For the Monomer-Dimer lambda_d(p), the Master Algebraic Conjecture

TL;DR

This paper explores the equivalence of two different power series expressions for lambda_d(p) in the monomer-dimer problem, proposing a conjecture that these expressions are identical, with a focus on the case where all J_i are zero.

Contribution

The paper formulates the Master Algebraic Conjecture asserting the equivalence of two series representations for lambda_d(p) and analyzes the special case where all J_i are zero.

Findings

Two series expressions for lambda_d(p) are shown to be equivalent in a special case.

The sets {b_i} and {J_i} can be derived from each other, linking cluster expansion and Mayer series.

The conjecture remains unproven in the general case, with analysis focused on the all J_i zero scenario.

Abstract

The author has recently presented two different expressions for lambda_d(p) of the monomer-dimer problem involving a power series in p, the first jointly with Shmuel Friedland. These two expressions are certainly equal, but this has not yet been proven rigorously. The first is naturally developed from quantities J_i, cluster expansion kernels. The second from the Mayer (or Virial) series of a dimer gas, in particular from the b_i coefficients in the Mayer series. The sets {b_i} and {J_i} can be derived from each other. Given an arbitrary set of values for either the b_i or the J_i, both expressions may be given in terms of a formal sum. The master algebraic conjecture is that these two expressions are equivalent. This is detailed in the special case all J_i are zero.