Thermodynamic Formalism for Transient Potential Functions

TL;DR

This paper extends thermodynamic formalism to transient potentials in Markov shifts, demonstrating existence of eigenfunctions and measures, and relating these to phase transitions in lattice gas models with infinite states.

Contribution

It introduces a framework for analyzing transient potentials, including eigenfunctions and eigenmeasures, and connects these to phase transitions in complex lattice models.

Findings

Existence of positive continuous eigenfunctions for the Ruelle operator.

Characterization of eigenmeasures via Martin kernels and escape directions.

Application to phase transitions in infinite-state lattice gas models.

Abstract

We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and positive Radon eigenmeasures in forms of Martin kernels. These eigenmeasures can be characterized in terms of the direction of escape to infinity of their orbits, when viewed inside a suitable Martin-like compactification of the underline shift space. We relate these results to first-order phase transitions in one-dimensional lattice gas models with infinite set of states. This work complements earlier works by Sarig who focused on the recurrent scenario.