Total variation distance and the Erd\H{o}s-Tur\'an law for random permutations with polynomially growing cycle weights

Julia Storm; Dirk Zeindler

Total variation distance and the Erd\H{o}s-Tur\'an law for random permutations with polynomially growing cycle weights

Probability 2014-10-21 v1

Authors: Julia Storm , Dirk Zeindler

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.

Keywords

random permutations total variation distance Erd\H{o}s-Tur\'an law cycle weights saddle-point analysis central limit theorem Poisson random variables

Cite

@article{arxiv.1410.5406,
  title  = {Total variation distance and the Erd\H{o}s-Tur\'an law for random permutations with polynomially growing cycle weights},
  author = {Julia Storm and Dirk Zeindler},
  journal= {arXiv preprint arXiv:1410.5406},
  year   = {2014}
}

Total variation distance and the Erd\H{o}s-Tur\'an law for random permutations with polynomially growing cycle weights

Abstract

Keywords

Cite

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