Quenched tail estimate for the random walk in random scenery and in random layered conductance

TL;DR

This paper derives the asymptotic probabilities of large deviations for a symmetric random walk in a heavy-tailed random environment, with applications to layered conductance models.

Contribution

It provides the first detailed quenched tail estimates for the random walk in random scenery with power-law tails, and extends these results to layered conductance models.

Findings

Identified the asymptotic behavior of upper deviation probabilities.

Established tail estimates depending on scenery distribution tails.

Applied results to random conductance models with layered structures.

Abstract

We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time aymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.