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

Storm, Julia and Zeindler, Dirk (2016) Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 52 (4). pp. 1614-1640. ISSN 0246-0203

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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 $(Z_1, Z_2, ..., Z_b)$ 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 Erd\H{o}s-Tur\'an Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.