Quenched Invariance Principle for the Random Walk on the Penrose Tiling

TL;DR

This paper proves that a simple random walk on Penrose tilings converges to Brownian motion, demonstrating a form of invariance principle for aperiodic tilings using the corrector method.

Contribution

It establishes a quenched invariance principle for random walks on Penrose tilings, a significant step in understanding stochastic processes on aperiodic structures.

Findings

Random walk path converges to Brownian motion for almost every Penrose tiling.

Uses the corrector method to prove the invariance principle.

Results apply under the invariant measure on the set of tilings.

Abstract

We consider the simple random walk on the graph corresponding to a Penrose tiling. We prove that the path distribution of the walk converges weakly to that of a non-degenerate Brownian motion for almost every Penrose tiling with respect to the appropriate invariant measure on the set of tilings. Our tool for this is the corrector method.