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

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.