Convergence of equilibrium measures corresponding to finite subgraphs of infinite graphs: new examples

TL;DR

This paper investigates the asymptotic behavior of equilibrium measures for increasing finite subgraphs of infinite graphs, introducing new examples and a geometric approach to extend previous results in thermodynamic formalism.

Contribution

It provides a solution for a new class of matrices using loaded graphs, expanding the understanding of equilibrium measures in infinite graph settings.

Findings

Established convergence results for a new matrix class

Introduced a geometric language of loaded graphs

Extended previous thermodynamic formalism results

Abstract

A problem from thermodynamic formalism for countable symbolic Markov chains is considered. It concerns asymptotic behavior of the equilibrium measures corresponding to increasing sequences of finite sub-matrices of an infinite nonnegative matrix $A$ when these sequences converge to $A$. After reviewing the results obtained up to now, a solution of the problem is given for a new matrix class, which differs from those studied previously in some essential feature. A geometric language of loaded graphs instead of the matrix language is used.