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diff --git a/Jauslin_IAS-Postdoc_2017.tex b/Jauslin_IAS-Postdoc_2017.tex new file mode 100644 index 0000000..c1b22a2 --- /dev/null +++ b/Jauslin_IAS-Postdoc_2017.tex @@ -0,0 +1,66 @@ +\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} |