From 743bbfd8a4a2a4173f309e1be98712f316fdf797 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Fri, 29 Sep 2017 03:55:25 +0000
Subject: As presented at the IAS on 2017-09-29

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+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Liquid crystals\par
+\smallskip
+\hfil and interacting dimers\par
+\vfil
+\large
+\hfil Ian Jauslin
+\normalsize
+\vfil
+\hfil\rm joint with {\bf Elliott H. Lieb}\par
+\vfil
+arXiv:{\tt \href{http://arxiv.org/abs/1709.05297}{1709.05297}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Nematic liquid crystals}
+\vfil
+\hfil\includegraphics[width=6cm]{nematic.png}
+\vfil\eject
+
+\title{Nematic liquid crystals}
+\vfil
+\begin{itemize}
+  \item {\bf Long range orientational order}: molecules tend to align, and maintain their alignment over macroscopic distances.
+  \item {\bf No positional order}: the locations of the centers of the molecules are decorrelated.
+\end{itemize}
+\vfil\eject
+
+\title{Heilmann-Lieb model}
+\hfil[Heilmann, Lieb, 1979]
+\vfil
+\hfil\includegraphics[width=5cm]{interaction.pdf}
+\vfil\eject
+
+\title{Heilmann-Lieb model}
+\begin{itemize}
+  \item Probability of a configuration (grand-canonical Gibbs distribution):
+  $$
+    \mathrm{Prob}(\mathrm{conf})=\frac1\Xi z^{\#\mathrm{particles}}e^{J\ \#\mathrm{interactions}}
+  $$
+  \begin{itemize}
+    \item $\Xi$: partition function
+    \item $z\geqslant 0$: activity
+    \item $J\geqslant 0$: interaction strength
+  \end{itemize}
+  \item Regime $J\gg z\gg 1$.
+\end{itemize}
+
+\title{[Heilmann, Lieb, 1979]}
+\vfil
+\begin{itemize}
+  \item {\it Theorem}: given $x,y\in\mathbb Z^2$, the probability that there is a horizontal dimer attached to $x$ and no horizontal dimer attached to $y$ tends to 0 as $J,z\to\infty$. ({\bf Orientational order}.)
+  \item {\it Conjecture}: given to edges $e$ and $e'$, the probability of finding a dimer on $e$ and another on $e'$ is independent of $e$ and $e'$, up to a term that decays {\it exponentially} in $\mathrm{dist}(e,e')$. ({\bf No positional order}.)
+\end{itemize}
+
+\end{document}
-- 
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