Noise sensitivity on virtually abelian groups

TL;DR

This paper investigates the noise sensitivity of aperiodic random walks with finite second moments on virtually abelian groups, establishing a precise criterion based on group homomorphisms to the infinite cyclic group.

Contribution

It provides a complete characterization of noise sensitivity for these random walks, linking it to the existence of nonzero homomorphisms onto the infinite cyclic group.

Findings

Noise sensitivity occurs if and only if no nonzero homomorphism to the infinite cyclic group exists.

The result applies to a broad class of virtually abelian groups and random walks.

The paper bridges group theory and probabilistic behavior of random walks.

Abstract

We show that aperiodic random walks with finite second moment on virtually abelian groups are noise sensitive in total variation if and only if the group admits no nonzero homomorphism onto the infinite cyclic group.