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

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.