Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy

Abstract

We define the matching measure of a lattice L as the spectral measure of the tree of self-avoiding walks in L. We connect this invariant to the monomer-dimer partition function of a sequence of finite graphs converging to L. This allows us to express the monomer-dimer free energy of L in terms of the measure. Exploiting an analytic advantage of the matching measure over the Mayer series then leads to new, rigorous bounds on the monomer-dimer free energies of various Euclidean lattices. While our estimates use only the computational data given in previous papers, they improve the known bounds significantly.

18 pages, 3 figures

References in corpus (3)

• Yang-Lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 <= d <= 7
• Approximating the monomer-dimer constants through matrix permanent
• Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices

Cited by in corpus (4)

• Spectral measures of factor of i.i.d. processes on vertex-transitive graphs
• Chromatic roots and limits of dense graphs
• Power series expansions for the planar monomer-dimer problem
• Atoms of the matching measure