Quenched invariance principle for random walks on Delaunay triangulations

TL;DR

This paper proves that simple random walks on Delaunay triangulations generated by various point processes in Euclidean space follow a quenched invariance principle, showing diffusive behavior under certain assumptions.

Contribution

It establishes a quenched invariance principle for random walks on Delaunay triangulations for a broad class of point processes, including Poisson and Matérn processes.

Findings

Random walks exhibit diffusive scaling behavior.

The invariance principle holds for processes with clustering or repulsiveness.

The method involves martingale decomposition and negligible corrector at large scales.

Abstract

We consider simple random walks on Delaunay triangulations generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on the point processes, we show that the random walk satisfies an almost sure (or quenched) invariance principle. This invariance principle holds for point processes which have clustering or repulsiveness properties including Poisson point processes, Mat{\'e}rn cluster and Mat{\'e}rn hardcore processes. The method relies on the decomposition of the process into a martingale part and a corrector which is proved to be negligible at the diffusive scale.