Precise asymptotics of longest cycles in random permutations without macroscopic cycles

Abstract

We consider Ewens random permutations of length n conditioned to have no cycle longer than ${\mathit{n}}^{\mathit{\beta }}$ with $0<\mathit{\beta }<1$ and study the asymptotic behaviour as $\mathit{n}\to \infty$. 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.