Localization of a one-dimensional simple random walk among power-law renewal obstacles

TL;DR

This paper studies how a one-dimensional random walk localizes among power-law distributed obstacles, revealing a phase transition in behavior depending on the tail exponent of the obstacle distribution.

Contribution

It characterizes the localization behavior of the walk conditioned on survival, identifying a dichotomy based on the renewal tail exponent and providing detailed probabilistic bounds.

Findings

Walk localizes in a unique obstacle-free gap with high probability.

Behavior differs depending on the tail exponent: complete trapping or partial excursions.

Provides bounds on the length and number of excursions outside the optimal gap.

Abstract

We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap. If the renewal tail exponent is larger than one then the random walk spends most of its time inside of the optimal gap but also performs short outward excursions, for which we provide matching upper and lower bounds on their length and cardinality. Our key tools include a Markov renewal interpretation of the survival probability as well as various comparison arguments for obstacle environments. Our results may also be rephrased in terms of localization properties for a directed polymer among multiple repulsive interfaces.