P. Butera, P. Federbush, M. Pernici
Higher order expansions for the entropy of a dimer or a monomer-dimer system on
d-dimensional lattices
(55K, LaTeX)
ABSTRACT. Recently an expansion as a power series in 1/d has been presented
for the specific entropy of a complete dimer covering of a
d-dimensional hypercubic lattice. This paper extends from 3 to 10
the number of terms known in the series. Likewise an expansion for
the dimer-density p-dependent entropy of a monomer-dimer system
involving a sum of a_k(d) p^k has been recently offered. We
herein extend the number of the known expansion coefficients from 6
to 20 for the hypercubic lattices of general dimensionality d and from
6 to 24 for the lattices of dimensionalities d < 5 .
We show that this extension can lead to accurate numerical estimates
of the p-dependent entropy for lattices with dimension d > 2.
The computations of this paper have led us to make the following marvelous
conjecture: In the case of the hypercubic lattices, all the
expansion coefficients, a_k(d) , are positive! This paper
results from a simple melding of two disparate research programs:
one computing to high orders the Mayer series coefficients of a
dimer gas, the other studying the development of entropy from these
coefficients. An effort is made to make the paper self-contained by
including a review of the earlier works.