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\begin{document}
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\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

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\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}