Upper bounds for the connective constant of weighted self-avoiding walks

TL;DR

This paper introduces a method to compute upper bounds for the weighted connective constant of self-avoiding walks on lattices, with applications in statistical physics and a publicly available software implementation.

Contribution

It develops a novel eigenvalue-based approach for bounding the connective constant and extends techniques for anisotropic contour models in statistical physics.

Findings

Upper bounds obtained as dominant eigenvalues of matrices

Method implemented in publicly available software

Potential applications in Peierls-type estimates for contour models

Abstract

Building on a work by Alm, we consider a model of weighted self-avoiding walks on a lattice and develop a method for computing upper bounds on the corresponding weighted connective constant, which we implement in a publicly available software package. The upper bounds are obtained as the dominant eigenvalues of certain matrices and provide detailed information about the convergence of the model's multivariate generating function. We discuss potential applications of our results to developing Peierls-type estimates for anisotropic contour models in statistical physics, generalizing a technique recently introduced by Fahrbach--Randall.