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Random permutations with logarithmic cycle weights

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<mark>Journal publication date</mark>1/10/2020
<mark>Journal</mark>Annales de l'institut Henri Poincare (B) Probability and Statistics
Issue number3
Number of pages26
Pages (from-to)1991-2016
Publication StatusPublished
Early online date26/06/20
<mark>Original language</mark>English


We consider random permutations on Sn with logarithmic growing cycles weights and study the asymptotic behavior as the length n tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form exp((− log(1 − z))k+1) with k ≥ 1, which have not yet been studied in the literature.