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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\par
\smallskip
\hfil and the 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{Liquid crystals}
\begin{itemize}
\item Orientational order and positional disorder.
\end{itemize}
\hfil\includegraphics[width=5cm]{nematic.png}
\hfil\includegraphics[width=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{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 $\mathbb R^2$.
\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}
\vfill
\begin{itemize}
\item \href{http://dx.doi.org/10.1007/s10955-015-1421-8}{[Alberici, 2016]}: different fugacities for horizontal and vertical dimers.
\vfill
\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:
$$
\left<A\right>_{\mathrm v}
:=
\lim_{\Lambda\to\mathbb Z^2}
\frac1{\Xi_{\Lambda,\mathrm v}(z)}
\sum_{\underline\delta\in\Omega_{\mathrm v}(\Lambda)}A(\underline\delta)z^{|\underline\delta|}\prod_{\delta\neq \delta'\in \underline\delta}e^{\frac12J\mathds 1_{\delta\sim\delta'}}
$$
\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{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
$$
\begin{array}{c}
\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J})
\\[0.3cm]
\left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v}
-\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}
\left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v}
=O(e^{-3J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})})
\end{array}
$$
\end{itemize}
\vfill
\eject
\title{1D system}
\begin{itemize}
\item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: mostly vertical dimers.
\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{Peierls argument}
\vfill
\hfil\includegraphics[width=5cm]{dimer_contour.pdf}
\vfill
\end{document}
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