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Precise Asymptotics of Longest Cycles in Random Permutations Without Macroscopic Cycles by Betz, Volker; Mühlbauer, Julian; Schäfer, Helge; Zeindler, Dirk is a scholarly article available to read on EtoBox.

What is Precise Asymptotics of Longest Cycles in Random Permutations Without Macroscopic Cycles about?

We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$ and to 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.

Author

Betz, Volker; Mühlbauer, Julian; Schäfer, Helge; Zeindler, Dirk

Published

2020

Language

EN