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\begin{document}
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\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

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\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}
