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Precise asymptotics of longest cycles in random permutations without macroscopic cycles

Research output: Contribution to journalJournal articlepeer-review

Forthcoming
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<mark>Journal publication date</mark>23/09/2020
<mark>Journal</mark>Bernoulli
Publication StatusAccepted/In press
<mark>Original language</mark>English

Abstract

We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$ and study the asymptotic behaviour as $n\to\infty$. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.