Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Julia Storm and Dirk Zeindler

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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 $1,2,\ldots,b$ and a process $(Z_{1},Z_{2},\ldots,Z_{b})$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ 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.

Résumé

Nous nous intéressons à un modèle de permutations aléatoires de $n$ éléments, avec des poids polynomiaux en les longueurs des cycles, qui a été étudié notamment par Ercolani et Ueltschi. En utilisant l’analyse des points selles des transformées de Fourier, nous montrons que la distance en variation totale entre la suite des nombres de cycles de tailles $1,2,\ldots,b$ et une certaine suite $(Z_{1},Z_{2},\ldots,Z_{b})$ de variables de Poisson indépendantes, converge vers $0$ quand $n\to\infty$ si et seulement si $b=o(\ell)$, où $\ell$ désigne la longueur d’un cycle typique de ce modèle. À l’aide de ce résultat, nous établissons ensuite un théorème central limite pour l’ordre de la permutation, étendant ainsi la loi d’Erdős–Turán. Enfin, nous démontrons le caractère brownien pour la limite des petits cycles.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1614-1640.

Dates
Received: 23 October 2014
Revised: 20 May 2015
Accepted: 31 May 2015
First available in Project Euclid: 17 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1479373242

Digital Object Identifier
doi:10.1214/15-AIHP692

Mathematical Reviews number (MathSciNet)
MR3573289

Zentralblatt MATH identifier
1382.60021

Subjects
Primary: 60C05: Combinatorial probability 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F17: Functional limit theorems; invariance principles

Keywords
Random permutations Polynomially growing cycle weights Total variation distance Erdős–Turán law

Citation

Storm, Julia; Zeindler, Dirk. Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1614--1640. doi:10.1214/15-AIHP692. https://projecteuclid.org/euclid.aihp/1479373242


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