Existence of generalized Busemann functions and Gibbs measures for random walks in random potentials

TL;DR

This paper proves the existence of generalized Busemann functions and Gibbs measures for a broad class of lattice random walks in random potentials, including models like directed polymers and percolation, under minimal assumptions.

Contribution

It establishes the existence of these functions and measures for a wide range of models, extending previous results to more general settings and dimensions.

Findings

Existence of generalized Busemann functions for various models.

Existence of Gibbs measures in complex random environments.

Applicable to static and dynamic random potentials across all dimensions.

Abstract

We establish the existence of generalized Busemann functions and Gibbs-Dobrushin-Landford-Ruelle measures for a general class of lattice random walks in random potential with finitely many admissible steps. This class encompasses directed polymers in random environments, first- and last-passage percolation, and elliptic random walks in both static and dynamic random environments in all dimensions and with minimal assumptions on the random potential.