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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:{\tt \href{http://arxiv.org/abs/1705.02032}{1705.02032}}\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
\eject

\setcounter{page}1
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\title{Non-sliding HCLP}
\begin{itemize}
  \item Lattice in $d\geqslant 2$ dimensions.
  \item Identical particles with pair hard-core interaction.
  \item Finite number of perfect packings, which are periodic, and related to each other by isometries.
  \item Non-sliding (defects are full of holes).
\end{itemize}
\eject

\title{Lattice}
\begin{itemize}
  \item Lattice $\Lambda_\infty$ of dimension $d\geqslant 2$
\end{itemize}
\vfill
\hfil\includegraphics[width=4.5cm]{grid.pdf}
\vfill
\eject

\title{Particles}
\begin{itemize}
  \item Identical particles: shape $\omega\subset\mathbb R^d$
\end{itemize}
\vfill
\hfil\includegraphics[width=4.5cm]{cross_example.pdf}
\vfill
\eject

\title{Perfect packings}
\begin{itemize}
  \item Perfect packing: $\forall x\in\Lambda_\infty$, $\exists! y$ such that $x\in\omega+y$
\end{itemize}
\vfill
\hfil\includegraphics[width=4cm]{cross_packing_r.pdf}
\hfil\includegraphics[width=4cm]{cross_packing_l.pdf}
\vfill
\eject

\title{Perfect packings}
\begin{itemize}
  \item Example with infinitely many perfect packings: $2\times2$ squares
\end{itemize}
\vfill
\hfil\includegraphics[width=3cm]{square_packing.pdf}
\hfil\includegraphics[width=3cm]{square_packing_slide.pdf}
\vfill
\eject

\title{Non-sliding condition}
\begin{itemize}
  \item Defects have an amount of empty space that is proportional to their volume.
  \item For every {\it connected} particle configuration $X$ that {\it cannot} be completed to a perfect packing, and for every particle configuration $Y\supset X$, at least one of the sites {\it neighboring} $X$ is empty.
\end{itemize}
\eject

\title{Non-sliding condition}
\begin{itemize}
  \item Example: the red area cannot be entirely covered.
\end{itemize}
\vfill
\hfil\includegraphics[width=2cm]{cross_sliding1.pdf}
\hfil\includegraphics[width=2.4cm]{cross_sliding2.pdf}
\vfill
\eject

\title{Non-sliding condition}
\begin{itemize}
  \item Counter-example: $2\times 2$ squares.
\end{itemize}
\vfill
\hfil\includegraphics[width=2.5cm]{square_sliding.pdf}
\hfil\includegraphics[width=2.5cm]{square_sliding_contra.pdf}
\vfill
\eject

\title{Examples}
\begin{itemize}
  \item (thick) crosses, diamonds, hexagons...
\end{itemize}
\medskip
\hfil\includegraphics[height=1.5cm]{cross.pdf}
\hfil\includegraphics[height=1.5cm]{cross3.pdf}
\hfil\includegraphics[height=1.5cm]{diamond.pdf}
\hfil\includegraphics[height=1.5cm]{hexagon.pdf}
\begin{itemize}
  \item Conjecture: $k$-nearest neighbor exclusion on $\mathbb Z^2$ with
  $$
    k\in\{1,3,6,7,8,9,12,13,\cdots\}.
  $$
\end{itemize}
\eject

\title{Partition function}
$$
  \Xi^{(\nu)}_\Lambda(z)=
  \sum_{X\subset\Lambda}
  z^{|X|}\prod_{x\neq x'\in X}\phi(x,x')
$$
\begin{itemize}
  \item $\Lambda\subset\Lambda_\infty$ is bounded.
  \item $z$: {\it fugacity}, $z=e^{\beta\mu}$.
  \item $\phi$: hard-core repulsion.
  \item $|X|\leqslant N_{\mathrm{max}}$, maximal density: $\rho_m:=\frac{N_{\mathrm{max}}}{|\Lambda|}$.
  \item Boundary condition: $\Lambda_\infty\setminus\Lambda$ is covered by a perfect covering, indexed by $\nu\in\{1,\cdots,\tau\}$.
\end{itemize}
\vfill
\eject

\title{Thermodynamical observables}
\begin{itemize}
  \item Pressure:
  $$
    p(z):=\lim_{\Lambda\to\Lambda_\infty}\frac1{|\Lambda|}\log\Xi_\Lambda^{(\nu)}(z)
  $$
  \item Correlation functions: for $\underline x\equiv\{x_1,\cdots,x_n\}\subset\Lambda_\infty$,
  $$
    \rho_n^{(\nu)}(\underline x)
    :=\lim_{\Lambda\to\Lambda_\infty}
    \frac1{\Xi_\Lambda^{(\nu)}(z)}
    \sum_{\displaystyle\mathop{\scriptstyle X\subset\Lambda}_{X\supset\underline x}}
    z^{|X|}\prod_{x\neq x'\in X}\phi(x,x')
  $$
\end{itemize}
\eject

\title{Result}
\begin{itemize}
  \item High fugacity regime: $y\equiv z^{-1}\ll 1$.
  \item {\bf Analyticity}: $p(z)-\rho_m\log z$ and $\rho_n^{(\nu)}(\underline x)$ are {\it analytic} functions of $y$ in a disk in the complex $y$-plane centered at $0$.
  \item {\bf Crystallization}: If $x$ is compatible with the $\nu$-th perfect packing, then
  $$
    \rho_1^{(\nu)}(x)=1+o(1)
  $$
  and if not, then
  $$
    \rho_1^{(\nu)}(x)=o(1)
  $$
\end{itemize}
\eject

\title{Low-fugacity expansion}
\begin{itemize}
  \item Formally,
  $$
    \frac1{|\Lambda|}\log\Xi_\Lambda^{(\nu)}(z)
    =
    \sum_{k=1}^\infty b_k(\Lambda)z^k
  $$
  where, if $Z_\Lambda(k_i)$ denotes the number of configurations with $k_i$ particles, then
  $$
    b_k(\Lambda):=\frac1{|\Lambda|}
    \sum_{j=1}^k\frac{(-1)^{j+1}}j
    \sum_{\displaystyle\mathop{\scriptstyle k_1,\cdots,k_j\geqslant 1}_{k_1+\cdots+k_j=k}}Z_\Lambda(k_1)\cdots Z_\Lambda(k_j)
  $$
\end{itemize}
\eject

\title{Low-fugacity expansion}
\begin{itemize}
  \item Second term:
  $$
    b_2(\Lambda)=\frac1{|\Lambda|}\left(Z_\Lambda(2)-\frac12Z_\Lambda^2(1)\right)
  $$
  \item $\frac12 Z_\Lambda^2(1)$: counts non-interacting particle configurations.
  \item $Z_\Lambda(2)$: counts interacting particle configurations.
  \item The terms of order $|\Lambda|^2$ cancel out!
\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 Inverse fugacity $y\equiv z^{-1}$:
  $$
    \Xi^{(\nu)}_\Lambda(z)=
    z^{N_{\mathrm{max}}}
    \sum_{X\subset\Lambda}
    y^{N_{\mathrm{max}}-|X|}\prod_{x\neq x'\in X}\phi(x,x')
  $$
\end{itemize}
\eject

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

\title{High-fugacity expansion}
\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. nearest neighbor exclusion in 1 dimension:
  $$
    c_2(\Lambda)=-\frac1{192}|\Lambda|(|\Lambda|^2+2)
  $$
\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}


\end{document}
