Differences between Lyapunov exponents for the simple random walk in Bernoulli potentials

TL;DR

This paper investigates how Bernoulli-distributed potentials affect the Lyapunov exponent of a simple random walk on a lattice, providing Lipschitz-type estimates for the differences in these exponents.

Contribution

It offers a detailed analysis of the potential's influence on the Lyapunov exponent, including deriving Lipschitz-type bounds, which advances understanding beyond known monotonicity.

Findings

Derived Lipschitz-type estimates for Lyapunov exponent differences

Confirmed strict monotonicity of Lyapunov exponent in Bernoulli parameter

Provided quantitative bounds on potential's impact

Abstract

We consider the simple random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ ($d \geq 1$), traveling in potentials which are Bernoulli distributed. The so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential, and it is known that the Lyapunov exponent is strictly monotone in the parameter of the Bernoulli distribution. Hence, the aim of this paper is to investigate the effect of the potential on the Lyapunov exponent more precisely, and we derive some Lipschitz-type estimates for the difference between the Lyapunov exponents.