Quenched invariance principles for random walks on percolation clusters

TL;DR

This paper establishes a quenched invariance principle for random walks on infinite percolation clusters in multi-dimensional integer lattices, providing a rigorous foundation for understanding their long-term behavior.

Contribution

It proves the almost sure quenched invariance principle for random walks on infinite Bernoulli percolation clusters in dimensions two and higher, advancing theoretical understanding.

Findings

Proves quenched invariance principle for $d \\geq 2$

Validates long-term diffusive behavior of random walks on clusters

Provides a rigorous mathematical framework for percolation-based random walks

Abstract

We prove the almost sure ('quenched') invariance principle for a random walker on an infinite Bernoulli percolation cluster in $\Z^d$ where $d$ is larger or equal than 2.