\documentclass{ian-presentation}

\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
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\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil High density phases\par
\smallskip
\hfil of hard-core lattice particle systems\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Joel L. Lebowitz} and {\bf Elliott H. Lieb}\par
\vfil
arXiv: \vbox{
  \hbox{\tt \href{http://arxiv.org/abs/1708.01912}{1708.01912}}
  \hbox{\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{Hard-core lattice particle (HCLP) systems}
\vfill
\hfil\includegraphics[width=1.2cm]{diamond.pdf}
\hfil\includegraphics[width=1.2cm]{cross.pdf}
\hfil\includegraphics[width=1.2cm]{hexagon.pdf}
\par
\vfill
\hfil\includegraphics[width=0.9cm]{V_triomino.pdf}
\hfil\includegraphics[width=0.9cm]{T_tetromino.pdf}
\hfil\includegraphics[width=0.9cm]{L_tetromino.pdf}
\hfil\includegraphics[width=0.9cm]{P_pentomino.pdf}
\vfill
\eject

\title{Non-sliding HCLPs}
\begin{itemize}
  \item There exist a {\bf finite} number $\tau$ of tilings which are \penalty-1000{\bf periodic} and {\bf isometric} to each other.
\end{itemize}
\hfil\includegraphics[width=4cm]{cross_packing_l.pdf}
\hfil\includegraphics[width=4cm]{cross_packing_r.pdf}
\vfill
\eject

\title{Non-sliding HCLPs}
\begin{itemize}
  \item Defects are {\bf localized}: for every connected particle configuration $X$ that is {\it not} the subset of a close packing and every $Y\supset X$, there is empty space in $Y$ neighboring $X$.
\end{itemize}
\vfill
\hfil\includegraphics[width=2.1cm]{cross_sliding_2.pdf}
\hfil\includegraphics[width=2.1cm]{cross_sliding_3a.pdf}
\hfil\includegraphics[width=2.4cm]{cross_sliding_3b.pdf}
\vfill
\eject

\title{Observables}
\begin{itemize}
  \item Gibbs measure:
  $$
    \left<A\right>_{\nu}
    :=
    \lim_{\Lambda\to\Lambda_\infty}
    \frac1{\Xi_{\Lambda,\nu}(z)}
    \sum_{X\subset\Lambda}A(X)z^{|X|}\mathfrak B_\nu(X)\prod_{x\neq x'\in X}\varphi(x,x')
  $$
  \vskip-10pt
  \begin{itemize}
    \item $\Lambda$: finite subset of lattice $\Lambda_\infty$.
    \item $z\geqslant 0$: fugacity.
    \item $\varphi(x,x')$: hard-core interaction.
    \item $\mathfrak B_\nu$: boundary condition: favors the $\nu$-th tiling.
  \end{itemize}
  \vskip-5pt

  \item Pressure:
  \vskip-10pt
  $$
    p(z):=\lim_{\Lambda\to\Lambda_\infty}\frac1{|\Lambda|}\log\Xi_{\Lambda,\nu}(z).
  $$
\end{itemize}
\vfill\eject

\title{Theorem}
\begin{itemize}
  \item $p(z)-\rho_m\log z$ and $\left<\mathds 1_{x_1}\cdots\mathds 1_{x_n}\right>_\nu$ are {\bf analytic} functions of $1/z$ for large values of $z$.
  \vfill

  \item There are $\tau$ distinct Gibbs states:
  $$
  \left<\mathds 1_x\right>_\nu=
  \left\{\begin{array}{ll}
    1+O(y)&\mathrm{\ if\ }x\in\mathcal L_\nu\\[0.3cm]
    O(y)&\mathrm{\ if\ not}
    .
  \end{array}\right.
  $$
\end{itemize}
\vfill
\eject

\title{High-fugacity expansion}
$$
  p(y)=-\rho_m\log y+\sum_{k=1}^\infty c_k y^k
$$
\begin{itemize}
  \item \href{http://dx.doi.org/10.1063/1.1697217}{[Gaunt, Fisher, 1965]}: diamonds: for $k\leqslant 9$.
  \item \href{http://dx.doi.org/10.1098/rsta.1988.0077}{[Joyce, 1988]}: hexagons (integrable, \href{http://dx.doi.org/10.1088/0305-4470/13/3/007}{[Baxter, 1980]}).
  \item \href{http://dx.doi.org/10.1209/epl/i2005-10166-3}{[Eisenberg, Baram, 2005]}: crosses: for $k\leqslant 6$.
  \item For sliding models, the high-fugacity expansion is ill-defined.
\end{itemize}
\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{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 model}
\begin{itemize}
  \item 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{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/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}, \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: nematic liquid crystal phase in systems of hard rods on $\mathbb Z^2$.
  \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{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}

\end{document}
