Noise Sensitivity and Noise Stability for Markov Chains: Existence Results

TL;DR

This paper extends the concepts of noise sensitivity and stability to continuous-time Markov chains, providing unified definitions and proving the existence of stable functions, linking to eigenvector localization issues.

Contribution

It introduces unified definitions for noise sensitivity and stability in continuous-time Markov chains and proves the existence of stable functions, connecting to eigenvector localization.

Findings

Unified framework for noise concepts in Markov chains

Existence of noise stable functions in various Markov chain classes

Connection to eigenvector localization problems

Abstract

During the past 15 years, several extensions of the concepts noise sensitivity and noise stability, first coined in~\cite{schramm2000}, has been studied. The purpose in this paper is to give definitions of this concepts in the setting of continuous time Markov chains, which then unifies many of the previously considered generalizations. In addition, a considerable amount of time is spent on proving the existence of sequences of noise stable and nondegenerate functions with respect to various classes of Markov chains, a problem which interestingly will appear to have close connections to the so called localization of eigenvectors, a problem which in the setting of random graphs has recently been given a lot of attention.