Monotonous subsequences and the descent process of invariant random permutations

Kammoun, Mohamed Slim

doi:10.1214/18-ejp244

Cited by 9 publications

(11 citation statements)

References 36 publications

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“…Our motivation to understand the cycle structure of random permutations is the relation, in the case of conjugation invariant permutations, to the longest common subsequence (LCS) of two permutations. For example, using [6,Theorem 1.2] and that LCS(σ, ρ) is equal to the length of the longest increasing subsequence of…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A product of invariant random permutations has the same small cycle structure as uniform

Kammoun¹,

Maïda²

2020

Electron. Commun. Probab.

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We use moment method to understand the cycle structure of the composition of two independent invariant permutations. We prove that under a good control on fixed points and cycles of length 2, the limiting joint distribution of the number of small cycles is the same as in the uniform case i.e. for any positive integer k, the number of cycles of length k converges to the Poisson distribution with parameter 1 k and is asymptotically independent of the number of cycles of length k = k.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A product of invariant random permutations has the same small cycle structure as uniform

Kammoun¹,

Maïda²

2020

Electron. Commun. Probab.

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“…These exactly solvable structures greatly facilitate the analysis of Mallows permutation models with Kendall's τ and Cayley distance; see e.g. [1,6,8,12,16,35,36,38,39,52,54,55,56,58] and [4,15,25,28,40,41,42] for various probabilistic properties of the two models.…”

Section: Mallows Permutation Modelmentioning

confidence: 99%

Mallows permutation models with $L^1$ and $L^2$ distances I: hit and run algorithms and mixing times

Zhong¹

2021

Preprint

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Mallows permutation model, introduced by Mallows in statistical ranking theory, is a class of non-uniform probability measures on the symmetric group Sn. The model depends on a distance metric d(σ, τ ) on Sn, which can be chosen from a host of metrics on permutations. In this paper, we focus on Mallows permutation models with L 1 and L 2 distances, respectively known in the statistics literature as Spearman's footrule and Spearman's rank correlation.Unlike most of the random permutation models that have been analyzed in the literature, Mallows permutation models with L 1 and L 2 distances do not have an explicit expression for their normalizing constants. This poses challenges to the task of sampling from these Mallows models. In this paper, we consider hit and run algorithms for sampling from both models. Hit and run algorithms are a unifying class of Markov chain Monte Carlo (MCMC) algorithms including the celebrated Swendsen-Wang and data augmentation algorithms. For both models, we show order log n mixing time upper bounds for the hit and run algorithms. This demonstrates much faster mixing of the hit and run algorithms compared to local MCMC algorithms such as the Metropolis algorithm. The proof of the results on mixing times is based on the path coupling technique, for which a novel coupling for permutations with one-sided restrictions is involved.Extensions of the hit and run algorithms to weighted versions of the above models, a two-parameter permutation model that involves the L 1 distance and Cayley distance, and lattice permutation models in dimensions greater than or equal to 2 are also discussed. The order log n mixing time upper bound pertains to the two-parameter permutation model.

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“…For more details, one can see [9]. We used the same techniques of proof with a different Markov operator.…”

Section: General Tools Related To the Longest Increasing Subsequencementioning

confidence: 99%

On the Longest Common Subsequence of Conjugation Invariant Random Permutations

Kammoun

2020

Electron. J. Combin.

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Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$. We prove in this paper that there exists a universal constant $n_1$ such that their conjecture is satisfied for any pair of i.i.d random permutations of size greater than $n_1$ with distribution invariant under conjugation. More generally, in the case where the laws of the two permutations are not necessarily the same, we give a lower bound for the expectation. In particular, we prove that if one of the permutations is invariant under conjugation and with a good control of the expectation of the number of its cycles, the limiting fluctuations of the length of the longest common subsequence are of Tracy-Widom type. This result holds independently of the law of the second permutation.

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Monotonous subsequences and the descent process of invariant random permutations

Cited by 9 publications

References 36 publications

A product of invariant random permutations has the same small cycle structure as uniform

A product of invariant random permutations has the same small cycle structure as uniform

Mallows permutation models with $L^1$ and $L^2$ distances I: hit and run algorithms and mixing times

On the Longest Common Subsequence of Conjugation Invariant Random Permutations

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