Limit profiles for reversible Markov chains

Nestoridi, Evita; Olesker-Taylor, Sam

doi:10.1007/s00440-021-01061-5

Cited by 11 publications

(11 citation statements)

References 16 publications

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“…It has a connection with a magnificent phenomenon in the theory of mixing times, which informally says, "occasionally certain aspects of a system mix much faster than the system as a whole" [19,22] and supports a conjecture of Nathanaël Berestycki [24, Conjecture 1.2]. Afterward, Nestoridi et al developed some methods to obtain the limit profile further to reversible Markov chains and applied it to some models [17,18]. In this article, we provide a Fourier analysis analogue of Nestoridi's comparison method [17].…”

Section: Introductionmentioning

confidence: 83%

“…The limit profile is known only for a handful number of Markov chains, viz. the riffle shuffle [1], the asymmetric exclusion process on the segment [2], the simple exclusion process on the cycle [13], the simple random walk on Ramanujan graphs [16], and a few random walks (random transposition [24] and star transposition shuffles [17]) on the symmetric group. Teyssier studied the limit profile for the classical random transposition model [24].…”

Section: Introductionmentioning

confidence: 99%

“…Afterward, Nestoridi et al developed some methods to obtain the limit profile further to reversible Markov chains and applied it to some models [17,18]. In this article, we provide a Fourier analysis analogue of Nestoridi's comparison method [17]. Our technique compares random walks on a finite group, and it does not assume the simultaneous diagonalisability of the transition matrices.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Total Variation Cutoff for the Transpose Top-2 with Random Shuffle

Ghosh

2019

J Theor Probab

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We consider a random walk on the hyperoctahedral group B n generated by the signed permutations of the forms (i, n) and (−i, n) for 1 ≤ i ≤ n. We call this the fliptranspose top with random shuffle on B n . We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order n log n. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group D n also has a cutoff at n − 1 2 log n.2010 Mathematics Subject Classification. 60J10, 60B15, 60C05.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Total Variation Cutoff for the Transpose Top-2 with Random Shuffle

Ghosh

2019

J Theor Probab

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“…This lemma characterises the limit profile of the total variation distance between two random variables B 1 and B 2 , following binomial distributions, when the difference between their means is of the same order of the standard deviation of B 1 /N . The proof can be found in the Appendix A.2 (Nestoridi and Olesker-Taylor, 2020), see also (Nestoridi and Olesker-Taylor, 2022), the published version of this paper.…”

Section: Neutral Multi-allelic Moran Type Process With Parent Indepen...mentioning

confidence: 98%

On the spectrum and ergodicity of a neutral multi-allelic Moran model

Corujo¹

2023

ALEA

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The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuoustime Markov chain with an unknown stationary distribution. Specifically, we consider N individuals such that each one of them is of one type among K possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.

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“…For a precise review on the results and its embedding in the literature of the discrete cutoff phenomenon we refer to the introductions in [8,9,12]. For standard texts for cutoff in discrete cutoff parameter we refer to [1,2,3,13,14,15,16,17,25,35,36,37,38,43,44,45,47,48,49,51,53,56,62,63,66]. Most recent developments in this active field are found in [18,19,20,21,26,40,42].…”

Section: Introductionmentioning

confidence: 99%

The cutoff phenomenon for the stochastic heat and the wave equation subject to small Lévy noise

Barrera¹,

Högele²,

Pardo³

2021

Preprint

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This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Lévy noises in the Wasserstein distance. For the additive noise case, we obtain analogous infinite dimensional results to the respective finite dimensional cases obtained recently by Barrera, Högele and Pardo (2021), that is, the (stronger) profile cutoff phenomenon for the stochastic heat equation and the (weaker) window cutoff phenomenon for the stochastic wave equation. For the multiplicative noise case, which is studied in this context for the first time, the stochastic heat equation also exhibits profile cutoff phenomenon, while for the stochastic wave equation the methods break down due to the lack of symmetry. The methods rely strongly on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the stochastic solution flows in terms of stochastic exponentials.

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Limit profiles for reversible Markov chains

Cited by 11 publications

References 16 publications

Total Variation Cutoff for the Transpose Top-2 with Random Shuffle

Total Variation Cutoff for the Transpose Top-2 with Random Shuffle

On the spectrum and ergodicity of a neutral multi-allelic Moran model

The cutoff phenomenon for the stochastic heat and the wave equation subject to small Lévy noise

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