Proof of Convergence for the Lattice Monomer-Dimer Cluster Expansion I, a Simplified Model

TL;DR

This paper discusses promising ideas to rigorously prove the convergence of a cluster expansion expression for the monomer-dimer problem on high-dimensional lattices, focusing on the $d$-dependence of expansion coefficients.

Contribution

It introduces a simplified model and proposes methods to establish the rigorous convergence of the cluster expansion for the monomer-dimer problem.

Findings

Convergence of the sum for small $p$ is already rigorously established.

The $d$-dependence of the coefficients $a_k(d)$ remains to be rigorously proven.

The paper suggests promising ideas for future rigorous proofs.

Abstract

We present some promising ideas to treat the problem of making completely rigorous the development of our expression for $\lambda_d(p)$ of the monomer-dimer problem on a $d$-dimensional hypercubic lattice \begin{equation}\label{abstract1} \lambda_d(p)=\frac{1}{2}\Big(p\ln(2d)-p\ln(p)-2(1-p)\ln(1-p)-p\Big) +\sum_{k=2}a_k(d)p^k \end{equation} where $a_k(d)$ is a sum of powers $(1/d)^r$ for \begin{equation}\label{abstract2} k-1\leq r\leq k/2 \end{equation} In fact as we will point out one has allready rigorously established the convergence of the sum in expression for $\lambda$ for small $p$. It is the $d$ dependence of $a_k(d)$ that has yet to be rigorously shown. We do not now know how to complete the proof.