On the number of cycles in a random permutation

Maples, Kenneth; Nikeghbali, Ashkan; Zeindler, Dirk

doi:10.1214/ecp.v17-1934

Cited by 17 publications

(24 citation statements)

References 8 publications

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“…103)). Maples et al (2012;Thorem 1.1) obtained the same convergence for the generalized Ewens sampling formula. In the case of θ = 1, we denote K n by K 0n .…”

Section: Introductionmentioning

confidence: 64%

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Edgeworth Expansions for the Number of Distinct Components Associated with the Ewens Sampling Formula

Yamato

2013

JJSS

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The Ewens sampling formula is well-known as a distribution of a random partition of the positive integer n. For the number of distinct components of the Ewens sampling formula, we derive its Edgeworth expansion. It is different from the Edgeworth expansion for the sum of independent and identically distributed random variables. It contains the digamma function of the parameter of the Ewens sampling formula. Especially, for the random permutation, the Edgeworth expansion contains Euler's constant. The Edgeworth expansion is examined numerically using its graph.

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“…103)). Maples et al (2012;Thorem 1.1) obtained the same convergence for the generalized Ewens sampling formula. In the case of θ = 1, we denote K n by K 0n .…”

Section: Introductionmentioning

confidence: 64%

“…For the generalized Ewens sampling formula, the probability, the large deviation estimate P (|(K n − µ n )/σ n − a| < ) is obtained by Maples et al (2012;Theorem 4.2).…”

Section: Introductionmentioning

confidence: 99%

Edgeworth Expansions for the Number of Distinct Components Associated with the Ewens Sampling Formula

Yamato

2013

JJSS

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“…The proof of Lemma 2.9 is analogue to the proof of Proposition 2.2 in [24]; one simply has to verify that all involved expression are uniform in s. This is straightforward and we thus omit the details.…”

Section: Generating Functionsmentioning

confidence: 99%

Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights

Storm¹,

Zeindler²

2016

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. We study the model of random permutations of n objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size 1, 2, ..., b and a process (Z1, Z2, ..., Z b ) of independent Poisson random variables converges to 0 if and only if b = o(ℓ) where ℓ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erdős-Turán Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.

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“…There has been significant activity [11,10,19,6,49,46,36,40,22,34,5,20] in recent years in studying random permutations in which the probability of a permutation is proportional to a product over its cycles of a weight depending on the length of the cycle. The spatial random permutation model has this general form, see Proposition 3.1, but differs from the models studied in this literature in that the weight assigned to a cycle depends both on the length of the cycle and on the size of the permutation.…”

Section: Random Permutations With Cycle Weightsmentioning

confidence: 99%

Limit Distributions for Euclidean Random Permutations

Elboim

Peled

2019

Commun. Math. Phys.

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We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length L, density ρ, dimension d and jump density ϕ, one samples ρL d particles in a d-dimensional torus of side length L, and a permutation π of the particles, with probability density proportional to the product of values of ϕ at the differences between a particle and its image under π. The distribution may be further weighted by a factor of θ to the number of cycles in π. Following Matsubara and Feynman, the emergence of macroscopic cycles in π at high density ρ has been related to the phenomenon of Bose-Einstein condensation. For each dimension d ≥ 1, we identify sub-critical, critical and super-critical regimes for ρ and find the limiting distribution of cycle lengths in these regimes. The results extend the work of Betz and Ueltschi. Our main technical tools are saddle-point and singularity analysis of suitable generating functions following the analysis by Bogachev and Zeindler of a related surrogate-spatial model.

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On the number of cycles in a random permutation

Abstract: We show that the number of cycles in a random permutation chosen according to generalized Ewens measure is normally distributed and compute asymptotic estimates for the mean and variance.Comment: 15 pages, no figure

Cited by 17 publications

References 8 publications

Edgeworth Expansions for the Number of Distinct Components Associated with the Ewens Sampling Formula

Edgeworth Expansions for the Number of Distinct Components Associated with the Ewens Sampling Formula

Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights

Limit Distributions for Euclidean Random Permutations

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