15-99 Paul Federbush

Proof of Convergence for the Lattice Monomer-Dimer Cluster Expansion I, a Simplified Model (31K, LaTeX) Sep 13, 15

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Abstract. In a set of papers we address 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 lambda_d(p)=(1/2)(pln(2d)-pln(p)-2(1-p)ln(1-p)-p) (A) +sum a_k(d)p^k where a_k(d) is a sum of powers (1/d)^r for k-1 >= r >= k/2 (B) In fact as we will point out one has already rigorously established the convergence of the sum in (A) for small p. It is the d dependence of a_k(d) that has yet to be rigorously shown. The study of these papers establishes the convergence of our cluster expansions, resulting in the said d dependence. In this paper we study a cute little problem, claiming the general case can be dealt with as a dressing up of this skeleton problem.

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