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August 2021 Precise asymptotics of longest cycles in random permutations without macroscopic cycles
Volker Betz, Julian Mühlbauer, Helge Schäfer, Dirk Zeindler
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Bernoulli 27(3): 1529-1555 (August 2021). DOI: 10.3150/20-BEJ1282


We consider Ewens random permutations of length n conditioned to have no cycle longer than nβ with 0<β<1 and study the asymptotic behaviour as n. 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.


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Volker Betz. Julian Mühlbauer. Helge Schäfer. Dirk Zeindler. "Precise asymptotics of longest cycles in random permutations without macroscopic cycles." Bernoulli 27 (3) 1529 - 1555, August 2021.


Received: 1 May 2020; Revised: 1 September 2020; Published: August 2021
First available in Project Euclid: 10 May 2021

Digital Object Identifier: 10.3150/20-BEJ1282

Keywords: Cycle structure , Ewens measure , Functional limit theorem , long cycles , Random permutations , total variation distance

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 3 • August 2021
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