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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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