Random walk on discrete point processes

TL;DR

This paper studies random walks on a specific type of random environment in high-dimensional lattices, proving zero velocity, characterizing recurrence and transience, and establishing a Central Limit Theorem under certain conditions.

Contribution

It introduces a new model of random walks on discrete point processes, providing the first proofs of zero velocity, recurrence/transience criteria, and a CLT for this setting.

Findings

Velocity of the random walk is almost surely zero.

Conditions under which the walk is recurrent or transient are characterized.

A Central Limit Theorem is proved under a specific distance condition.

Abstract

We consider a model for random walks on random environments (RWRE) with random subset of the d-dimensional Euclidean lattice as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We prove that the velocity of such random walks is almost surely 0, and give partial characterization of transience and recurrence for the different dimensions. Finally we prove Central Limit Theorem for such random walks, under a condition on the distance between nearest coordinate nearest points.