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

## TL;DR

This paper advances the understanding of the tail behavior of the random walk in random scenery and layered conductance, providing bounds on deviations and asymptotics of key probabilities and functions.

## Contribution

It completes the analysis of upper deviations and establishes bounds on lower deviations, extending previous work and deriving asymptotics for return probability, moderate deviations, and the Green function.

## Key findings

- Bounds on upper deviation probabilities
- Bounds on lower deviation probabilities
- Asymptotic formulas for return probability and Green function

## Abstract

This is a continuation of our earlier work [Stochastic Processes and their Applications, 129(1), pp.102--128, 2019] on the random walk in random scenery and in random layered conductance. We complete the picture of upper deviation of the random walk in random scenery, and also prove a bound on lower deviation probability. Based on these results, we determine asymptotics of the return probability, a certain moderate deviation probability, and the Green function of the random walk in random layered conductance.