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

Charles Bordenave; Yanqi Qiu; Yiwei Zhang

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arXiv · 1805.06205

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

Charles Bordenave·Yanqi Qiu·Yiwei Zhang

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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.

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BibTeX RIS

Charles Bordenave, Yanqi Qiu, Yiwei Zhang. 2018-05-16. Spectral gap of sparse bistochastic matrices with exchangeable rows with application to shuffle-and-fold maps. https://arxiv.org/abs/1805.06205

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