Random permutations with logarithmic cycle weights

Robles, Nicolas; Zeindler, Dirk

doi:10.1214/19-aihp1025

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…This is the most general form of h [34,35,36,37,43,73,149,207,225] and references cited 28 ). The link comes from the first equality in (34) that defines the following probability measure on the set Y n of partitions of n, or equivalently on the symmetric group S n for positive specialisations of (p k ) k :…”

Section: A3 Probabilistic Representations Of Hmentioning

confidence: 94%

A new approach to the characteristic polynomial of a random unitary matrix

Barhoumi-Andréani¹

2020

Preprint

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Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance:• its value in 1 (Keating-Snaith theorem),• the truncation of its Fourier series up to any fraction of its degree,• the computation of the relative volume of the Birkhoff polytope,• its products and ratios taken in different points,• the product of its iterated derivatives in different points,• functionals in relation with sums of divisor functions in F q [X].• its mid-secular coefficients,• the "moments of moments", etc.We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional. Y. BARHOUMI-ANDRÉANI 2. Notations and prerequisites 2.1. General notations and conventions 2.2. Reminders on symmetric functions 2.3. Scalar products and unitary integrals 2.4. Classical tricks 3. Theory 3.1. The ubiquitous Schur function 3.2. The Schur-CUE connection 3.3. A plethystic-RKHS perspective on duality 3.4. CFKRS with RKHS 3.5. Duality in H N 4. Applications 4.1. The Keating-Snaith theorem 4.2. Autocorrelations of the characteristic polynomial in the microscopic setting 4.3. Ratios of the characteristic polynomial in the microscopic setting 4.4. The mid-secular coefficients 4.5. Back to the autocorrelations : the randomisation paradigm 4.6. The truncated characteristic polynomial 4.7. The Birkhoff polytope 4.8. Iterated derivatives of the characteristic polynomial 4.9. Sums of divisor functions in F q [X] 4.10. The moments of moments 5. Ultimate remarks 5.1. The Hankel form of the limit 5.2. Expansions 5.3. Other approaches to s λ [A] 5.4. Summary of the encountered functionals and limits 5.5. Partial conclusion and future work Appendix A. Probabilistic representations A.1. A symmetric function extension of Gegenbauer polynomials A.2. An integral representation of h n [A] A.3. Probabilistic representations of h (κ) c,∞ A.4. Integrability of h (κ) c,∞ A.5. Integrability in c A.6. The supersymmetric case Appendix B. Truncated ζ function in the microscopic scaling B.1. Reminders B.2. A phase transition Acknowledgements References

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Section: A3 Probabilistic Representations Of Hmentioning

confidence: 94%

A new approach to the characteristic polynomial of a random unitary matrix

Barhoumi-Andréani¹

2020

Preprint

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“…Another well studied case are the weights k ∼ k , see [6,10]. Further studied weights are for instance k = log m (k) and k = {k≤n } , see [19] and [3]. An overview can be found in [9].…”

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confidence: 99%

Long cycle of random permutations with polynomially growing cycle weights

Zeindler

2020

Random Struct Algorithms

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We study random permutations of n objects with respect to multiplicative measures with polynomial growing cycle weights. We determine in this paper the asymptotic behavior of the long cycles under this measure and also prove that the cumulative cycle numbers converge in the region of the long cycles to a Poisson process. KEYWORDScycle counts, long cycles, Poisson process, random permutations, saddle point method INTRODUCTIONLet n be the symmetric group of all permutations on elements {1, … , n}. For any permutation ∈ n , denote by C m = C m ( ) the cycle counts, that is, the number of cycles of length m = 1, … , n in the cycle decomposition of ; clearly(1.1) Random permutationsClassical probability measures studied on n are the uniform measure and the Ewens measure. The uniform measure is well studied and has a long history (see e.g., the first chapter of [1] for a detailed account with references). The Ewens measure originally appeared in population genetics, see [11], but has also various applications through its connection with Kingman's coalescent process, see [13].Classical results about uniform and Ewens random permutations include convergence of joint cycle counts towards independent Poisson random variables in total variation distance [2] and a central limit theorem for cumulative cycle counts [7]. Furthermore, the longest cycles have order of magnitude n and it was established by Kingman ([14]) and by Vershik and Shimdt ([20]) that the vector of renormalized and ordered length of the cycles converges in law to a Poisson-Dirichlet distribution.In this paper, we study random permutations with respect to the probability measure 726

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Limit Shapes for Gibbs Partitions of Sets

Fatkullin

Xue

2021

J Stat Phys

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Random permutations with logarithmic cycle weights

Cited by 4 publications

References 21 publications

A new approach to the characteristic polynomial of a random unitary matrix

A new approach to the characteristic polynomial of a random unitary matrix

Long cycle of random permutations with polynomially growing cycle weights

Limit Shapes for Gibbs Partitions of Sets

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