Infinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts

TL;DR

This paper investigates DLR measures on countable Markov shifts, establishing their relation to conformal measures, and analyzes phase transitions related to measure finiteness at different temperatures, especially on the renewal shift.

Contribution

It extends the understanding of DLR measures in countable Markov shifts, showing their connection to conformal measures and characterizing phase transitions for specific shifts.

Findings

DLR measures include conformal measures for Walters potentials.

In BIP cases, DLR and conformal measures coincide.

Critical temperature for volume-type phase transition is identified.

Abstract

We consider the natural definition of DLR measure in the setting of $\sigma$-finite measures on countable Markov shifts. We prove that the set of DLR measures contains the set of conformal measures associated with Walters potentials. In the BIP case, or when the potential normalizes the Ruelle's operator, we prove that the notions of DLR and conformal coincide. On the standard renewal shift, we study the problem of describing the cases when the set of the eigenmeasures jumps from finite to infinite measures when we consider high and low temperatures, respectively. For this particular shift, we prove that there always exist finite DLR measures, and we have an expression to the critical temperature for this volume-type phase transition, which occurs only for potentials with the infinite first variation.