Precise asymptotics of longest cycles in random permutations without macroscopic cycles

Betz, Volker and Mühlbauer, Julian and Schäfer, Helge and Zeindler, Dirk (2021) Precise asymptotics of longest cycles in random permutations without macroscopic cycles. Bernoulli, 27 (3). pp. 1529-1555. ISSN 1350-7265

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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.