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\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}
