The Aesthetic Asymptotics of the Mayer Series Coefficients for a Dimer Gas on a Regular Lattice

Abstract

We conjecture that for all regular lattices b(n) is asymptotically of the form in eq.(A1). (-1)^{n+1} b(n) = exp( k(-1) n + k(0) ln(n) + k(1) / n + k(2) / n^(2)...) (A1) We restrict testing this to lattices for which we know the first 20 Mayer series coefficients, the b(n). This includes the infinite number of rectangular lattices, one for each dimension, the tetrahedral lattice ( in this one case we know only the first 19 coefficients ), and the (bipartite) body centered cubic lattices, in dimensions 3 through 7. In this paper we will detail results for the rectangular lattices in dimensions 2,3,5,11,and 20, for the tetrahedral lattice, and for the body centered cubic lattices in dimensions 3,4, and 5. These are all bipartite, unfortunately we do not have an example of a non-bipartite regular lattice for which we know enough of the b(n) to work with. For the triangular lattice, regular and non-bipartite, we know the first 14 b(n). We feel this is not enough terms to make any judgement, hopefully someone may compute more terms. We work with an 'approximation' that keeps the first four terms, in k{-1), k(0), k(1), k(2), in the exponent in eq.(A1). Agreement will be striking. At the end of Part 1 there is a digression on a conjecture in line with recent applications of the renormalization group to study phase transitions and the ideas of Cardy, [10]. In Part 7 there is some study of susceptibility series for the Ising model on the 2d rectangular lattice, triangular lattice, and honeycomb lattice; where there is surprising similarity to Mayer series on regular graphs, as studied herein. Also in Part 7 we show, mirabile dictu, that the number of partitions function. p(n), has the 'magic' property. A CHALLENGE TO COMBINATORISTS, EXPLAIN THIS.