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diff --git a/Jauslin_Trieste_2016.tex b/Jauslin_Trieste_2016.tex new file mode 100644 index 0000000..24b8eab --- /dev/null +++ b/Jauslin_Trieste_2016.tex @@ -0,0 +1,247 @@ +\documentclass{kiss} +\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} |