Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^2$

TL;DR

This paper investigates directed polymer models on the 2D lattice, establishing new results on the existence, uniqueness, and asymptotic behavior of semi-infinite measures, and analyzing Busemann functions and measure non-existence.

Contribution

It introduces covariant cocycles to study semi-infinite polymer measures and proves new results on their existence, uniqueness, and asymptotic directions, along with non-existence of bi-infinite measures.

Findings

Existence of covariant cocycles for directed polymers.

Conditions for uniqueness and non-uniqueness of semi-infinite measures.

Almost sure existence of Busemann function limits in regular directions.

Abstract

We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity.