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

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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.

### Most cited references2

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(1984)
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### Chaotic Mixing in a Torus Map

(2002)
The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.
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### Author and article information

###### Journal
16 May 2018
###### Article
1805.06205