Variational formulas and cocycle solutions for directed polymer and percolation models

TL;DR

This paper develops variational formulas for the asymptotic behavior of directed polymer and percolation models, applicable in arbitrary dimensions and with general path steps, linking cocycle solutions and invariant measures.

Contribution

It introduces new variational formulas involving cocycles and invariant measures for directed polymers and percolation, extending existing results to more general models and environments.

Findings

Variational formulas valid in arbitrary dimensions.

Connection between cocycles and Busemann functions.

Application to 1+1 dimensional exactly solvable models.

Abstract

We discuss variational formulas for the limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under shifts, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder.