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Diffstat (limited to 'Giuliani_Jauslin_Lieb_2015.tex')
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diff --git a/Giuliani_Jauslin_Lieb_2015.tex b/Giuliani_Jauslin_Lieb_2015.tex index 68b3fd1..43dc9da 100644 --- a/Giuliani_Jauslin_Lieb_2015.tex +++ b/Giuliani_Jauslin_Lieb_2015.tex @@ -50,7 +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 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 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 the product of two Pfaffians. \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. @@ -62,6 +62,7 @@ in particular for graphs that may not admit a transfer matrix treatment. {\bf Remarks}: \listparpenalty \begin{itemize} +\item The asymptotic behavior of monomer pair correlations on the square lattice have been computed explicitly [\cite{FS63}, \cite{Ha66}, \cite{FH69}] for monomers on a row, column or diagonal (note that, as mentionned in~[\cite{AP84}], [\cite{Ha66}] contains small mistakes). In addition, the general bulk monomer pair correlations have been shown~[\cite{AP84}] to be expressible in terms of two critical Ising correlation functions. \item An alternative approach for the boundary MD problem on an $N\times M$ rectangle (by which we mean $N$ vertices times $M$ vertices) with monomers allowed only on the upper and lower sides would be to use the transfer matrix technique [\cite{Li67}]. In that case, the boundary MD partition function is written as $x_M\cdot V^{M-1}x_1$ where $V$ is the ($N\times N$) transfer matrix and $x_1$ and $x_M$ are vectors determined by the boundary condition at the boundaries $y=1$ and $y=M$ respectively. |