From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

TL;DR

This paper links the Lifshitz tail phenomenon to the quenched survival probabilities of diffusing particles in random traps, providing sharp bounds and asymptotics applicable to various trap distributions.

Contribution

It introduces a simple method to derive quenched survival asymptotics from Lifshitz tail effects, extending results to general trap configurations and specific lattice perturbations.

Findings

Upper bounds for survival probabilities are established in general settings.

Sharp asymptotics are proved for Brownian motion in Poissonian obstacles.

Quenched asymptotics are derived for traps with lattice perturbations.

Abstract

The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion in the Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion in traps distributed according to a random perturbation of the lattice.