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

TL;DR

This paper introduces the matching measure for lattices, linking it to the monomer-dimer free energy, and uses it to derive improved bounds for Euclidean lattices through spectral analysis and convergence techniques.

Contribution

It establishes a novel connection between the matching measure and monomer-dimer free energy, providing a new approach to bounding these energies on lattices.

Findings

Derived rigorous bounds on monomer-dimer free energies for Euclidean lattices.

Connected spectral measures to partition functions through Benjamini-Schramm convergence.

Improved existing bounds significantly using spectral analysis and computational data.

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.