An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Abstract

Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).

15 pages, much more about d=1,2,3

References in corpus (7)

• Validations of the Asymptotic Matching Conjectures
• Computation of Terms in the Asymptotic Expansion of Dimer lambda_d for High Dimension
• Convergence of the Formal Expansion for lambda_d(p) of the Monomer-Dimer Problem for Small p
• Dimer lambda_d Expansion, A Contour Integral Stationary Point Argument
• Dimer lambda_d Expansion Computer Computations
• Dimer lambda_d Expansion, Dimensional Dependence of J_n Kernels
• Hidden Structure in Tilings, Conjectured Asymptotic Expansion for lambda_d in Multidimensional Dimer Problem

Cited by in corpus (6)

• Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices
• A positivity property of the dimer entropy of graphs
• Convergence of the Formal Expansion for lambda_d(p) of the Monomer-Dimer Problem for Small p
• Power series expansions for the planar monomer-dimer problem
• Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems
• Matchings in vertex-transitive bipartite graphs