As presented at Harvard University on 2018-10-03HEADv1.0master

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+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{dsfont}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Liquid crystals and the\par
+\smallskip
+\hfil Heilmann-Lieb model\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{Gas-liquid-crystal}
+\vfill
+\hfil
+\includegraphics[width=3cm]{gas.png}
+\includegraphics[width=3cm]{liquid.png}
+\includegraphics[width=3cm]{crystal.png}
+\vfill
+\eject
+
+\title{Liquid crystals}
+\begin{itemize}
+  \item Orientational order and positional disorder.
+\end{itemize}
+\hfil\includegraphics[width=4.5cm]{nematic.png}
+\hfil\includegraphics[width=4.5cm]{chiral.png}
+\vfill
+\eject
+
+\title{History}
+\begin{itemize}
+  \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$.
+  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers.
+\vphantom{
+  \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$ with a {\it finite} number of orientations.
+  \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length).
+  \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$.
+}
+\end{itemize}
+\vfill
+\eject
+
+
+\title{Heilmann-Lieb model}
+\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
+\vfil
+\hfil\includegraphics[width=5cm]{grid.pdf}
+\vfil\eject
+
+\addtocounter{page}{-1}
+\title{Heilmann-Lieb model}
+\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
+\vfil
+\hfil\includegraphics[width=5cm]{dimers.pdf}
+\vfil\eject
+
+\addtocounter{page}{-1}
+\title{Heilmann-Lieb model}
+\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}
+\vfil
+\hfil\includegraphics[width=5cm]{interaction.pdf}
+\vfil\eject
+
+\title{Heilmann-Lieb conjecture}
+\begin{itemize}
+  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: proved orientational order using reflection positivity.
+  \item HL Conjecture: absence of positional order.
+\vphantom{
+  \item \href{http://dx.doi.org/10.1007/s10955-015-1421-8}{[Alberici, 2016]}: different fugacities for horizontal and vertical dimers.
+  \item \href{http://dx.doi.org/10.1103/PhysRevB.89.035128}{[Papanikolaou, Charrier, Fradkin, 2014]}: numerics.
+}
+\end{itemize}
+\vfill
+\eject
+
+\title{History}
+\begin{itemize}
+  \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$.
+  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers.
+  \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$ with a {\it finite} number of orientations.
+  \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length).
+  \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$.
+\end{itemize}
+\vfill
+\eject
+
+\title{Heilmann-Lieb conjecture}
+\begin{itemize}
+  \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: proved orientational order using reflection positivity.
+  \item HL Conjecture: absence of positional order.
+  \item \href{http://dx.doi.org/10.1007/s10955-015-1421-8}{[Alberici, 2016]}: different fugacities for horizontal and vertical dimers.
+  \item \href{http://dx.doi.org/10.1103/PhysRevB.89.035128}{[Papanikolaou, Charrier, Fradkin, 2014]}: numerics.
+\end{itemize}
+\vfill
+\eject
+
+
+\title{Heilmann-Lieb model}
+\begin{itemize}
+  \item Grand-canonical Gibbs measure:
+  $$
+    \left<A\right>_{\mathrm v}
+    :=
+    \lim_{\Lambda\to\mathbb Z^2}
+    \frac1{\Xi_{\mathrm v}(\Lambda)}
+    \sum_{\underline\delta\in\Omega_{\mathrm v}(\Lambda)}A(\underline\delta)z^{|\underline\delta|}\prod_{\delta\neq \delta'\in \underline\delta}e^{\frac12J\mathds 1_{\delta\sim\delta'}}
+  $$
+  \vskip-15pt
+  \begin{itemize}
+    \item $\Lambda$: finite box.
+    \item $\Omega_{\mathrm v}(\Lambda)$: non-overlapping dimer configurations satisfying the boundary condition.
+    \item $z\geqslant 0$: fugacity.
+    \item $J\geqslant 0$: interaction strength.
+    \item $\mathds 1_{\delta\sim\delta'}$ indicator that dimers are adjacent and aligned.
+  \end{itemize}
+\end{itemize}
+\vfill
+\eject
+
+\title{Boundary condition}
+\begin{itemize}
+  \item Fix length $\ell_0:=e^{\frac32J}\sqrt z$,
+\end{itemize}
+\hfil\includegraphics[width=5cm]{boundary.pdf}
+
+\title{Theorem}
+For $1\ll z\ll J$, $\|(x,y)\|_{\mathrm{HL}}:=J|x|+e^{-\frac32J}z^{-\frac12}|y|$,
+\begin{itemize}
+  \item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and
+  $$
+    \begin{array}{c}
+      \left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12}))
+      \\[0.3cm]
+      \left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
+      -\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}
+      \left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
+      =O(e^{-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})})
+    \end{array}
+  $$
+  \vskip-5pt
+  \item Given two horizontal edges $e_{\mathrm h},f_{\mathrm h}$, $\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm h}$ and
+  $$
+    \begin{array}{c}
+      \left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J})
+      \\[0.3cm]
+      \left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
+      -\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}
+      \left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
+      =O(e^{-6J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm h},f_{\mathrm h})})
+    \end{array}
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{1D system}
+\begin{itemize}
+  \item {\it Only} vertical dimers: integrable.
+  \item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and
+  $$
+    \begin{array}{c}
+      \left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12}))
+      \\[0.3cm]
+      \left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
+      -\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}
+      \left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
+      =O(e^{-c\ \mathrm{dist}_{\mathrm{1D}}(e_{\mathrm v},f_{\mathrm v})})
+    \end{array}
+  $$
+  with $\|(x,y)\|_{\mathrm{1D}}:=e^{-\frac32J}z^{-\frac12}|y|$.
+\end{itemize}
+\vfill
+\eject
+
+\title{Loop model}
+\vfill
+\hfil\includegraphics[width=5cm]{dimer_contour.pdf}
+\begin{itemize}
+  \item Weight of a loop of length $|l|$: $e^{-\frac12J|l|}$.
+\end{itemize}
+\vfill
+\eject
+
+\title{Difficulty: loops interact}
+\vskip-10pt
+\begin{itemize}
+  \item Correlated dimers induce an interaction between loops, which decays exponentially with a rate $e^{-\frac32J}z^{-\frac12}$.
+\end{itemize}
+\hfil\includegraphics[width=3.5cm]{segments.pdf}
+\begin{itemize}
+  \item Vertical-to-horizontal boundaries and horizontal-to-vertical ones have different geometries.
+\end{itemize}
+\vfill
+\eject
+
+\title{Pirogov-Sinai}
+\hfil\includegraphics[width=5cm]{dimer_contour.pdf}
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Pirogov-Sinai}
+\hfil\includegraphics[width=5cm]{dimer_pirogov_sinai.pdf}
+$$
+  \Xi_{\mathrm v}(\Lambda)=\Xi_{\mathrm v}(\mathrm{Out})\ \Xi_{\mathrm h}(\mathrm{In})
+$$
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Pirogov-Sinai}
+\hfil\includegraphics[width=5cm]{dimer_pirogov_sinai_preflip.pdf}
+$$
+  \Xi_{\mathrm v}(\Lambda)=\Xi_{\mathrm v}(\mathrm{Out})\ \Xi_{\mathrm h}(\mathrm{In})
+$$
+\vfill
+\eject
+
+\addtocounter{page}{-1}
+\title{Pirogov-Sinai}
+\hfil\includegraphics[width=5cm]{dimer_pirogov_sinai_flip.pdf}
+$$
+  \Xi_{\mathrm v}(\Lambda)=\Xi_{\mathrm v}(\mathrm{Out})\ \Xi_{\mathrm v}(\mathrm{In})\eta_{\mathrm h,\mathrm v}(\mathrm{In})
+  ,\quad
+  \eta_{\mathrm h,\mathrm v}(\mathrm{In})=
+  \frac{\Xi_{\mathrm h}(\mathrm{In})}{\Xi_{\mathrm v}(\mathrm{In})}
+$$
+\vfill
+\eject
+
+\title{Pirogov-Sinai}
+\begin{itemize}
+  \item Boundary term:
+  $$
+    \eta_{\mathrm h,\mathrm v}(\mathrm{In})
+    \leqslant e^{c|\partial\mathrm{In}|}
+  $$
+  \item Energy gain:
+  $$
+    e^{-\frac12J|\partial\mathrm{In}|}e^{c|\partial\mathrm{In}|}\ll 1
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Entropy of contours}
+\begin{itemize}
+  \item Loops at a distance $<e^{\frac32J}z^{\frac12}\equiv\ell_0$ form a single object: a {\it contour}.
+\end{itemize}
+\hfil\includegraphics[width=5cm]{contours.pdf}
+\vfill
+\eject
+
+\title{Entropy of contours}
+\begin{itemize}
+  \item Weight of a contour:
+  \begin{itemize}
+    \item $e^{-\frac12J|l|}$ for each loop $l$.
+    \item $e^{-\ell_0^{-1}|\sigma|}$ for each segment $\sigma$.
+  \end{itemize}
+  \item Each new loop in a contour contributes $e^{-3J}\ell_0\equiv e^{-\frac32J}z^{\frac12}\ll 1$
+\end{itemize}
+\hfil\includegraphics[height=3cm]{smallest.pdf}
+\vfill
+\eject
+
+\title{Parameter regime}
+\vfill
+\hfil\includegraphics[width=8cm]{regime.pdf}
+
+
+\end{document}