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Random permutations without macroscopic cycles

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<mark>Journal publication date</mark>1/06/2020
<mark>Journal</mark>Annals of Applied Probability
Issue number3
Volume30
Number of pages22
Pages (from-to)1484-1505
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We consider uniform random permutations of length n conditioned to have no cyclelonger than nβ with 0 < β < 1, in the limit of large n. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.