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

TL;DR

This paper establishes a sharp lower bound on the annealed Lyapunov exponent for a random walk in a non-integrable potential, revealing new insights into the decay of the Green function of the Anderson Hamiltonian as eta approaches zero.

Contribution

It provides the first sharp lower bound on the annealed Lyapunov exponent without assuming potential integrability, extending understanding of random walks in complex potentials.

Findings

Sharp lower bound on annealed Lyapunov exponent for small eta

Results applicable to non-integrable potentials

Insights into decay of Green function of Anderson Hamiltonian

Abstract

We consider the simple random walk on Z^d, d > 2, evolving in a potential of the form \beta V, where (V(x), x \in Z^d) are i.i.d. random variables taking values in [0,+\infty), and \beta\ > 0. When the potential is integrable, the asymptotic behaviours as \beta\ tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small \beta. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -\Delta\ + \beta V.