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.