Extinction time for a random walk in a random environment

TL;DR

This paper analyzes the survival time of a random walk with death in a dynamic environment influenced by particle flux, providing an exponential upper bound on survival probability uniform in system size.

Contribution

It introduces a model of a random walk in a time-dependent environment with death, and establishes a uniform exponential upper bound on survival probability.

Findings

Survival probability decays exponentially with time.

Upper bound is uniform in the size of the system.

The decay rate scales with the inverse square of the system size.

Abstract

We consider a random walk with death in $[-N,N]$ moving in a time dependent environment. The environment is a system of particles which describes a current flux from $N$ to $-N$. Its evolution is influenced by the presence of the random walk and in turn it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in $N$) for the survival probability up to time $t$ which goes as $c\exp\{-bN^{-2}t\}$, with $c$ and $b$ positive constants.