Quenched invariance principle for simple random walk on discrete point processes

TL;DR

This paper proves that a simple random walk on certain random graphs generated by stationary, finitely dependent discrete point processes converges to Brownian motion, establishing a quenched invariance principle.

Contribution

It demonstrates the quenched invariance principle for random walks on discrete point process-generated graphs under specific dependence and stationarity conditions.

Findings

Quenched invariance principle established for the model

Random walk converges to Brownian motion almost surely

Applicable to graphs from finitely dependent, stationary point processes

Abstract

We consider the simple random walk on random graphs generated by discrete point processes. This random graph has a random subset of a cubic lattice as the vertices and lines between any consecutive vertices on lines parallel to each coordinate axis as the edges. Under the assumption that discrete point processes are finitely dependent and stationary, we prove that the quenched invariance principle holds, that is, for almost every configuration of a point process, the path distribution of the walk converges weakly to that of a Brownian motion.