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\begin{document}
\pagestyle{empty}
\hbox{}\vfil
\bf\Large
\hfil Crystalline ordering\par
\smallskip
\hfil in hard-core lattice particle systems\par
\vfil
\large
\hfil Ian Jauslin
\normalsize
\vfil
\hfil\rm joint with {\bf Joel L. Lebowitz}\par
\vfil
arXiv: \vbox{
  \hbox{\tt \href{http://arxiv.org/abs/1705.02032}{1705.02032}}
  \hbox{\tt \href{http://arxiv.org/abs/1708.01912}{1708.01912}}
}
\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

\setcounter{page}1
\pagestyle{plain}

\title{Crystallization in particle systems}
\begin{itemize}
  \item Gas-liquid-solid paradigm: not yet understood mathematically in realistic particle models.
  \item Crystallization: long range positional order.
  \item Here: discuss a class of hard-core lattice particle models for which we can prove crystallization.
\end{itemize}
\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 $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ which are \penalty-1000{\bf periodic} and {\bf isometric} to each other.
\end{itemize}
\hfil\includegraphics[width=4cm]{cross_packing_1.pdf}
\hfil\includegraphics[width=4cm]{cross_packing_1.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Non-sliding HCLPs}
\begin{itemize}
  \item There exist a {\bf finite} number $\tau$ of tilings $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ which are \penalty-1000{\bf periodic} and {\bf isometric} to each other.
\end{itemize}
\hfil\includegraphics[width=4cm]{cross_packing_l_2.pdf}
\hfil\includegraphics[width=4cm]{cross_packing_r_2.pdf}
\vfill
\eject

\addtocounter{page}{-1}
\title{Non-sliding HCLPs}
\begin{itemize}
  \item There exist a {\bf finite} number $\tau$ of tilings $\{\mathcal L_1,\cdots,\mathcal L_\tau\}$ 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{Example of a sliding HCLP}
\begin{itemize}
  \item $2\times2$ squares:
\end{itemize}
\hfil\includegraphics[height=4.5cm]{square_packing_slide.pdf}
\hfil\includegraphics[height=4.5cm]{hole_example_square.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:
  $$
  \left<\mathds 1_x\right>_\nu=
  \left\{\begin{array}{ll}
    1+O(z^{-1})&\mathrm{\ if\ }x\in\mathcal L_\nu\\[0.3cm]
    O(z^{-1})&\mathrm{\ if\ not}
    .
  \end{array}\right.
  $$
\end{itemize}
\vfill
\eject

\title{Low-fugacity expansion}
\begin{itemize}
  \item Partition function: $Z_\Lambda(n)$: number of configurations with $n$ particles:
  $$
    \Xi_\Lambda(z)
    =
    \sum_{X\subset\Lambda}z^{|X|}\prod_{x\neq x'\in X}\varphi(x,x')
    =\sum_{n=0}^\infty z^n Z_\Lambda(n)
  $$
  \vskip-10pt
  \item Formally,
  $$
    \frac1{|\Lambda|}\log\Xi_\Lambda(z)
    =
    \sum_{k=1}^\infty b_k(\Lambda)z^k
  $$
  where, if $Z_\Lambda(k_i)$ denotes the number of configurations with $k_i$ particles, then
  $$
    b_k(\Lambda):=\frac1{|\Lambda|}
    \sum_{j=1}^k\frac{(-1)^{j+1}}j
    \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Z_\Lambda(k_1)\cdots Z_\Lambda(k_j)
  $$
  \vskip-10pt
\end{itemize}
\eject

\title{Low-fugacity expansion}
\begin{itemize}
  \item Partition function:
  $$
    \Xi_\Lambda(z)
    =1+z Z_\Lambda(1)+z^2 Z_\Lambda(2)+\cdots
  $$
  \item Second term:
  $$
    b_2(\Lambda)=\frac1{|\Lambda|}\left(Z_\Lambda(2)-\frac12Z_\Lambda^2(1)\right)
  $$
  \item $Z_\Lambda(2)$: counts interacting particle configurations: $O(\Lambda^2)$.
  \item $\frac12 Z_\Lambda^2(1)$: counts non-interacting particle configurations: $O(\Lambda^2)$.
  \item The terms of order $|\Lambda|^2$ cancel out, so $b_2(\Lambda)$ has a limit as $\Lambda\to\Lambda_\infty$.
\end{itemize}
\eject

\title{Low-fugacity expansion}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1017/S0305004100011191}{[Ursell, 1927]}, \href{http://dx.doi.org/10.1063/1.1749933}{[Mayer, 1937]}: $b_k(\Lambda)\to b_k$.
  \item \href{http://dx.doi.org/10.1016/0031-9163(62)90198-1}{[Groeneveld, 1962]}, \href{http://dx.doi.org/10.1016/0003-4916(63)90336-1}{[Ruelle, 1963]}, \href{http://dx.doi.org/10.1063/1.1703906}{[Penrose, 1963]}: 
  $$
    p(z)=\sum_{k=1}^\infty b_kz^k
  $$
  which has a positive radius of convergence.
\end{itemize}
\eject

\title{High-fugacity expansion}
\begin{itemize}
  \item Partition function: $Z_\Lambda(n)$: number of configurations with $n$ particles:
  $$
    \Xi_\Lambda(z)
    =
    \sum_{X\subset\Lambda}z^{|X|}\prod_{x\neq x'\in X}\varphi(x,x')
    =\sum_{n=0}^{N_{\mathrm{max}}} z^n Z_\Lambda(n)
  $$
  \item Inverse fugacity $y\equiv z^{-1}$:
  $$
    \Xi_\Lambda(z)=
    z^{N_{\mathrm{max}}}
    \sum_{X\subset\Lambda}
    y^{N_{\mathrm{max}}-|X|}\prod_{x\neq x'\in X}\varphi(x,x')
    =z^{N_{\mathrm{max}}}\sum_{n=0}^{N_{\mathrm{max}}}y^n Q_\Lambda(n)
  $$
  with $Q_\Lambda(n)\equiv Z_\Lambda(N_{\mathrm{max}}-n)$.
\end{itemize}
\eject

\title{High-fugacity expansion}
\begin{itemize}
  \item Formally,
  $$
    \frac1{|\Lambda|}\log\Xi_\Lambda
    =
    \rho_m\log z
    +
    \sum_{k=1}^\infty c_k(\Lambda)y^k
  $$
  where $\rho_m=\frac{N_{\mathrm{max}}}{|\Lambda|}$,
  $$
    c_k(\Lambda):=\frac1{|\Lambda|}
    \sum_{j=1}^k\frac{(-1)^{j+1}}{j\tau^j}
    \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Q_\Lambda(k_1)\cdots Q_\Lambda(k_j)
  $$
\end{itemize}
\eject

\title{High-fugacity expansion}
\vfill
\hfil\includegraphics[width=2cm]{gf_diamond1.pdf}
\hfil\includegraphics[width=2.33cm]{gf_diamond2.pdf}
\par\vfill
\hfil\includegraphics[width=2.33cm]{gf_diamond3.pdf}
\vfill\eject

\title{High-fugacity expansion}
\begin{itemize}
  \item \href{http://dx.doi.org/10.1063/1.1697217}{[Gaunt, Fisher, 1965]}: diamonds: $c_k(\Lambda)\to c_k$ 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: $c_k(\Lambda)\to c_k$ for $k\leqslant 6$.
  \item Cannot be done {\it systematically}: there exist counter-examples: e.g. hard $2\times2$ squares on $\mathbb Z^2$:
  $$
    c_1(\Lambda)\propto\sqrt{|\Lambda|}
  $$
\end{itemize}
\eject

\title{Holes interact}
\begin{itemize}
  \item Total volume of holes: $\in\rho_m^{-1}\mathbb N$.
\end{itemize}
\vfill
\hfil\includegraphics[height=4.5cm]{hole_example_cross.pdf}
\hfil\includegraphics[height=4.5cm]{hole_example_square.pdf}
\vfill
\eject

\title{Non-sliding condition}
\begin{itemize}
  \item Distinct defects are decorrelated.
\end{itemize}
\vfill
\hfil\includegraphics[height=5cm]{hole_example_cross_decorrelated.pdf}
\vfill
\eject

\title{Gaunt-Fisher configurations}
\begin{itemize}
  \item Group together empty space and neighboring particles.
\end{itemize}
\vfill
\hfil\includegraphics[width=2.5cm]{gaunt_fisher2.pdf}
\hfil\includegraphics[width=4cm]{gaunt_fisher3.pdf}
\vfill
\eject

\title{Defect model}
\vskip-5pt
\begin{itemize}
  \item Map particle system to a model of defects:
  $$
    \Xi_{\Lambda,\nu}(z)=z^{\rho_m|\Lambda|}\sum_{\underline\gamma\subset\mathfrak C_\nu(\Lambda)}
    \left(\prod_{\gamma\neq\gamma'\in\underline\gamma}\Phi(\gamma,\gamma')\right)
    \prod_{\gamma\in\underline\gamma}\zeta_\nu^{(z)}(\gamma)
  $$
  \begin{itemize}
    \item $\Phi$: hard-core repulsion of defects.
    \item $\zeta_\nu^{(z)}(\gamma)$: activity of defect.
  \end{itemize}
  \item The activity of a defect is exponentially small: $\exists\epsilon\ll 1$
  $$
    \zeta_\nu^{(z)}(\gamma)<\epsilon^{|\gamma|}
  $$
  \vskip-5pt
  \item Low-fugacity expansion for defects.
\end{itemize}
\eject

\title{Crystallization}
\vfill
\begin{itemize}
  \item Peierls argument: in order to have a particle at $x$ that is not compatible with the $\nu$-th perfect packing, it must be part of or surrounded by a defect.
  \vfill
  \item Note: a naive Peierls argument requires the partition function to be independent from the boundary condition. This is not necessarily the case here, and we need elements from Pirogov-Sinai theory.
\end{itemize}
\vfill
\eject

\title{Lee-Yang zeros}
\begin{itemize}
  \item Lee-Yang zeros: roots of $\Xi_\Lambda(z)$ $\Longleftrightarrow$ singularities of $p_\Lambda(z)$.
  \item Whenever the high fugacity expansion has a radius of convergence $\tilde R$, there are no Lee-Yang zeros outside of a disc of radius $\tilde R^{-1}$.
\end{itemize}

\hfil\includegraphics[height=4cm]{lee_yang.pdf}



\end{document}
