From b1090e768378b0f563e2bffd29a81ded180f6f1e Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@roma1.infn.it>
Date: Thu, 22 Sep 2016 00:15:35 +0000
Subject: As presented at Rutgers University on 2016-09-22

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+\documentclass{kiss}
+\usepackage{presentation}
+\usepackage{header}
+\usepackage{toolbox}
+\usepackage{tikz}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Emergence of a nematic phase\par
+\smallskip
+\hfil in a system of hard plates\par
+\vfil
+\large
+\hfil Ian Jauslin
+\normalsize
+\vfil
+\hfil\rm with {\bf Margherita Disertori} and {\bf Alessandro Giuliani}\par
+\vfil
+\hfil\href{http://ian.jauslin.org}{\tt http://ian.jauslin.org}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Hard plates}
+  \begin{itemize}
+    \item Parallelepiped $k\times k^\alpha\times 1$, $\alpha\in[0,1]$, $k\gg1$
+  \end{itemize}
+  \bigskip
+  \includegraphics[width=\textwidth]{figs/plate.pdf}
+  \begin{itemize}
+    \item Center in $\mathbb R^3$
+  \end{itemize}
+\eject
+
+\title{Hard plates}
+  \begin{itemize}
+    \item 6 orientations
+  \end{itemize}
+  \vfil
+  \hfil\includegraphics[width=220pt]{figs/plates.pdf}
+\eject
+
+\title{Heuristics}
+  \begin{itemize}
+    \item For $\frac12\leqslant\alpha\leqslant1$
+  \end{itemize}
+  \vfil
+  \includegraphics[width=\textwidth]{figs/phase_largealpha.pdf}
+  \begin{itemize}
+    \itemptchange{$\scriptstyle\blacktriangleright$}
+    \begin{itemize}
+      \item $N_b$: biaxial nematic
+      \item $N_-$: plate-like nematic
+      \item $I$: isotropic
+      \item $?$: ?
+    \end{itemize}
+    \itemptreset
+  \end{itemize}
+\eject
+
+\title{Heuristics}
+  \begin{itemize}
+    \item For $0\leqslant\alpha\leqslant\frac12$
+  \end{itemize}
+  \vfil
+  \includegraphics[width=\textwidth]{figs/phase_smallalpha.pdf}
+  \begin{itemize}
+    \itemptchange{$\scriptstyle\blacktriangleright$}
+    \begin{itemize}
+      \item $N_b$: biaxial nematic
+      \item $N_+$: rod-like nematic
+      \item $I$: isotropic
+      \item $?$: ?
+    \end{itemize}
+    \itemptreset
+  \end{itemize}
+\eject
+
+
+\title{Result}
+  \begin{itemize}
+    \item $\frac56<\alpha<1$ (we can also do $\alpha=1$).  \end{itemize}
+  \vfil
+  \includegraphics[width=\textwidth]{figs/phase_result.pdf}
+\eject
+
+\title{Previous results}
+  \begin{itemize}
+    \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: 2-dimensional hard rods (on a lattice).
+    \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: 2-dimensional hard needles.
+    \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: elongated molecules in 3 dimensions: 1st order phase transition! (non-rigorous)
+  \end{itemize}
+\eject
+
+\title{Setup}
+  \begin{itemize}
+    \item {\it Type} of a plate: $q\in\{1,2,3\}$.
+    \item (Grand-canonical) Gibbs average, in a box $\Lambda$, with $q$-boundary conditions:
+  \end{itemize}
+  $$
+    \left<A\right>_{\Lambda,q}:=\frac1{Z(\Lambda|q)}\int_{\Omega_{\Lambda,q}}\kern-10pt dP\ z^{|P|}\varphi(P)A(P)
+  $$
+  \begin{itemize}
+    \itemptchange{$\scriptstyle\blacktriangleright$}
+    \begin{itemize}
+      \item $\Omega_{\Lambda,q}$: plate configurations with $q$-boundary conditions, $|P|$: number of plates in $P$,
+      \item $z$: {\it activity}: $z=e^{\beta\mu}$,
+      \item $\varphi(P)$: hard-core potential,
+      \item $Z(\Lambda|q)$: partition function (normalization),
+      \item $A$: {\it local} observable.
+    \end{itemize}
+    \itemptreset
+  \end{itemize}
+\eject
+
+\title{Setup}
+  \begin{itemize}
+    \item Boundary condition: any plate, centered at $x\in\Lambda$, that satisfies
+      $$
+	d_\infty(x,\ \mathbb R^3\setminus\Lambda)\leqslant\left(\frac4{1-\alpha}+3\right)\frac k2
+      $$
+      is of type $q$.
+
+    \item Local observable:
+      $$
+	A(P)=\sum_{p\in P}a(p)
+      $$
+      and $a$ is {\it compactly supported}.
+  \end{itemize}
+\eject
+
+\title{Theorem}
+  \begin{itemize}
+    \item If $zk^{3-\alpha}\ll1\ll zk^{5\alpha-2}$
+    \itemptchange{$\scriptstyle\blacktriangleright$}
+    \begin{itemize}
+      \item $\mathds1_x(P)$: indicator that $\exists p\in P$: $d_\infty(x,p)\leqslant\frac12$:
+	$$
+	  \left<\mathds1_x(P)\right>_{\Lambda,q}\equiv\rho=z(1+o(1))
+	$$
+      \vskip5pt
+      \item $\mathcal N_{x,q}(P)$: number of plates $p\in P$ of type $q$ with $d_\infty(p,x)\leqslant\frac k4$: for $m\neq q$,
+	$$
+	  \left<\mathcal N_{x,q}(P)\right>_{\Lambda,q}\geqslant C zk^3\gg1,\quad
+	  \left<\mathcal N_{x,m}(P)\right>_{\Lambda,q}=o(1)
+	$$
+      \item There exists $\eta_k\to0$ as $k\to\infty$ such that
+	$$
+	  \left<\mathds1_x(P);\mathds1_y(P)\right>_{\Lambda,q}^T\leqslant C\rho^2 \eta_k^{\frac{|x-y|}k}
+	$$
+    \end{itemize}
+    \itemptreset
+  \end{itemize}
+\eject
+
+\title{Cluster expansion}
+  \begin{itemize}
+    \item For $\mathbf A\equiv(A_1,\cdots,A_n)$, $\mathbf s\equiv(s_1,\cdots,s_n)\in\mathbb R^n$
+      $$
+	F_{\Lambda,q}(\mathbf s\cdot\mathbf A):=\log\int_{\Omega_{\Lambda,q}}\kern-10pt dP\ z^{|P|}\varphi(P)e^{\sum_{i=1}^ns_iA_i(P)}
+      $$
+    \item Generating function:
+      $$
+	\left<A_1,\cdots,A_n\right>_{\Lambda,q}^T=\left.\partial_{s_1}\cdots\partial_{s_n}F_{\Lambda,q}(\mathbf s\cdot\mathbf A)\right|_{\mathbf s=0}
+      $$
+  \end{itemize}
+\eject 
+
+\title{Cluster expansion}
+  $$
+    F_{\Lambda,q}(\mathbf s\cdot\mathbf A)=F^{(0)}_{\Lambda,q}(\mathbf s\cdot\mathbf A)+\sum_{\mathcal X\in\Xi(\Lambda)}\phi^T(\mathcal X)\prod_{X\in\mathcal X} K_{\Lambda,q}^{(\mathbf s\cdot\mathbf A)}(X)
+  $$
+  \begin{itemize}
+    \itemptchange{$\scriptstyle\blacktriangleright$}
+    \begin{itemize}
+      \item $F_{\Lambda,q}^{(0)}(\mathbf s\cdot\mathbf A)$: {\it all} plates are of type $q$,
+      \item $\Xi(\Lambda)$: collections of {\it polymers}: connected unions of $\frac k2\times\frac k2\times\frac k2$ cubes,
+      \item $\phi^T$: {\it Mayer coefficient},
+      \item $K_{\Lambda,q}^{(\mathbf s\cdot\mathbf A)}(X)$: {\it activity} of $X$.
+    \end{itemize}
+    \itemptreset
+  \end{itemize}
+\eject
+
+\title{Cluster expansion}
+  \begin{itemize}
+    \item Absolutely convergent expansion: $\exists\epsilon_k\to0$ such that, for $m\geqslant0$,
+      $$
+	\sum_{\mAthop{\mathcal X\in\Xi(\Lambda)}_{|\mathcal X|\ge m}}\left|\phi^T(\mathcal X)\prod_{X\in\mathcal X}K_{\Lambda,q}^{(\mathbf s\cdot\mathbf A)}(X)\right|\le
+	\epsilon_k^m
+      $$
+  \end{itemize}
+
+\end{document}
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