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diff --git a/GLaMP_2017.tex b/GLaMP_2017.tex new file mode 100644 index 0000000..417100a --- /dev/null +++ b/GLaMP_2017.tex @@ -0,0 +1,312 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{dsfont} + +\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 +\pagestyle{plain} + +\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} |