From 023ea227e93af47ae9a62f6d1813d59c1583ca0f Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Wed, 5 Aug 2020 00:58:37 -0400
Subject: As presented on 2020-08-05 at Harvard University

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+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{amssymb}
+\usepackage{dsfont}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Crystalline ordering\par
+\smallskip
+\hfil in hard-core lattice particle systems\par
+\vfil
+\large
+\hfil Ian Jauslin
+\normalsize
+\vfil
+\hfil\rm joint with {\bf Joel L. Lebowitz}\par
+\vfil
+arXiv: \vbox{
+  \hbox{\tt \href{http://arxiv.org/abs/1705.02032}{1705.02032}}
+  \hbox{\tt \href{http://arxiv.org/abs/1708.01912}{1708.01912}}
+}
+\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Crystallization in particle systems}
+\begin{itemize}
+  \item Gas-liquid-solid paradigm: not yet understood mathematically in realistic particle models.
+  \item Crystallization: long range positional order.
+  \item Here: discuss a class of hard-core lattice particle models for which we can prove crystallization.
+\end{itemize}
+\vfill
+\eject
+
+\title{Hard-core lattice particle (HCLP) systems}
+\vfill
+\hfil\includegraphics[width=1.2cm]{diamond.pdf}
+\hfil\includegraphics[width=1.2cm]{cross.pdf}
+\hfil\includegraphics[width=1.2cm]{hexagon.pdf}
+\par
+\vfill
+\hfil\includegraphics[width=0.9cm]{V_triomino.pdf}
+\hfil\includegraphics[width=0.9cm]{T_tetromino.pdf}
+\hfil\includegraphics[width=0.9cm]{L_tetromino.pdf}
+\hfil\includegraphics[width=0.9cm]{P_pentomino.pdf}
+\vfill
+\eject
+
+\title{Non-sliding HCLPs}
+\begin{itemize}
+  \item There exist a {\bf finite} number $\tau$ of tilings $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ which are \penalty-1000{\bf periodic} and {\bf isometric} to each other.
+\end{itemize}
+\hfil\includegraphics[width=4cm]{cross_packing_1.pdf}
+\hfil\includegraphics[width=4cm]{cross_packing_1.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Non-sliding HCLPs}
+\begin{itemize}
+  \item There exist a {\bf finite} number $\tau$ of tilings $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ which are \penalty-1000{\bf periodic} and {\bf isometric} to each other.
+\end{itemize}
+\hfil\includegraphics[width=4cm]{cross_packing_l_2.pdf}
+\hfil\includegraphics[width=4cm]{cross_packing_r_2.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Non-sliding HCLPs}
+\begin{itemize}
+  \item There exist a {\bf finite} number $\tau$ of tilings $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ which are \penalty-1000{\bf periodic} and {\bf isometric} to each other.
+\end{itemize}
+\hfil\includegraphics[width=4cm]{cross_packing_l.pdf}
+\hfil\includegraphics[width=4cm]{cross_packing_r.pdf}
+\vfill
+\eject
+
+\title{Non-sliding HCLPs}
+\begin{itemize}
+  \item Defects are {\bf localized}: for every connected particle configuration $X$ that is {\it not} the subset of a close packing and every $Y\supset X$, there is empty space in $Y$ neighboring $X$.
+\end{itemize}
+\vfill
+\hfil\includegraphics[width=2.1cm]{cross_sliding_2.pdf}
+\hfil\includegraphics[width=2.1cm]{cross_sliding_3a.pdf}
+\hfil\includegraphics[width=2.4cm]{cross_sliding_3b.pdf}
+\vfill
+\eject
+
+\title{Example of a sliding HCLP}
+\begin{itemize}
+  \item $2\times2$ squares:
+\end{itemize}
+\hfil\includegraphics[height=4.5cm]{square_packing_slide.pdf}
+\hfil\includegraphics[height=4.5cm]{hole_example_square.pdf}
+\vfill
+\eject
+
+\title{Observables}
+\begin{itemize}
+  \item Gibbs measure:
+  $$
+    \left<A\right>_{\nu}
+    :=
+    \lim_{\Lambda\to\Lambda_\infty}
+    \frac1{\Xi_{\Lambda,\nu}(z)}
+    \sum_{X\subset\Lambda}A(X)z^{|X|}\mathfrak B_\nu(X)\prod_{x\neq x'\in X}\varphi(x,x')
+  $$
+  \vskip-10pt
+  \begin{itemize}
+    \item $\Lambda$: finite subset of lattice $\Lambda_\infty$.
+    \item $z\geqslant 0$: fugacity.
+    \item $\varphi(x,x')$: hard-core interaction.
+    \item $\mathfrak B_\nu$: boundary condition: favors the $\nu$-th tiling.
+  \end{itemize}
+  \vskip-5pt
+
+  \item Pressure:
+  \vskip-10pt
+  $$
+    p(z):=\lim_{\Lambda\to\Lambda_\infty}\frac1{|\Lambda|}\log\Xi_{\Lambda,\nu}(z).
+  $$
+\end{itemize}
+\vfill\eject
+
+\title{Theorem}
+\begin{itemize}
+  \item $p(z)-\rho_m\log z$ and $\left<\mathds 1_{x_1}\cdots\mathds 1_{x_n}\right>_\nu$ are {\bf analytic} functions of $1/z$ for large values of $z$.
+  \vfill
+
+  \item There are $\tau$ distinct Gibbs states:
+  $$
+  \left<\mathds 1_x\right>_\nu=
+  \left\{\begin{array}{ll}
+    1+O(z^{-1})&\mathrm{\ if\ }x\in\mathcal L_\nu\\[0.3cm]
+    O(z^{-1})&\mathrm{\ if\ not}
+    .
+  \end{array}\right.
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Low-fugacity expansion}
+\begin{itemize}
+  \item Partition function: $Z_\Lambda(n)$: number of configurations with $n$ particles:
+  $$
+    \Xi_\Lambda(z)
+    =
+    \sum_{X\subset\Lambda}z^{|X|}\prod_{x\neq x'\in X}\varphi(x,x')
+    =\sum_{n=0}^\infty z^n Z_\Lambda(n)
+  $$
+  \vskip-10pt
+  \item Formally,
+  $$
+    \frac1{|\Lambda|}\log\Xi_\Lambda(z)
+    =
+    \sum_{k=1}^\infty b_k(\Lambda)z^k
+  $$
+  where, if $Z_\Lambda(k_i)$ denotes the number of configurations with $k_i$ particles, then
+  $$
+    b_k(\Lambda):=\frac1{|\Lambda|}
+    \sum_{j=1}^k\frac{(-1)^{j+1}}j
+    \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Z_\Lambda(k_1)\cdots Z_\Lambda(k_j)
+  $$
+  \vskip-10pt
+\end{itemize}
+\eject
+
+\title{Low-fugacity expansion}
+\begin{itemize}
+  \item Partition function:
+  $$
+    \Xi_\Lambda(z)
+    =1+z Z_\Lambda(1)+z^2 Z_\Lambda(2)+\cdots
+  $$
+  \item Second term:
+  $$
+    b_2(\Lambda)=\frac1{|\Lambda|}\left(Z_\Lambda(2)-\frac12Z_\Lambda^2(1)\right)
+  $$
+  \item $Z_\Lambda(2)$: counts interacting particle configurations: $O(\Lambda^2)$.
+  \item $\frac12 Z_\Lambda^2(1)$: counts non-interacting particle configurations: $O(\Lambda^2)$.
+  \item The terms of order $|\Lambda|^2$ cancel out, so $b_2(\Lambda)$ has a limit as $\Lambda\to\Lambda_\infty$.
+\end{itemize}
+\eject
+
+\title{Low-fugacity expansion}
+\begin{itemize}
+  \item \href{http://dx.doi.org/10.1017/S0305004100011191}{[Ursell, 1927]}, \href{http://dx.doi.org/10.1063/1.1749933}{[Mayer, 1937]}: $b_k(\Lambda)\to b_k$.
+  \item \href{http://dx.doi.org/10.1016/0031-9163(62)90198-1}{[Groeneveld, 1962]}, \href{http://dx.doi.org/10.1016/0003-4916(63)90336-1}{[Ruelle, 1963]}, \href{http://dx.doi.org/10.1063/1.1703906}{[Penrose, 1963]}: 
+  $$
+    p(z)=\sum_{k=1}^\infty b_kz^k
+  $$
+  which has a positive radius of convergence.
+\end{itemize}
+\eject
+
+\title{High-fugacity expansion}
+\begin{itemize}
+  \item Partition function: $Z_\Lambda(n)$: number of configurations with $n$ particles:
+  $$
+    \Xi_\Lambda(z)
+    =
+    \sum_{X\subset\Lambda}z^{|X|}\prod_{x\neq x'\in X}\varphi(x,x')
+    =\sum_{n=0}^{N_{\mathrm{max}}} z^n Z_\Lambda(n)
+  $$
+  \item Inverse fugacity $y\equiv z^{-1}$:
+  $$
+    \Xi_\Lambda(z)=
+    z^{N_{\mathrm{max}}}
+    \sum_{X\subset\Lambda}
+    y^{N_{\mathrm{max}}-|X|}\prod_{x\neq x'\in X}\varphi(x,x')
+    =z^{N_{\mathrm{max}}}\sum_{n=0}^{N_{\mathrm{max}}}y^n Q_\Lambda(n)
+  $$
+  with $Q_\Lambda(n)\equiv Z_\Lambda(N_{\mathrm{max}}-n)$.
+\end{itemize}
+\eject
+
+\title{High-fugacity expansion}
+\begin{itemize}
+  \item Formally,
+  $$
+    \frac1{|\Lambda|}\log\Xi_\Lambda
+    =
+    \rho_m\log z
+    +
+    \sum_{k=1}^\infty c_k(\Lambda)y^k
+  $$
+  where $\rho_m=\frac{N_{\mathrm{max}}}{|\Lambda|}$,
+  $$
+    c_k(\Lambda):=\frac1{|\Lambda|}
+    \sum_{j=1}^k\frac{(-1)^{j+1}}{j\tau^j}
+    \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Q_\Lambda(k_1)\cdots Q_\Lambda(k_j)
+  $$
+\end{itemize}
+\eject
+
+\title{High-fugacity expansion}
+\vfill
+\hfil\includegraphics[width=2cm]{gf_diamond1.pdf}
+\hfil\includegraphics[width=2.33cm]{gf_diamond2.pdf}
+\par\vfill
+\hfil\includegraphics[width=2.33cm]{gf_diamond3.pdf}
+\vfill\eject
+
+\title{High-fugacity expansion}
+\begin{itemize}
+  \item \href{http://dx.doi.org/10.1063/1.1697217}{[Gaunt, Fisher, 1965]}: diamonds: $c_k(\Lambda)\to c_k$ for $k\leqslant 9$.
+  \item \href{http://dx.doi.org/10.1098/rsta.1988.0077}{[Joyce, 1988]}: hexagons (integrable, \href{http://dx.doi.org/10.1088/0305-4470/13/3/007}{[Baxter, 1980]}).
+  \item \href{http://dx.doi.org/10.1209/epl/i2005-10166-3}{[Eisenberg, Baram, 2005]}: crosses: $c_k(\Lambda)\to c_k$ for $k\leqslant 6$.
+  \item Cannot be done {\it systematically}: there exist counter-examples: e.g. hard $2\times2$ squares on $\mathbb Z^2$:
+  $$
+    c_1(\Lambda)\propto\sqrt{|\Lambda|}
+  $$
+\end{itemize}
+\eject
+
+\title{Holes interact}
+\begin{itemize}
+  \item Total volume of holes: $\in\rho_m^{-1}\mathbb N$.
+\end{itemize}
+\vfill
+\hfil\includegraphics[height=4.5cm]{hole_example_cross.pdf}
+\hfil\includegraphics[height=4.5cm]{hole_example_square.pdf}
+\vfill
+\eject
+
+\title{Non-sliding condition}
+\begin{itemize}
+  \item Distinct defects are decorrelated.
+\end{itemize}
+\vfill
+\hfil\includegraphics[height=5cm]{hole_example_cross_decorrelated.pdf}
+\vfill
+\eject
+
+\title{Gaunt-Fisher configurations}
+\begin{itemize}
+  \item Group together empty space and neighboring particles.
+\end{itemize}
+\vfill
+\hfil\includegraphics[width=2.5cm]{gaunt_fisher2.pdf}
+\hfil\includegraphics[width=4cm]{gaunt_fisher3.pdf}
+\vfill
+\eject
+
+\title{Defect model}
+\vskip-5pt
+\begin{itemize}
+  \item Map particle system to a model of defects:
+  $$
+    \Xi_{\Lambda,\nu}(z)=z^{\rho_m|\Lambda|}\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
+    \left(\prod_{\gamma\neq\gamma'\in\underline\gamma}\Phi(\gamma,\gamma')\right)
+    \prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
+  $$
+  \begin{itemize}
+    \item $\Phi$: hard-core repulsion of defects.
+    \item $\zeta_\nu^{(z)}(\gamma)$: activity of defect.
+  \end{itemize}
+  \item The activity of a defect is exponentially small: $\exists\epsilon\ll 1$
+  $$
+    \zeta_\nu^{(z)}(\gamma)<\epsilon^{|\gamma|}
+  $$
+  \vskip-5pt
+  \item Low-fugacity expansion for defects.
+\end{itemize}
+\eject
+
+\title{Crystallization}
+\vfill
+\begin{itemize}
+  \item Peierls argument: in order to have a particle at $x$ that is not compatible with the $\nu$-th perfect packing, it must be part of or surrounded by a defect.
+  \vfill
+  \item Note: a naive Peierls argument requires the partition function to be independent from the boundary condition. This is not necessarily the case here, and we need elements from Pirogov-Sinai theory.
+\end{itemize}
+\vfill
+\eject
+
+\title{Lee-Yang zeros}
+\begin{itemize}
+  \item Lee-Yang zeros: roots of $\Xi_\Lambda(z)$ $\Longleftrightarrow$ singularities of $p_\Lambda(z)$.
+  \item Whenever the high fugacity expansion has a radius of convergence $\tilde R$, there are no Lee-Yang zeros outside of a disc of radius $\tilde R^{-1}$.
+\end{itemize}
+
+\hfil\includegraphics[height=4cm]{lee_yang.pdf}
+
+
+
+\end{document}
-- 
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