Lyapunov exponents of random walks in small random potential: the upper bound

TL;DR

This paper analyzes the asymptotic behavior of Lyapunov exponents for simple random walks in a non-integrable, rescaled random potential on high-dimensional lattices, extending previous work to small potential scales.

Contribution

It provides the first detailed asymptotic analysis of Lyapunov exponents in this setting for dimensions three and higher, completing prior research.

Findings

Asymptotic formulas for Lyapunov exponents as potential scale tends to zero

Results apply to both annealed and quenched cases

Extension of previous work to non-integrable potentials

Abstract

We consider the simple random walk on $\mathbb{Z}^d$ evolving in a random i.i.d. potential taking values in $[0,+\infty)$. The potential is not assumed integrable, and can be rescaled by a multiplicative factor $\lambda > 0$. Completing the work started in a companion paper, we give the asymptotic behaviour of the Lyapunov exponents for $d \ge 3$, both annealed and quenched, as the scale parameter $\lambda$ tends to zero.