Strict comparison for the Lyapunov exponents of the simple random walk in random potentials

TL;DR

This paper proves that the Lyapunov exponent for a simple random walk in i.i.d. nonnegative potentials on a lattice is strictly monotone with respect to the potential's distribution, impacting the understanding of travel costs in random environments.

Contribution

It establishes the strict monotonicity of the Lyapunov exponent in the distribution function of the potential, a novel comparison result in this context.

Findings

Lyapunov exponent is strictly monotone under distribution dominance.

Comparison results extend to the rate function of the model.

Provides a new understanding of travel costs in random potentials.

Abstract

We consider the simple random walk in i.i.d. nonnegative potentials on the $d$-dimensional cubic lattice $\mathbb{Z}^d$ ($d \geq 1$). In this model, the so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential. The Lyapunov exponent depends on the distribution function of the potential, and the aim of this article is to prove that the Lyapunov exponent is strictly monotone in the distribution function of the potential with the order according to strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.