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\bf\Large
\hfil Liquid crystals\par
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\hfil and interacting dimers\par
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\hfil Ian Jauslin
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\hfil\rm joint with {\bf Elliott H. Lieb}\par
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arXiv:{\tt \href{http://arxiv.org/abs/1709.05297}{1709.05297}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
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\title{Nematic liquid crystals}
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\hfil\includegraphics[width=6cm]{nematic.png}
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\title{Nematic liquid crystals}
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\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}
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\title{Heilmann-Lieb model}
\hfil[Heilmann, Lieb, 1979]
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\hfil\includegraphics[width=5cm]{interaction.pdf}
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\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]}
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\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}

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