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Reference: [Ay15]

Comment: Wick rule only at close packing
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Ian Jauslin 2015-11-06 11:20:21 +00:00
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Giuliani_Jauslin_Lieb_2015.tex
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@ -53,7 +53,7 @@ Clearly, $\Xi(z)$ is a polynomial in $z$, and $z$ is called the {\it monomer fug
\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 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. \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. By differentiation with respect to the monomer weights, we also obtain a fermionic Wick rule for boundary monomer correlations at 0-monomer density (see~(\ref{eqcorrdef}) for a precise definition).
There are two main novelties in our result. First, we obtain the Pfaffian formula for the partition function with arbitrary weights rather than for There are two main novelties in our result. First, we obtain the Pfaffian formula for the partition function with arbitrary weights rather than for
the problem with monomers at fixed locations. Second, we study the boundary MD problem on {\it any} planar graph, the problem with monomers at fixed locations. Second, we study the boundary MD problem on {\it any} planar graph,
in particular for graphs that may not admit a transfer matrix treatment. in particular for graphs that may not admit a transfer matrix treatment.
@ -152,6 +152,7 @@ does not require the graph to be treatable via a transfer matrix approach.
\item In addition, we have developed an alternative algorithm (see appendix~\ref{appinout}) to express the {\it full} MD partition function on Hamiltonian planar graphs as a derivative of the product of just two Pfaffians. From a computational point of view, this approach is even slower than the previous one, but it is nonetheless conceptually interesting. Note that this algorithm can be adapted to non-planar graphs as well. \item In addition, we have developed an alternative algorithm (see appendix~\ref{appinout}) to express the {\it full} MD partition function on Hamiltonian planar graphs as a derivative of the product of just two Pfaffians. From a computational point of view, this approach is even slower than the previous one, but it is nonetheless conceptually interesting. Note that this algorithm can be adapted to non-planar graphs as well.
\item Finally, we have also computed upper and lower bounds for the {\it full} partition function, see theorems~\ref{theolowerbound} and~\ref{theoupperbound}. \item Finally, we have also computed upper and lower bounds for the {\it full} partition function, see theorems~\ref{theolowerbound} and~\ref{theoupperbound}.
\item As a side remark, note that Monte Carlo methods methods can be even faster, i.e., polynomial in the size of the system~[\cite{KRS96}], but they provide correct results only with high probability rather than with certainty. \item As a side remark, note that Monte Carlo methods methods can be even faster, i.e., polynomial in the size of the system~[\cite{KRS96}], but they provide correct results only with high probability rather than with certainty.
\item A result similar to theorem~\ref{theomain} has recently been established~[\cite{Ay15}] for another model, called the {\it monopole-dimer} model, for which the partition function can be written as a determinant.
\end{itemize} \end{itemize}
\unlistparpenalty \unlistparpenalty
\bigskip \bigskip
@ -171,6 +172,8 @@ M_n(i_1,\cdots,i_{2n})=\mathrm{pf}(\mathfrak M_{i_1,\ldots,i_{2n}}),
\label{eqwick}\end{equation} \label{eqwick}\end{equation}
where $\mathfrak M_{i_1,\ldots,i_{2n}}$ is the $2n\times 2n$ antisymmetric matrix whose $(j,j')$-th entry with $j<j'$ is $M_1(i_j,i_{j'})$. where $\mathfrak M_{i_1,\ldots,i_{2n}}$ is the $2n\times 2n$ antisymmetric matrix whose $(j,j')$-th entry with $j<j'$ is $M_1(i_j,i_{j'})$.
\endtheo \endtheo
{\bf Remark}: Away from close packing (i.e. omitting $\ell_1=\cdots=\ell_{|g|}$ in~(\ref{eqcorrdef})), {\it the Wick rule does not hold}. This can be checked immediately by considering a graph consisting of a square with an extra edge on the diagonal.
\bigskip \bigskip
\indent As stated in theorem~\ref{theomain}, the edges and vertices of $g$ must be directed and labeled in a special way. In particular, the direction of the edges must satisfied a so called {\it Kasteleyn} condition, and the labeling must satisfy a {\it positivity} condition. The positivity condition ensures that the terms that appear in the Pfaffian add up constructively and reproduce the MD partition function. The Kasteleyn condition is used to prove the positivity of a graph: if such a condition holds, then it suffices to look at a single dimer covering of $g$ to prove its positivity. \indent As stated in theorem~\ref{theomain}, the edges and vertices of $g$ must be directed and labeled in a special way. In particular, the direction of the edges must satisfied a so called {\it Kasteleyn} condition, and the labeling must satisfy a {\it positivity} condition. The positivity condition ensures that the terms that appear in the Pfaffian add up constructively and reproduce the MD partition function. The Kasteleyn condition is used to prove the positivity of a graph: if such a condition holds, then it suffices to look at a single dimer covering of $g$ to prove its positivity.

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@ -3,6 +3,7 @@
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