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Addition: relation between the Wick rule and the Pfaffian formula

References: [GBK87], [WTI11], [Te74]

References: [AP84] and mention minor mistakes in [Ha66]
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Ian Jauslin 2015-10-26 22:58:21 +00:00
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Giuliani_Jauslin_Lieb_2015.tex
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@ -50,9 +50,7 @@ Clearly, $\Xi(z)$ is a polynomial in $z$, and $z$ is called the {\it monomer fug
\indent In the pure dimer case, where $z=0$, $\Xi$ has been shown by Temperley and Fisher (for the square lattice) [\cite{TF61}] and by Kasteleyn (for general planar graphs) [\cite{Ka63}] to be expressible as a Pfaffian (which is convenient since Pfaffians can be computed as square roots of determinants). However, when monomers are allowed to appear, such a Pfaffian formula is thought to be impossible (at least a Pfaffian formula for the {\it full} MD problem on {\it any} planar graphs): indeed it has been shown [\cite{Je87}] that the enumeration of MD coverings of generic planar graphs is ``computationally intractable'', whereas Pfaffians can be computed in polynomial time. More precisely, [\cite{Je87}] proves that the enumeration of MD coverings of generic planar graphs is
``$\# P$ complete'', which implies that it is believed not to be computable in polynomial time.
\indent However, by introducing restrictions on the location of monomers, such a result can be proven in some cases. Namely, in [\cite{TW03}, \cite{Wu06}], the authors use a Pfaffian formula to compute the partition function of a system with a {\it single} monomer located on the boundary of a finite square lattice. In [\cite{PR08}], the MD problem is studied on the square lattice on the half-plane with the restriction that the monomers are {\it fixed} on points of the boundary. They derive a Pfaffian formula for this case, and use it to compute the scaling limit of the multipoint boundary monomer correlations. Finally, in [\cite{AF14}], it is shown that if the monomers are {\it fixed} at any position in a square lattice, then the partition function can also be written as a Pfaffian. In addition, the authors use their formula for
computing the asymptotics of monomer correlations, thus generalizing the classical results by Fisher, Hartwig and Stephenson [\cite{FS63}, \cite{Ha66}, \cite{FH69}]
for monomer correlations along a row, column or main diagonal.
\indent However, by introducing restrictions on the location of monomers, such a result can be proven in some cases. Namely, in [\cite{TW03}, \cite{Wu06}], the authors derive a Pfaffian formula, based on the ``Temperley bijection'' [\cite{Te74}], for the partition function of a system with a {\it single} monomer located on the boundary of a finite square lattice, and in [\cite{WTI11}], on a cylinder of odd width (which is a nonbipartite lattice). In [\cite{PR08}], the MD problem is studied on the square lattice on the half-plane with the restriction that the monomers are {\it fixed} on points of the boundary. They derive a Pfaffian formula for this case, and use it to compute the scaling limit of the multipoint boundary monomer correlations. Finally, in [\cite{AF14}], it is shown that if the monomers are {\it fixed} at any position in a square lattice, then the partition function can also be written as a Pfaffian. In addition, the authors use their formula for computing the asymptotics of monomer correlations, thus generalizing the classical results by Fisher, Hartwig and Stephenson [\cite{FS63}, \cite{FH69}] and [\cite{Ha66}] (which is claimed to have small mistakes in~[\cite{AP84}], which also contains the corrected formula) for monomer correlations along a row, column or main diagonal.
\indent In the present work, we prove that, on an {\it arbitrary} planar graph, the {\it boundary} MD partition function (in which the monomers are restricted to the boundary of the graph, but are not necessarily fixed at prescribed locations) with arbitrary dimer and monomer weights can be written as a Pfaffian.
By differentiation with respect to the monomer weights, we also obtain a fermionic Wick rule for boundary monomer correlations.
@ -85,11 +83,11 @@ planar graphs is via the random current representation developed by
Aizenman [\cite{Ai82}]. It has been recently observed
[\cite{AD}] that this representation, adapted to planar lattices, implies, for
purely geometrical
reasons, the validity of the fermionic Wick rule both for boundary spin
correlations in the nearest neighbor Ising model, and might imply the
reasons, the validity of the fermionic Wick rule for boundary spin
correlations in the nearest neighbor Ising model (which has already been proved by J.~Groeneveld, R.J.~Boel and P.W.~Kasteleyn~[\cite{GBK78}]), and might imply the
same for boundary monomer correlations in the dimer model.
Their method also suggests a stochastic geometric perspective on the
emergence of planarity at the critical points of non-planar 2D models, in the sense of the previous item.
emergence of planarity at the critical points of non-planar 2D models, in the sense of the previous item. Note that our Pfaffian formula (see theorem~\ref{theomain}) goes beyond the Wick rule, see the \ref{rkWick} at the end of section~\ref{subsecpfaffian}.
\item It may be worth noting that the MD partition function can be computed exactly in some cases, e.g. on the complete graph [\cite{ACM14}].
\end{itemize}
\unlistparpenalty
@ -521,11 +519,18 @@ M_n(i_1,\cdots,i_{2n})=&
(-1)^n\mathrm{pf}(\{a^{-1}(\mathbf d)\}_{\mathcal I})\\[0.3cm]
=&\frac{1}{2^nn!}\sum_{\pi\in\mathcal S_{2n}}(-1)^\pi\prod_{j=1}^n(-a^{-1}(\mathbf d))_{i_{\pi(2j-1)},i_{\pi(2j)}},
\end{array}\label{eqpfMconc}\end{equation}
which concludes the proof, by noting that $(-a^{-1}(\mathbf d))_{i,j}=M_1(i,j)$.
which concludes the proof, by noting that
\begin{equation}
(-a^{-1}(\mathbf d))_{i,j}=M_1(i,j).
\label{eqM1a}\end{equation}
\hfill$\square$
\bigskip
\makelink{rkWick}{remark}{\bf Remark}: We have shown that theorem~\ref{theomain} implies corollary~\ref{corrmain}. It turns out that the converse is also true assuming~(\ref{eqM1a}) holds (with the sign that appears in~(\ref{eqM1a})). More precisely, corollary~\ref{corrmain} and~(\ref{eqM1a}) imply theorem~\ref{theomain}. Indeed, (\ref{eqM1a}) and corollary~\ref{corrmain} imply
$$
\Xi_\partial(\bm\ell,\mathbf d)=\Xi_\partial(\mathbf0,\mathbf d)\sum_{\mathcal I\subset\{1,\cdots,|g|\}}\mathrm{pf}(\{-a^{-1}\}_{\mathcal I})\prod_{j\in\mathcal I}\ell_j
$$
which, by~(\ref{eqkasteleyntheo}), (\ref{eqpfminors}) and theorem~\ref{theolieb}, yields~(\ref{eqtheo}). As a consequence, if one were to prove the Wick rule for boundary monomers (possibly by extending the analysis of~[\cite{GBK78}] to the monomer-dimer problem) then the Pfaffian formula~(\ref{eqtheo}) could be recovered by directing and labeling the graph $g$ in such a way that~(\ref{eqM1a}) holds, which can be achieved by ensuring that $[g]_{\{v,v'\}}$ is positive and Kasteleyn for every $v,v'\in\mathcal V(\partial g)$ (in the present paper, we prove the Pfaffian formula~(\ref{eqtheo}) without first proving the Wick rule, and ensuring that $[g]_{\mathcal I}$ is Kasteleyn and positive for every $\mathcal I\subset\mathcal V(\partial g)$, which does not seem to be harder than proving it for sets of cardinality 2). In other words, the Pfaffian formula~(\ref{eqtheo}) that counts MD coverings with any number of monomers on the boundary can be seen as a consequence of a similar Pfaffian formula for the MD coverings with 2 monomers on the boundary and the Wick rule.
\section{Simple graphs}\label{secsimple}
@ -1450,10 +1455,9 @@ replace the edge according to figure~\ref{figreplodd}.
\hfil{\Large\bf Acknowledgments}\par
\bigskip
We thank Michael Aizenman and Hugo Duminil-Copin for discussing their work in progress on the random current representation for planar lattice models with us. We gratefully acknowledge financial support from the A*MIDEX project ANR-11-IDEX-0001-02 (A.G.),
from the PRIN National Grant {\it Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions} (A.G. and I.J.), and
NSF grant PHY-1265118 (E.H.L.).
We also would like to thank Tom Spencer and Joel Lebowitz for their hospitality at the IAS in Princeton and their continued interest in this problem.
\indent Thanks go to Jan Philip Solovej and Lukas Schimmer for devoting their time to some preliminary calculations that helped put us on the right track. We also would like to thank Tom Spencer and Joel Lebowitz for their hospitality at the IAS in Princeton and their continued interest in this problem. In addition, we thank Michael Aizenman and Hugo Duminil-Copin for discussing their work in progress on the random current representation for planar lattice models with us. We thank Jacques Perk and Fa Yueh Wu for very useful historical comments.
\indent We gratefully acknowledge financial support from the A*MIDEX project ANR-11-IDEX-0001-02 (A.G.), from the PRIN National Grant {\it Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions} (A.G. and I.J.), and NSF grant PHY-1265118 (E.H.L.).
\vfill
\eject

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@ -2,6 +2,7 @@
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@ -10,6 +11,7 @@
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@ -29,5 +31,7 @@
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@ -76,6 +76,12 @@
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