Spectral gap of sparse bistochastic matrices with exchangeable rows with application to shuffle-and-fold maps

TL;DR

This paper analyzes the spectral properties of sparse bistochastic matrices formed by permutation and fixed matrices, providing bounds on eigenvalues and applications to random walks and fluid mixing.

Contribution

It establishes bounds on the second largest eigenvalue of such matrices under sparsity conditions and applies results to random walks and mixing maps.

Findings

Second eigenvalue bounded by normalized Hilbert-Schmidt norm of Q

Results apply to random regular digraphs

Implications for fluid mixing protocols

Abstract

We consider a random bistochastic matrix of size $n$ of the form $M Q$ where $M$ is a uniformly distributed permutation matrix and $Q$ is a given bistochastic matrix. Under mild sparsity and regularity assumptions on $Q$, we prove that the second largest eigenvalue of $MQ$ is essentially bounded by the normalized Hilbert-Schmidt norm of $Q$ when $n$ grows large. We apply this result to random walks on random regular digraphs and to shuffle-and-fold maps of the unit interval popularized in fluid mixing protocols.