Cutoff for Random Walks on Upper Triangular Matrices

Hermon, Jonathan; Olesker-Taylor, Sam

doi:10.48550/arxiv.1911.02974

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…Interesting recent work of Hermon and Olesker-Taylor [8] considers abelian groups, showing that if the number of generators becomes unbounded, then the cutoff phenomenon typically does occur. In a different paper [10], the same authors examined groups of upper triangular matrices, showing that cutoff also typically occurs for walks on these nonabelian groups when the number of generators grows. In a more general con-text, Salez [14] recently showed that cutoff is a consequence of a certain nonnegative curvature condition, which is often satisfied in the case of Cayley graphs of abelian groups.…”

Section: Introductionmentioning

confidence: 99%

No cutoff for circulants: an elementary proof

Abrams¹,

Babson²,

Landau³

et al. 2022

PMS

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We give an elementary proof of a result due to Diaconis and Saloff-Coste (1994) that families of symmetric simple random walks on Cayley graphs of abelian groups with a bound on the number of generators never have sharp cutoff. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion; sharp cutoff (or the cutoff phenomenon) has been shown to occur in families such as random walks on a hypercube (Diaconis, 1996) in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion (Diaconis and Saloff-Coste, 1993).

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Section: Introductionmentioning

confidence: 99%

No cutoff for circulants: an elementary proof

Abrams¹,

Babson²,

Landau³

et al. 2022

PMS

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“…This was later extended to random walks on random graphs with given degrees [3,4,5], and to random directed graphs [7]. Other notable examples are random lifts of graphs [6,13], random Cayley graphs for Abelian groups [19], and for upper triangular matrices [20].…”

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confidence: 99%

Cutoff for permuted Markov chains

Ben-Hamou¹,

Peres

2023

Ann. Inst. H. Poincaré Probab. Statist.

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Cutoff for permuted Markov chains

Ben-Hamou,

Peres

2021

Preprint

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Let P be a bistochastic matrix of size n, and let Π be a permutation matrix of size n. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by Q = P Π. In other words, the chain alternates between random steps governed by P and deterministic steps governed by Π. We show that if the permutation Π is chosen uniformly at random, then under mild assumptions on P , with high probability, the chain Q exhibits cutoff at time log n h , where h is the entropic rate of P . Moreover, for deterministic permutations, we show that the upper bound on the mixing time obtained by Chatterjee and Diaconis [11] may be improved using a result of Chen and Saloff-Coste [12] that allows to do away with the dependence on the laziness parameter.

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Cutoff for Random Walks on Upper Triangular Matrices

Cited by 3 publications

References 17 publications

No cutoff for circulants: an elementary proof

No cutoff for circulants: an elementary proof

Cutoff for permuted Markov chains

Cutoff for permuted Markov chains

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