Quenched Point-to-Point Free Energy for Random Walks in Random Potentials

TL;DR

This paper proves the existence and regularity of quenched point-to-point free energy for random walks in ergodic random potentials, establishing a variational formula and linking it to large deviations and infinite volume free energy.

Contribution

It introduces a variational formula for quenched free energy in random environments with unbounded potentials, extending previous results to more general settings.

Findings

Existence of quenched point-to-point free energy.

Regularity properties and variational formula established.

Quenched large deviation principle with continuous rate function.

Abstract

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and some steps of the walk. The potential can be unbounded, but it is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of an entropy. We establish regularity properties of the point-to-point free energy, as a function of the potential and as a function on the convex hull of the admissible steps of the walk, and link it to the infinite volume free energy and quenched large deviations of the endpoint of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.