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

2014/10/20 by Julia Storm, Storm, Julia, Dirk Zeindler +1
Computer Science · Mathematics · Medicine · #60B15 #60C05 #60F17 #Bayesian Methods and Mixture Models #Data-Driven Disease Surveillance #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60B15 #msc:60C05 #msc:60F17

paper · pdf · doi:10.48550/arxiv.1410.5406

arxiv created 2014/10/20 · openalex publication_date 2014/10/20 · arxiv updated 2014/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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