A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

TL;DR

This paper establishes a shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential, applicable across various models including polymers and percolation, under broad conditions.

Contribution

It introduces a unified framework for analyzing the quenched Lyapunov exponent and Green's function in diverse random walk models with a general potential.

Findings

Proves a shape theorem for the quenched Lyapunov exponent.

Derives a variational formula for the limiting behavior.

Applies results to models like polymers and percolation at zero temperature.

Abstract

We prove a shape theorem and derive a variational formula for the limiting quenched Lyapunov exponent and the Green's function of random walk in a random potential on a square lattice of arbitrary dimension and with an arbitrary finite set of steps. The potential is a function of a stationary environment and the step of the walk. This potential is subject to a moment assumption whose strictness is tied to the mixing of the environment. Our setting includes directed and undirected polymers, random walk in static and dynamic random environment, and, when the temperature is taken to zero, our results also give a shape theorem and a variational formula for the time constant of both site and edge directed last-passage percolation and standard first-passage percolation.