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diff --git a/Jauslin_Rutgers_2017.tex b/Jauslin_Rutgers_2017.tex new file mode 100644 index 0000000..13099f9 --- /dev/null +++ b/Jauslin_Rutgers_2017.tex @@ -0,0 +1,208 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{dsfont} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil High density phases\par +\smallskip +\hfil of hard-core lattice particle systems\par +\vfil +\large +\hfil Ian Jauslin +\normalsize +\vfil +\hfil\rm joint with {\bf Joel L. Lebowitz} and {\bf Elliott H. Lieb}\par +\vfil +arXiv: \vbox{ + \hbox{\tt \href{http://arxiv.org/abs/1708.01912}{1708.01912}} + \hbox{\tt \href{http://arxiv.org/abs/1709.05297}{1709.05297}} +} +\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}} +\eject + +\setcounter{page}1 +\pagestyle{plain} + +\title{Gas-liquid-crystal} +\vfill +\hfil +\includegraphics[width=3cm]{gas.png} +\includegraphics[width=3cm]{liquid.png} +\includegraphics[width=3cm]{crystal.png} +\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 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{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: + $$ + \left<\mathds 1_x\right>_\nu= + \left\{\begin{array}{ll} + 1+O(y)&\mathrm{\ if\ }x\in\mathcal L_\nu\\[0.3cm] + O(y)&\mathrm{\ if\ not} + . + \end{array}\right. + $$ +\end{itemize} +\vfill +\eject + +\title{High-fugacity expansion} +$$ + p(y)=-\rho_m\log y+\sum_{k=1}^\infty c_k y^k +$$ +\begin{itemize} + \item \href{http://dx.doi.org/10.1063/1.1697217}{[Gaunt, Fisher, 1965]}: diamonds: 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: for $k\leqslant 6$. + \item For sliding models, the high-fugacity expansion is ill-defined. +\end{itemize} +\vfill +\eject + +\title{Liquid crystals} +\begin{itemize} + \item Orientational order and positional disorder. +\end{itemize} +\hfil\includegraphics[width=4.5cm]{nematic.png} +\hfil\includegraphics[width=4.5cm]{chiral.png} +\vfill +\eject + +\title{Heilmann-Lieb model} +\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]} +\vfil +\hfil\includegraphics[width=5cm]{interaction.pdf} +\vfil\eject + +\title{Heilmann-Lieb model} +\begin{itemize} + \item Gibbs measure: + $$ + \left<A\right>_{\mathrm v} + := + \lim_{\Lambda\to\mathbb Z^2} + \frac1{\Xi_{\Lambda,\mathrm v}(z)} + \sum_{\underline\delta\in\Omega_{\mathrm v}(\Lambda)}A(\underline\delta)z^{|\underline\delta|}\prod_{\delta\neq \delta'\in \underline\delta}e^{\frac12J\mathds 1_{\delta\sim\delta'}} + $$ + \vskip-15pt + \begin{itemize} + \item $\Lambda$: finite box. + \item $\Omega_{\mathrm v}(\Lambda)$: non-overlapping dimer configurations satisfying the boundary condition. + \item $z\geqslant 0$: fugacity. + \item $J\geqslant 0$: interaction strength. + \item $\mathds 1_{\delta\sim\delta'}$ indicator that dimers are adjacent and aligned. + \end{itemize} +\end{itemize} +\vfill +\eject + +\title{Heilmann-Lieb conjecture} +\begin{itemize} + \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: proved orientational order using reflection positivity. + \item HL Conjecture: absence of positional order. + \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}, \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: nematic liquid crystal phase in systems of hard rods on $\mathbb Z^2$. + \item \href{http://dx.doi.org/10.1007/s10955-015-1421-8}{[Alberici, 2016]}: different fugacities for horizontal and vertical dimers. + \item \href{http://dx.doi.org/10.1103/PhysRevB.89.035128}{[Papanikolaou, Charrier, Fradkin, 2014]}: numerics. +\end{itemize} +\vfill +\eject + +\title{Theorem} +For $1\ll z\ll J$, $\|(x,y)\|_{\mathrm{HL}}:=J|x|+e^{-\frac32J}z^{-\frac12}|y|$, +\begin{itemize} + \item Given two vertical edges $e_{\mathrm v},f_{\mathrm v}$, $\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm v}$ and + $$ + \begin{array}{c} + \left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12})) + \\[0.3cm] + \left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v} + -\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v} + \left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v} + =O(e^{-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})}) + \end{array} + $$ + \vskip-5pt + \item Given two horizontal edges $e_{\mathrm h},f_{\mathrm h}$, $\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}$ is {\it independent} of $e_{\mathrm h}$ and + $$ + \begin{array}{c} + \left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J}) + \\[0.3cm] + \left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v} + -\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v} + \left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v} + =O(e^{-3J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm v},f_{\mathrm v})}) + \end{array} + $$ +\end{itemize} + +\end{document} |