The random k cycle walk on the symmetric group

Hough, Bob

doi:10.1007/s00440-015-0636-6

Cited by 11 publications

(34 citation statements)

References 13 publications

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“…Thus, in principle, the techniques developed here could be used to study a random walk generated by elements of SO(k) ⊂ SO(2n + 1) instead of by rotations in a single plane. A similar formula for character ratios on the symmetric group appears in [11].…”

Section: Introductionmentioning

confidence: 70%

Cut-off phenomenon in the uniform plane Kac walk

Hough¹,

Jiang²

2017

Ann. Probab.

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Abstract. We consider an analogue of the Kac random walk on the special orthogonal group SO(N ), in which at each step a random rotation is performed in a randomly chosen 2-plane of R N . We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large N limit. In the special case where the angle of rotation is deterministic this confirms a conjecture of Rosenthal [19]. Under mild conditions we also establish a cut-off for convergence of the walk to stationarity under the L 2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L 2 norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.

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Section: Introductionmentioning

confidence: 70%

Cut-off phenomenon in the uniform plane Kac walk

Hough¹,

Jiang²

2017

Ann. Probab.

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“…We establish the limit profile for 2 ≤ k n. There are parity constraints. To handle such parity constraints, we follow the set-up used by Hough [14]:…”

Section: Definition Bmentioning

confidence: 99%

“…The case of random transpositions, ie k = 2, was one of the first Markov chains studied using representation theory; cutoff was established by Diaconis and Shahshahani [10]. For general 2 ≤ k n, cutoff was established for by Hough [14] using representation theory. Berestycki, Schramm and Zeitouni [3] previously established the same result for k independent of n, using probabilistic arguments instead of representation theory.…”

Section: Theorem B (Random K-cyclesmentioning

confidence: 99%

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Limit profiles for reversible Markov chains

Nestoridi

Olesker-Taylor

2021

Probab. Theory Relat. Fields

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In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

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“…Moreover, by the same argument as for The cut-off parameter is the same as in the classical case (see [7]). We can even consider the conjugacy class of m-cycles for any integer m and, for N large enough, the cut-off will appear at N ln(N)/m steps, again as in the classical case [8].…”

Section: 1mentioning

confidence: 99%

Cut-off phenomenon for random walks on free orthogonal quantum groups

Freslon

2018

Probab. Theory Relat. Fields

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We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the uniform plane Kac walk on this quantum group has a cutoff at N ln(N )/2(1 − cos(θ)). This is the first result of this type for genuine compact quantum groups. We also obtain similar results for mixtures of rotations and quantum permutations.2010 Mathematics Subject Classification. 46L53, 60J05 20G42.

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The random k cycle walk on the symmetric group

Cited by 11 publications

References 13 publications

Cut-off phenomenon in the uniform plane Kac walk

Cut-off phenomenon in the uniform plane Kac walk

Limit profiles for reversible Markov chains

Cut-off phenomenon for random walks on free orthogonal quantum groups

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