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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.

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          Stirring by chaotic advection

           Hassan Aref (1984)
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            Chaotic Mixing in a Torus Map

            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

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

              Custom metadata
              5 Figures 34 pages
              math.DS math-ph math.MP math.PR

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