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diff --git a/Jauslin_UBC_colloquium_2020.tex b/Jauslin_UBC_colloquium_2020.tex new file mode 100644 index 0000000..5d4a384 --- /dev/null +++ b/Jauslin_UBC_colloquium_2020.tex @@ -0,0 +1,219 @@ +\documentclass{ian-presentation} + +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{dsfont} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf\Large +\hfil Liquid crystals and the\par +\smallskip +\hfil Heilmann-Lieb model\par +\vfil +\large +\hfil Ian Jauslin +\normalsize +\vfil +\hfil\rm joint with {\bf Elliott H. Lieb}\par +\vfil +arXiv: {\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{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{History} +\begin{itemize} + \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$. + \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers. +\vphantom{ + \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$ with a {\it finite} number of orientations. + \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length). + \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$. + \item \href{http://dx.doi.org/10.1007/s00220-019-03543-z}{[Disertori, Giuliani, Jauslin, 2019]}: hard plates in $\mathbb Z^3$. +} +\end{itemize} +\vfill +\eject + + +\title{Heilmann-Lieb model} +\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]} +\vfil +\hfil\includegraphics[width=5cm]{grid.pdf} +\vfil\eject + +\addtocounter{page}{-1} +\title{Heilmann-Lieb model} +\hfil\href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]} +\vfil +\hfil\includegraphics[width=5cm]{dimers.pdf} +\vfil\eject + +\addtocounter{page}{-1} +\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 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. +\vphantom{ + \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{History} +\begin{itemize} + \item \href{http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x}{[Onsager, 1949]}: mean field model for hard needles in $\mathbb R^3$. + \item \href{http://dx.doi.org/10.1007/BF01009518}{[Heilmann, Lieb, 1979]}: interacting dimers. + \item \href{http://dx.doi.org/10.1007/BF00535264}{[Bricmont, Kuroda, Lebowitz, 1984]}: hard needles in $\mathbb R^2$ with a {\it finite} number of orientations. + \item \href{http://dx.doi.org/10.1007/s10955-005-8085-8}{[Ioffe, Velenik, Zahradn\'\i k, 2006]}: hard rods in $\mathbb Z^2$ (variable length). + \item \href{http://dx.doi.org/10.1007/s00220-013-1767-1}{[Disertori, Giuliani, 2013]}: hard rods in $\mathbb Z^2$. + \item \href{http://dx.doi.org/10.1007/s00220-019-03543-z}{[Disertori, Giuliani, Jauslin, 2019]}: hard plates in $\mathbb Z^3$. +\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/s10955-015-1421-8}{[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{Heilmann-Lieb model} +\begin{itemize} + \item Grand-canonical Gibbs measure: + $$ + \left<A\right>_{\mathrm v} + := + \lim_{\Lambda\to\mathbb Z^2} + \frac1{\Xi_{\mathrm v}(\Lambda)} + \sum_{\underline\delta\in\Omega_{\mathrm v}(\Lambda)}A(\underline\delta)z^{|\underline\delta|}\prod_{\delta\neq \delta'\in \underline\delta}e^{\frac12J\mathds 1_{\delta\sim\delta'}} + $$ + \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{Boundary condition} +\begin{itemize} + \item Fix length $\ell_0:=e^{\frac32J}\sqrt z$, +\end{itemize} +\hfil\includegraphics[width=5cm]{boundary.pdf} + +\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 + $$ + \begin{array}{c} + \left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v}=O(e^{-3J}) + \\[0.3cm] + \left<\mathds 1_{e_{\mathrm h}}\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v} + -\left<\mathds 1_{e_{\mathrm h}}\right>_{\mathrm v} + \left<\mathds 1_{f_{\mathrm h}}\right>_{\mathrm v} + =O(e^{-6J-c\ \mathrm{dist}_{\mathrm{HL}}(e_{\mathrm h},f_{\mathrm h})}) + \end{array} + $$ +\end{itemize} +\vfill +\eject + +\title{1D system} +\begin{itemize} + \item {\it Only} vertical dimers: integrable. + \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 + $$ + \begin{array}{c} + \left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v}=\frac12(1+O(e^{-\frac12J}z^{-\frac12})) + \\[0.3cm] + \left<\mathds 1_{e_{\mathrm v}}\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v} + -\left<\mathds 1_{e_{\mathrm v}}\right>_{\mathrm v} + \left<\mathds 1_{f_{\mathrm v}}\right>_{\mathrm v} + =O(e^{-c\ \mathrm{dist}_{\mathrm{1D}}(e_{\mathrm v},f_{\mathrm v})}) + \end{array} + $$ + with $\|(x,y)\|_{\mathrm{1D}}:=e^{-\frac32J}z^{-\frac12}|y|$. +\end{itemize} +\vfill +\eject + +\title{Loop model} +\vfill +\hfil\includegraphics[width=5cm]{dimer_contour.pdf} +\begin{itemize} + \item Weight of a loop of length $|l|$: $e^{-\frac12J|l|}$. +\end{itemize} +\vfill +\eject + +\title{Difficulty: loops interact} +\vskip-10pt +\begin{itemize} + \item Correlated dimers induce an interaction between loops, which decays exponentially with a rate $e^{-\frac32J}z^{-\frac12}$. +\end{itemize} +\hfil\includegraphics[width=3.5cm]{segments.pdf} +\begin{itemize} + \item Vertical-to-horizontal boundaries and horizontal-to-vertical ones have different geometries. +\end{itemize} +\vfill +\eject + +\title{Parameter regime} +\vfill +\hfil\includegraphics[width=8cm]{regime.pdf} + + +\end{document} |