,\cdots,x_N)\geqslant \frac{N^2}{2 \gamma^{-d}(2\pi)^{\frac d2}}A-\frac N2\phi_0(0) + . + \end{equation} + + In summary, $H$ is superstable with $D\equiv \max\{0,\frac12\phi_0(0)\}$ and $C$ can be chosen arbitrarily close to $\frac12(2\pi)^{-\frac d2}\hat\phi_0(0)\int \dd{k}f(|k|)$ (at the expense of making $\gamma$ smaller). +\end{proof} + +We now have all the ingredients to easily prove the following proposition. + +\begin{proposition}\label{prop:convergence_assumptions} + If the potential is superstable and tempered, then the conditions of Section \ref{sec:convergence_assumptions} are satisfied. +\end{proposition} + +\begin{proof} + Conditions 1(a), and 2(a) follow immediately from (\ref{ruelle_lim1}) and Corollary \ref{cor:uniform}. + Condition 1(b) follows from (\ref{ruelle_lim2}). + Condition 2(b) an immediate consequence of Lemma \ref{lemma:superstable}. + We are left with condition 1(c). + + Suppose the potential has a hard-core: $\phi(x)=\infty$ for $|x|0$ in the vicinity of the origin. + + \item + In three dimensions, the + Lennard--Jones potential: $\phi(x)=4 \epsilon((R/|x|)^{12}-(R/|x|)^6)$. + + \item + In three dimensions, the + Morse potential: $\phi(x)=\epsilon(e^{-2 \alpha(|x|-R)}-2e^{-\alpha(|x|-R)})$ for $e^{\alpha R}>16$. + \end{itemize} +\end{proposition} + +It is proved in \cite{ruelle1963classical} that these potentials are superstable (the first bullet point is only stated in three dimensions, but its proof extends trivially to arbitrary dimensions). +The fact that they are tempered is obvious. + +\section{Continuity results} + +We prove here a useful continuity property of Gibbs measures of the $(J,\omega)$-spin models, used in the proof of Theorem \ref{thm:DS theorem}, namely that any (subsequential) limit of these measures in distribution along a convergent sequence $((J_j,\omega_j))_{j\ge 1}$, $(J_j,\omega_j)\rightarrow(J,\omega)$, is a \emph{Gibbs measure} of the $(J,\omega)$-spin model. +We note that a result of a similar flavor is proven in \cite[Theorem 4.17]{georgii2011gibbs}, although it does not apply directly to our situation. + +\begin{proposition} +\label{prop:convergence-of-Gibbs-measures} +Let $J_j,J\ge 0$, and $\omega_j,\omega$, $j\ge 1$, be Borel measures on $\R^n$ with finite, positive total measure. +Suppose that $J_j\rightarrow J$ and $\omega_j\rightarrow\omega$ in the sense of \eqref{eqn:good regions_continuity}. +For each~$j$, let $\P_j$ be a Gibbs measures of the $(J_j,\omega_j)$-spin model, and suppose that $(\P_j)_{j\ge 1}$ converges in distribution to $\P$. +Then, $\P$ is a Gibbs measure of the $(J, \omega)$-spin model. +\end{proposition} + +To prove Proposition \ref{prop:convergence-of-Gibbs-measures}, it is necessary to make the notion of prescribed boundary conditions, alluded to after \eqref{eq:box model finite-volume partition function periodic}, more explicit. +Given a finite, non-empty $\Lambda\subset\Z^d$ and a configuration $\tau:\Lambda^c\to\R^n$, the set of configurations with prescribed boundary conditions $\tau$ is +\begin{equation} + \Omega^{\Lambda,\tau}:=\set{\eta:\Z^d\to\R^n\mid\eta_v=\tau_v\text{ for all }v\in\Lambda^c}, +\end{equation} +the corresponding finite-volume Hamiltonian $H^{\Lambda,\tau}_J:\Omega^{\Lambda,\tau}\to\R$ is defined by +\begin{equation} + H^{\Lambda,\tau}_J(\eta) := J\sum_{\substack{v\sim w\\\set{v,w}\cap\Lambda\ne\emptyset}}\norm{\eta_v-\eta_w}^2, +\end{equation} +and the corresponding finite-volume Gibbs measure is the probability measure $\P_{J,\omega}^{\Lambda,\tau}$ on $\Omega^{\Lambda,\tau}$ given by +\begin{equation} + \P_{J,\omega}^{\Lambda,\tau}(\dd{\eta}) := \frac1{\Xi^{\Lambda,\tau}_{J,\omega}} e^{-H^{\Lambda,\tau}_J(\eta)}\prod_{v\in\Lambda} \omega(\dd{\eta}_v) +\end{equation} +where $\Xi^{\Lambda,\tau}_{J,\omega}$ is the normalization constant which makes $\P^{\Lambda,\tau}_{J,\omega}$ into a probability measure. + +We start by making the following observation. +\begin{lemma} +\label{lem:convergence-of-Gibbs-measures_continuity} + Let $\Lambda\subset\Z^d$ be finite and $f:\Omega\rightarrow\R$ be bounded and continuous. + Then, $\E_{J,\omega}^{\Lambda,\tau}[f]$ is continuous respectively in $(J,\omega)$ and in $\tau$. +\end{lemma} + +\begin{proof} + Recall the normalization $\bar{\omega}$ of $\omega$ from \eqref{eqn:normalized probability measure}. + Write + \begin{align} + \E_{J,\omega}^{\Lambda,\tau}[f] + {}&=\frac{1}{\bar{\Xi}_{J,\omega}^{\Lambda,\tau}}\int\prod_{v\in\Lambda}\bar{\omega}(\dd{\eta}_v)e^{-H_{J}^{\Lambda,\tau}(\eta)}f(\eta), + \label{eqn:convergence-of-Gibbs-measures_auxiliary_expectation} + \\ + \bar{\Xi}_{J,\omega}^{\Lambda,\tau}{}&:= + \int\prod_{v\in\Lambda}\bar{\omega}(\dd{\eta}_v)e^{-H_{J}^{\Lambda,\tau}(\eta)}. + \end{align} + The continuity of $\E_{J,\omega}^{\Lambda,\tau}[f]$ in $\tau$ follows from the bounded convergence theorem and the continuity of $H_{J}^{\Lambda,\tau}$ and $f$ in $\tau$. + For the continuity of $\E_{J,\omega}^{\Lambda,\tau}[f]$ in $(J,\omega)$, we rely on the following elementary observation: given $00$ such that for all $x,y\in[a,b]$ and $t\ge 0$, $\abs{e^{-tx}-e^{-ty}}\le C\abs{x-y}$. + Let $((J_j,\omega_j))_{j\ge 1}$ be a sequence converging to $(J,\omega)$ as $j\rightarrow\infty$. + We write + \begin{equation} + \label{eqn:convergence-of-Gibbs-measures_auxiliary_partition-function} + \bar{\Xi}_{J_j,\omega_j}^{\Lambda,\tau} + =\int\prod_{v\in\Lambda}\bar{\omega}_{j}(\dd{\eta}_v)\left[e^{-H_{J_j}^{\Lambda,\tau}(\eta)}-e^{-H_{J}^{\Lambda,\tau}(\eta)}\right] + +\int\prod_{v\in\Lambda}\bar{\omega}_{j}(\dd{\eta}_v)e^{-H_{J}^{\Lambda,\tau}(\eta)}. + \end{equation} + We bound the first integral as follows. + Recalling the form \eqref{Jw_ham} of the Hamiltonian and using that the sequence $(J_j)_{j\ge 1}$ is necessarily bounded, we deduce using the earlier observation that there exists a constant $C>0$ such that + \begin{equation} + \abs{\exp{-H_{J_j}^{\Lambda,\tau}(\eta)}-\exp{-H_{J}^{\Lambda,\tau}(\eta)}}\le C\abs{J_{j}-J}, + \end{equation} + so the first integral of \eqref{eqn:convergence-of-Gibbs-measures_auxiliary_partition-function} is bounded in modulus by $C\abs{J_j-J}$, which vanishes as $j\rightarrow\infty$. + For the second integral of \eqref{eqn:convergence-of-Gibbs-measures_auxiliary_partition-function}, we note that $\bar{\omega}_{j}\rightarrow\bar{\omega}$ in distribution implies the convergence in distribution of the corresponding product measures \cite[Theorem 2.8]{billingsley2013convergence}: $\prod_{v\in\Lambda}\bar{\omega}_{j}\rightarrow\prod_{v\in\Lambda}\bar{\omega}$. + By the continuity and non-negativity of the Hamiltonian, we conclude that the second integral converges to $\bar{\Xi}_{J,\omega}^{\Lambda,\tau}$ as $j\rightarrow\infty$. + Having thus shown the continuity of $\bar{\Xi}_{J,\omega}^{\Lambda,\tau}$ in $(J,\omega)$, we note that the same argument applies to the integral in \eqref{eqn:convergence-of-Gibbs-measures_auxiliary_expectation}, which completes the proof. +\end{proof} + +We now deduce Proposition \ref{prop:convergence-of-Gibbs-measures}. + +\begin{proof}[Proof of Proposition \ref{prop:convergence-of-Gibbs-measures}] +Our goal is to prove that $\P$ verifies the DLR condition \cite[Definition 2.9]{georgii2011gibbs} with the Gibbsian specifications of the $(J,\omega)$-spin model. +By \cite[Chapter 3, Proposition 4.6(b)]{ethier2009markov}, it suffices to show that, for all finite $\Lambda\subset\Z^d$ and bounded, continuous $f:\Omega\rightarrow\R$, +\begin{equation} + \E[f\mid\eta_{\Lambda^c}]=\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f], +\end{equation} +which is, in turn, verified if for all bounded, continuous, and $\mathscr{F}_{\Lambda^c}$-measurable $g:\Omega\rightarrow\R$, +\begin{equation} +\label{eqn:convergence-of-Gibbs-measures_goal} + \E[g\cdot\E[f\mid\eta_{\Lambda^c}]] + =\E[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f]]. +\end{equation} + +As $g$ is $\mathscr{F}_{\Lambda^c}$-measurable, the LHS of \eqref{eqn:convergence-of-Gibbs-measures_goal} reduces to $\E[f\cdot g]$. +In the meantime, we write the RHS of \eqref{eqn:convergence-of-Gibbs-measures_goal} as +\begin{equation} +\label{eqn:convergence-of-Gibbs-measures_rhs} +\begin{multlined} + \E[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f]] + =\left(\E[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f]] + -\E_{j}[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f]]\right) + \\ + +\E_{j}[g\cdot(\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f]-\E^{\Lambda,\eta_{\Lambda^c}}_{J_j,\omega_j}[f])] + +\E_{j}[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J_j,\omega_j}[f]], +\end{multlined} +\end{equation} +where the last term further reduces to +\begin{equation} + \E_{j}[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J_j,\omega_j}[f]] + =\E_{j}[g\cdot\E_{j}[f\mid\eta_{\Lambda^c}]] + =\E_{j}[f\cdot g]. +\end{equation} +By Lemma \ref{lem:convergence-of-Gibbs-measures_continuity}, the first two terms on the RHS of \eqref{eqn:convergence-of-Gibbs-measures_rhs} both vanish as $j\rightarrow\infty$, so +\begin{equation} + \E[g\cdot\E[f\mid\eta_{\Lambda^c}]] + =\E[f\cdot g] + =\lim_{j\rightarrow\infty}\E_{j}[f\cdot g] + =\E[g\cdot\E^{\Lambda,\eta_{\Lambda^c}}_{J,\omega}[f]], +\end{equation} +as required. +\end{proof} + +\end{document} diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..63fcd9f --- /dev/null +++ b/Makefile @@ -0,0 +1,54 @@ +PROJECTNAME=$(basename $(wildcard *.tex)) +AUX=$(addsuffix .aux, $(PROJECTNAME)) +FIGS=$(notdir $(wildcard figs/*.fig)) +BBLS=$(addsuffix .bbl, $(PROJECTNAME)) + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) +SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): $(FIGS) $(BBLS) + pdflatex -file-line-error $(patsubst %.pdf, %.tex, $@) + pdflatex -file-line-error $(patsubst %.pdf, %.tex, $@) + pdflatex -synctex=1 $(patsubst %.pdf, %.tex, $@) + +$(AUX): $(FIGS) + pdflatex -file-line-error -draftmode $(patsubst %.aux, %.tex, $@) + + +$(SYNCTEXS): $(FIGS) + pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@) + + +bib: $(BBLS) + +$(BBLS): $(AUX) + bibtex $(patsubst %.bbl, %.aux, $@) + +figs: $(FIGS) + +$(FIGS): + make -C figs/$@ + for pdf in $$(find figs/$@/ -name '*.pdf'); do ln -fs "$$pdf" ./ ; done + +clean-aux: clean-figs-aux + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + rm -f $(addsuffix .out, $(PROJECTNAME)) + rm -f $(addsuffix .toc, $(PROJECTNAME)) + +clean-figs: + $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; ) + rm -f $(notdir $(wildcard figs/*.fig/*.pdf)) + +clean-figs-aux: + $(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; ) + +clean-tex: + rm -f $(PDFS) $(SYNCTEXS) + +clean-bibliography: + rm -f $(addprefix .bbl, $PROJECTNAME) + +clean: clean-aux clean-tex clean-figs diff --git a/README b/README new file mode 100644 index 0000000..994cc99 --- /dev/null +++ b/README @@ -0,0 +1,44 @@ +This directory contains the source files to typeset the article, and generate +the figures and bibliography. This can be accomplished by running + make + + +* Dependencies: + + pdflatex + TeXlive packages: + amsmath + amssymb + amsthm + array + authblk + comment + dsfont + enumitem + geometry + hyperref + mathrsfs + mathtools + natbib + physics + pgf + pgfplots + standalone + subcaption + ulem + xargs + xcolor + GNU make + BibTeX + +* Files: + + He_Jauslin_Lebowitz_Peled_2025.tex: + main LaTeX file + + bibliography.bib: + BibTeX database + + figs: + source code for the figures + diff --git a/bibliography.bib b/bibliography.bib new file mode 100644 index 0000000..bc91a0f --- /dev/null +++ b/bibliography.bib @@ -0,0 +1,630 @@ +@book{billingsley2013convergence, + title={Convergence of probability measures}, + author={Billingsley, Patrick}, + year={2013}, + publisher={John Wiley \& Sons} +} + +@article{frohlich1978phase, + title={Phase transitions in anisotropic lattice spin systems}, + author={Fr{\"o}hlich, J{\"u}rg and Lieb, Elliott H}, + journal={Communications in Mathematical Physics}, + 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The Bakerian Lecture.---On the continuity of the gaseous and liquid states of matter}}, + author={Andrews, Thomas}, + journal={Philosophical Transactions of the Royal Society of London}, + number={159}, + pages={575--590}, + year={1869}, + publisher={The Royal Society London} +} + +@article{biskup2003rigorous, + title={Rigorous analysis of discontinuous phase transitions via mean-field bounds}, + author={Biskup, Marek and Chayes, Lincoln}, + journal={Communications in mathematical physics}, + volume={238}, + number={1}, + pages={53--93}, + year={2003}, + publisher={Springer} +} + +@article{biskup2006mean, + title={Mean-field driven first-order phase transitions in systems with long-range interactions}, + author={Biskup, Marek and Chayes, Lincoln and Crawford, Nicholas}, + journal={Journal of Statistical Physics}, + volume={122}, + number={6}, + pages={1139--1193}, + year={2006}, + publisher={Springer} +} \ No newline at end of file diff --git a/figs/double_tangent.fig/Makefile b/figs/double_tangent.fig/Makefile new file mode 100644 index 0000000..e98abe6 --- /dev/null +++ b/figs/double_tangent.fig/Makefile @@ -0,0 +1,20 @@ +PROJECTNAME=$(basename $(wildcard *.tex)) + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): + pdflatex $(patsubst %.pdf, %.tex, $@) + + + +clean-aux: + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + rm -f $(addsuffix .out, $(PROJECTNAME)) + +clean-tex: + rm -f $(PDFS) $(SYNCTEXS) + +clean: clean-aux clean-tex diff --git a/figs/double_tangent.fig/double_tangent.tikz.tex b/figs/double_tangent.fig/double_tangent.tikz.tex new file mode 100644 index 0000000..8e9745e --- /dev/null +++ b/figs/double_tangent.fig/double_tangent.tikz.tex @@ -0,0 +1,18 @@ +\documentclass[tikz]{standalone} + +\begin{document} + +\begin{tikzpicture} + \draw[->,thick] (-0.5,0.5) -- (9,0.5) node[right] {$\rho$}; + \draw[->,thick] (0,-4.5) -- (0,1.5) node[above] {$f(T,\rho)$}; + + \draw[thick, domain=0.6:8.5, smooth] plot (\x, {-2.282*(1.3*(\x-0.6))+0.926*(1.3*(\x-0.6))^2-0.153*(1.3*(\x-0.6))^3+0.0081*(1.3*(\x-0.6))^4}); + \draw[dashed,thick] (1.8,-1.84) -- (6.7,-4.12); + \node[above] at (8.2,-1.3) {$T,thick] (-0.5,0) -- (9,0) node[right] {$1/\rho$}; + \draw[->,thick] (0,-0.5) -- (0,5.5) node[above] {$p(T,\rho)$}; + + \begin{scope}[scale=1.8] + \fill[liq2, opacity=0.5] + (0.604,0.64) -- + plot[domain=0.604:2.403, smooth, samples=100] (\x, {7.2/(3*\x-1)-3/(\x^2)}) -- + (2.403,0.64) -- cycle; + \begin{scope} + \clip (0, 0) rectangle (5,3); + \draw[thick, domain=0.4:4.5, smooth,samples=100] plot (\x, {9/(3*\x-1)-3/(\x^2)}); + \draw[ice2,thick, domain=0.4:4.5, smooth,samples=100] plot (\x, {8/(3*\x-1)-3/(\x^2)}); + \draw[thick, domain=0.4:4.5, smooth,samples=100] plot (\x, {7.2/(3*\x-1)-3/(\x^2)}); + \end{scope} + \node[above] at (4,0.7) {$T>T_c$}; + \node[ice2,right] at (4.5,0.5) {$T=T_c$}; + \node[above] at (4,0.1) {$T,thick] (0,0) -- (11.4,0) node [right] {$T$}; + \draw[->,thick] (0,0) -- (0,8.3) node [left] {$P$}; + + \draw (7,1.5) node {vapor}; + \draw (5,4.5) node {liquid}; + \draw (1,2.5) node {solid}; + + \begin{scope} + \clip (0,0) rectangle (11,8); + \begin{scope}[shift={(9,6)}] + \begin{axis}[ + opacity=70, + axis equal=true, + hide axis, + domain=15:75, + domain y=0:1, + view={0}{90}, + shader=interp, + colormap={liqvap}{ + color=(vap1) color=(liq2) + }, + anchor=origin, + data cs=polar, + ] + \addplot 3 [surf] {x}; + \end{axis} + \end{scope} + \end{scope} + + \fill (9,6) circle (0.1); + + \end{tikzpicture} +\end{document}