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author | Ian Jauslin <ian.jauslin@roma1.infn.it> | 2016-01-18 21:35:40 +0000 |
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committer | Ian Jauslin <ian.jauslin@roma1.infn.it> | 2016-01-18 21:35:40 +0000 |
commit | dae6a9add7289043d363010320b7fcc4b8e1c49c (patch) | |
tree | 1d7def9e54277ddd6c17cb17fb07fc29f4bcaa46 |
24 files changed, 491 insertions, 0 deletions
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+\usepackage{presentation} +\usepackage{header} + +\begin{document} +\pagestyle{empty} +\hbox{}\vfil +\bf +\large +\hfil A Pfaffian formula\par +\smallskip +\hfil for monomer-dimer partition functions\par +\vfil +\hfil Ian Jauslin +\rm +\normalsize + +\vfil +\small +\hfil joint with {\normalsize\bf A.~Giuliani} and {\normalsize\bf E.H.~Lieb}\par +\vskip10pt +arXiv: {\tt1510.05027}\hfill{\tt http://ian.jauslin.org/} +\eject + +\pagestyle{plain} +\setcounter{page}{1} + +\title{Monomer-dimer system} +\bigskip +\hfil\includegraphics{Figs/even_example.pdf} +\begin{itemize} +\item Monomer: occupies a single vertex. +\item Dimer: occupies an edge and its end-vertices. +\item Monomer-Dimer (MD) covering: every vertex is occupied exactly once. +\end{itemize} +\eject + +\addtocounter{page}{-1} +\title{Monomer-dimer system} +\bigskip +\hfil\includegraphics{Figs/MD_example.pdf} +\begin{itemize} +\item Monomer: occupies a single vertex. +\item Dimer: occupies an edge and its end-vertices. +\item Monomer-Dimer (MD) covering: every vertex is occupied exactly once. +\end{itemize} +\eject + +\title{Partition function} +\begin{itemize} +\item Weights: edges $d_e$, vertices $\ell_v$ (for simplicity, assume they are $\ge0$). +\item Partition function: +$$ +\Xi(\bm\ell,\mathbf d)=\sum_{\mathrm{MD\ coverings}}\prod_{\mAthop{e:}_{\mAthop{\mathrm{occupied}}_{\mathrm{by\ dimer}}}}d_e\prod_{\mAthop{v:}_{\mAthop{\mathrm{occupied}}_{\mathrm{\ by\ monomer}}}}\ell_v. +$$ +\end{itemize} +\eject + +\title{Kasteleyn's theorem} +\begin{itemize} +\item If there are {\bf no monomers} (i.e. $\ell_v=0$ for all $v$), then $\Xi$ counts pairs of neighboring vertices: +$$ +\Xi(\mathbf0,\mathbf d)=\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}\prod_{i=1}^nd_{(\pi(2i-1),\pi(2i))}. +$$ +\end{itemize} +\hfil\includegraphics[width=3cm]{Figs/even_example_label_nodir.pdf} +\eject + +\title{Kasteleyn's theorem} +\begin{itemize} +\item If there are {\bf no monomers} (i.e. $\ell_v=0$ for all $v$), then $\Xi$ counts pairs of neighboring vertices: +$$ +\Xi(\mathbf0,\mathbf d)=\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}\prod_{i=1}^nd_{(\pi(2i-1),\pi(2i))}. +$$ +\end{itemize} +\hfil\includegraphics[width=3cm]{Figs/even_example_label_nodir_dimers.pdf} +\eject + +\title{Kasteleyn's theorem} +\begin{itemize} +\item Assume, in addition, that the graph is {\bf planar}. +\item [Kasteleyn, 1963]: {\color{blue}Direct} the graph in such a way that, for every face, moving along the boundary of the face in the counterclockwise direction, the number of arrows going against the motion is odd. +\end{itemize} +\hfil\includegraphics{Figs/even_example_directed.pdf} +\eject + +\title{Kasteleyn's theorem} +\begin{itemize} +\item Recall +$$ +\Xi(\mathbf0,\mathbf d)=\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}\prod_{i=1}^nd_{(\pi(2i-1),\pi(2i))}. +$$ +\item Kasteleyn's theorem: if $s_{i,j}:=+1$ if $i\rightarrow j$ and $-1$ if $j\rightarrow i$, then +$$ +\Xi(\mathbf0,\mathbf d)=\left|\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}(-1)^\pi\prod_{i=1}^nd_{(\pi(2i-1),\pi(2i))}s_{\pi(2i-1),\pi(2i)}\right|. +$$ +\item In other words, $(-1)^\pi\prod_{i=1}^ns_{\pi(2i-1),\pi(2i)}$ is {\color{blue} independent} of $\pi$. +\end{itemize} +\eject + +\title{Kasteleyn's theorem} +\begin{itemize} +\item $(-1)^\pi\prod_{i=1}^ns_{\pi(2i-1),\pi(2i)}$: +\end{itemize} +\hfil\includegraphics{Figs/even_example_label_dimers.pdf} +\eject + +\title{Kasteleyn's theorem} +\begin{itemize} +\item Recall +$$ +\Xi(\mathbf0,\mathbf d)=\left|\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}(-1)^\pi\prod_{i=1}^nd_{(\pi(2i-1),\pi(2i))}s_{\pi(2i-1),\pi(2i)}\right|. +$$ +\item Introducing an antisymmetric matrix $a$ with entries $a_{i,j}:=d_{(i,j)}s_{i,j}$ for $i<j$, +$$ +\Xi(\mathbf0,\mathbf d)=|\mathrm{pf}(a)| +$$ +with +$$ +\mathrm{pf}(a)=\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}(-1)^\pi\prod_{i=1}^na_{\pi(2i-1),\pi(2i)}. +$$ +\item Determinantal relation: $\mathrm{pf(a)}^2=\det(a)$. +\end{itemize} +\eject + +\title{Boundary monomers} +\begin{itemize} +\item Assume the monomers are on the {\bf boundary} of the graph. +\item If the monomers are fixed, the MD partition function reduces to a pure dimer partition function on a sub-graph. +\end{itemize} +\hfil\includegraphics{Figs/even_example_monomer_shadow.pdf} +\eject + +\title{Boundary monomers} +\begin{itemize} +\item By Kasteleyn's theorem +$$ +\Xi(\bm\ell,\mathbf d)=\sum_{\mathcal M:\ \mathrm{monomers}}\left|\mathrm{pf}([a]_{\mathcal M})\right|\prod_{v\in\mathcal M}\ell_v +$$ +where $[a]_{\mathcal M}$ is obtained from $a$ by removing the lines and columns corresponding to vertices in $\mathcal M$. +\item[Lieb, 1968]: if $A_{i,j}=a_{i,j}+(-1)^{i+j}\ell_i\ell_j$ for $i<j$ and $A_{j,i}=-A_{i,j}$, then +$$\mathrm{pf}(A)=\sum_{\mathcal M}\mathrm{pf}([a]_{\mathcal M})\prod_{v\in\mathcal M}\ell_v.$$ +\end{itemize} +\eject + +\title{Boundary monomers} +\begin{itemize} +\item Question: is the sign of $\mathrm{pf}([a]_{\mathcal M})$ independent of $\mathcal M$? +\item In general, no: +\end{itemize} +\hfil\includegraphics{Figs/even_example_counterexample_label.pdf} +\eject + +\addtocounter{page}{-1} +\title{Boundary monomers} +\begin{itemize} +\item Question: is the sign of $\mathrm{pf}([a]_{\mathcal M})$ independent of $\mathcal M$? +\item In general, no: +\end{itemize} +\hfil\includegraphics{Figs/even_example_counterexample_label_dimers.pdf} +\eject + +\addtocounter{page}{-1} +\title{Boundary monomers} +\begin{itemize} +\item Question: is the sign of $\mathrm{pf}([a]_{\mathcal M})$ independent of $\mathcal M$? +\item In general, no: +\end{itemize} +\hfil\includegraphics{Figs/even_example_counterexample_monomer.pdf} +\eject + +\addtocounter{page}{-1} +\title{Boundary monomers} +\begin{itemize} +\item Question: is the sign of $\mathrm{pf}([a]_{\mathcal M})$ independent of $\mathcal M$? +\item In general, no: +\end{itemize} +\hfil\includegraphics{Figs/even_example_counterexample_monomer_dimers.pdf} +\eject + +\title{Boundary monomers} +\begin{itemize} +\item The vertices must be labeled and the edges directed correctly. +\end{itemize} +\hfil\includegraphics{Figs/even_example_phantom.pdf} +\eject + +\addtocounter{page}{-1} +\title{Boundary monomers} +\begin{itemize} +\item The vertices must be labeled and the edges directed correctly. +\end{itemize} +\hfil\includegraphics{Figs/even_example_label_pre.pdf} +\eject + +\addtocounter{page}{-1} +\title{Boundary monomers} +\begin{itemize} +\item The vertices must be labeled and the edges directed correctly. +\end{itemize} +\hfil\includegraphics{Figs/even_example_label.pdf} +\eject + +\title{Main theorem} +\vfill +\begin{framed} +Every {\bf planar} graph can be labeled and directed in such a way that the {\bf boundary} monomer-dimer partition function is +$$ +\Xi(\bm\ell,\mathbf d)=\mathrm{pf}(A) +$$ +with $A_{i,j}=d_{(i,j)}s_{i,j}+(-1)^{i+j}\ell_i\ell_j$ for $i<j$. +\end{framed} +\vfill +\eject + +\title{Sketch of the proof} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_phantom.pdf} +\eject + +\addtocounter{page}{-1} +\title{Sketch of the proof} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_auxiliary_thin.pdf} +\eject + +\addtocounter{page}{-1} +\title{Sketch of the proof} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_cover_phantom.pdf} +\eject + +\addtocounter{page}{-1} +\title{Sketch of the proof} +\hfil\includegraphics[width=6cm]{Figs/enclosed_example_auxiliary_cover.pdf} +\eject + +\title{Boundary monomer correlations} +\begin{itemize} +\item Monomer correlations at close packing: +$$ +M_n(i_1,\cdots,i_{2n})=\frac1{\Xi(\mathbf0,\mathbf d)}\left.\frac{\partial^{2n}\Xi(\bm\ell,\mathbf d)}{\partial\ell_{i_1}\cdots\partial\ell_{i_{2n}}}\right|_{\bm\ell=\mathbf0}. +$$ +\item Fermionic Wick rule: +$$ +M_n(i_1,\cdots,i_{2n})=\frac1{n!2^n}\sum_{\pi\in\mathcal S_{2n}}(-1)^\pi\prod_{j=1}^nM_1(i_{\pi(2j-1)},i_{\pi(2j)}). +$$ +\end{itemize} + +\end{document} @@ -0,0 +1,35 @@ +* Typeset + +In order to typeset the LaTeX presentation, run + pdflatex Jauslin_Trieste_2015.tex + pdflatex Jauslin_Trieste_2015.tex + + +* Files + + Jauslin_Rutgers_2015.tex + body of the presentation. + + Figs : + figures. + + header.sty : + packages and local definitions + + presentation.sty : + presentation style file. + + kiss.cls : + barebones class file + + toolbox.sty : + useful commands + + +* Comment on the presentation style + +This presentation is typeset using the 'presentation.sty' style file, in which +the margins and sizes are set to make the output look like a presentation, and +several useful commands are defined. + +The LaTeX 'beamer' package is not used. diff --git a/header.sty b/header.sty new file mode 100644 index 0000000..e097e71 --- /dev/null +++ b/header.sty @@ -0,0 +1,16 @@ +%% +%% Load packages +%% + +\usepackage{color} +\usepackage[hidelinks]{hyperref} +\usepackage{graphicx} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{array} +\usepackage{etoolbox} +\usepackage{dsfont} +\usepackage{bm} +\usepackage{toolbox} +\usepackage{framed} + diff --git a/kiss.cls b/kiss.cls new file mode 100644 index 0000000..7f0029f --- /dev/null +++ b/kiss.cls @@ -0,0 +1,51 @@ +%% +%% Barebones class declaration +%% + +\NeedsTeXFormat{LaTeX2e}[1995/12/01] +\ProvidesClass{kiss} + +\setlength\paperheight {297mm} +\setlength\paperwidth {210mm} + +%% fonts +\input{size11.clo} +\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} +\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} +\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} +\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} +\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} +\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} +\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} + +%% something is wrong with \thepage, redefine it +\gdef\thepage{\the\c@page} + +%% array lines +\setlength\arraycolsep{5\p@} +\setlength\arrayrulewidth{.4\p@} + + +%% default offsets: 1in, correct with \hoffset and \voffset +%\hoffset=0pt +\hoffset=-50pt +%\voffset=0pt +\voffset=-72pt +%% horizontal margins +%\oddsidemargin=31pt +%\evensidemargin=31pt +%% vertical margin +%\topmargin=20pt +%% body size +%\textwidth=390pt +\textwidth=460pt +%\textheight=592pt +\textheight=704pt +%% header size and margin +%\headheight=12pt +%\headsep=25pt +%% footer size +%\footskip=30pt +%% margin size and margin +%\marginparwidth=35pt +%\marginparsep=10pt diff --git a/presentation.sty b/presentation.sty new file mode 100644 index 0000000..e55928f --- /dev/null +++ b/presentation.sty @@ -0,0 +1,110 @@ +%% +%% Presentation style +%% + +%% can call commands even when they are not defined +\def\safe#1{% +\ifdefined#1% +#1% +\else% +{\color{red}\bf?}% +\fi% +} + + +%% paper size +\setlength\paperheight{240pt} +\setlength\paperwidth{320pt} + +%% body size +% height=paperheight-2xtopmargin-footskip +\textheight=208pt +% width=paperwidth-2xoddsidemargin +\textwidth=272pt + +%% margins +\voffset=-1in +\hoffset=-1in +\oddsidemargin=24pt +\evensidemargin=24pt +\topmargin=8pt +\headheight=0pt +\headsep=0pt +\marginparsep=0pt +\marginparwidth=0pt +\footskip=16pt +\skip\footins=0pt + +%% reset skips +\parskip=0pt +\parindent=0pt +\baselineskip=0pt +\AtBeginDocument{ +} + +%% footer +\def\ps@plain{ + \def\@oddhead{} + \def\@evenhead{\@oddhead} + \def\@oddfoot{\tiny\hfill\thepage/\safe\slidecount\hfill} + \def\@evenfoot{\@oddfoot} +} +\def\ps@empty{ + \def\@oddhead{} + \def\@evenhead{\@oddhead} + \def\@oddfoot{} + \def\@evenfoot{\@oddfoot} +} + +%% save total slide count +\AtEndDocument{ + \immediate\write\@auxout{\noexpand\gdef\noexpand\slidecount{\thepage}} +} + +\pagestyle{plain} + +%% title of slide +\def\title#1{ + \hfil{\bf\large #1}\par + \hfil\vrule width0.75\textwidth height0.3pt\par + \vskip5pt +} + +%% itemize +\newlength\itemizeskip +% left margin for items +\setlength\itemizeskip{20pt} +% item symbol +\def\itemizept{\textbullet} +\newlength\itemizeseparator +% space between the item symbol and the text +\setlength\itemizeseparator{5pt} + +\newlength\current@itemizeskip +\setlength\current@itemizeskip{0pt} +\def\itemize{ + \vfil + \addtolength\current@itemizeskip{\itemizeskip} + \leftskip\current@itemizeskip +} +\def\enditemize{ + \addtolength\current@itemizeskip{-\itemizeskip} + \par\leftskip\current@itemizeskip + \vfil +} +\newlength\itempt@total +\def\item{ + \settowidth\itempt@total{\itemizept} + \addtolength\itempt@total{\itemizeseparator} + \par + \vfil + \hskip-\itempt@total\itemizept\hskip\itemizeseparator +} + +\def\itemptchange#1{ + \let\itempt@prev\itemizept + \def\itemizept{#1} +} +\def\itemptreset{ + \def\itemizept{\itempt@prev} +} diff --git a/toolbox.sty b/toolbox.sty new file mode 100644 index 0000000..04c8acd --- /dev/null +++ b/toolbox.sty @@ -0,0 +1,32 @@ +%% +%% A collection of useful commands +%% + +%% can call commands even when they are not defined +\def\safe#1{% +\ifdefined#1% +#1% +\else% +{\color{red}\bf?}% +\fi% +} + + +%% larger skip +\newskip\hugeskipamount + \hugeskipamount=24pt plus8pt minus8pt +\def\hugeskip{\vskip\hugeskipamount} + + +%% stack relations in subscript or superscript +\def\mAthop#1{\displaystyle\mathop{\scriptstyle #1}} + +%% array spanning the entire line +\newlength\largearray@width +\setlength\largearray@width\textwidth +\addtolength\largearray@width{-10pt} +\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\largearray@width}}\\[-.5cm]} +\def\endlargearray{\end{array}} + +%% qedsquare +\def\qed{\penalty10000\hfill\penalty10000$\square$} |