On random walks and switched random walks on homogeneous spaces

TL;DR

This paper establishes new mixing rate bounds for random walks on homogeneous spaces influenced by finite group distributions, introduces switched random walks, and provides algorithms to estimate their spectral properties.

Contribution

It introduces the concept of switched random walks on homogeneous spaces and links their long-term behavior to the Fourier joint spectral radius, along with algorithms for its estimation.

Findings

Derived new mixing rate estimates for random walks on homogeneous spaces.

Introduced switched random walks and characterized their asymptotic behavior.

Developed hermitian sum-of-squares algorithms for spectral radius estimation.

Abstract

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$. We introduce the switched random walk determined by a finite set of probability distributions on $G$, prove that its long-term behavior is determined by the Fourier joint spectral radius of the distributions and give hermitian sum-of-squares algorithms for the effective estimation of this quantity.