Survival time of random walk in random environment among soft obstacles

TL;DR

This paper investigates the survival time of a random walk in a random environment with obstacles, providing bounds and detailed results in various models, including one-dimensional cases and branching random walks.

Contribution

It offers new bounds on survival and hitting times for RWRE among obstacles, including refined results for one-dimensional models and mixed probability measures.

Findings

Derived quenched and annealed tail bounds for survival times in general dimensions.

Obtained finer tail results for one-dimensional models with independent transition probabilities and obstacles.

Extended methods to bounds on hitting times of Branching Random Walks in Random Environment.

Abstract

We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.\ random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general $d$-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the "mixed" probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE).