Weak quenched limit theorems for a random walk in a sparse random environment

TL;DR

This paper investigates the quenched behavior of a symmetric random walk with random drifts at sparse, randomly chosen sites, revealing the absence of strong quenched limit laws under certain conditions and exploring weak convergence of measures.

Contribution

It introduces a model with sparse random perturbations and demonstrates the non-existence of strong quenched limit laws when gaps are regularly varying with small index.

Findings

No strong quenched limit laws under specified conditions

Weak convergence of random measures is studied

Gaps between marked sites significantly affect limit behavior

Abstract

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.