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

Abstract

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.

Citation

Download Citation

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. https://doi.org/10.3150/20-BEJ1282

Information

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

Rights: Copyright © 2021 ISI/BS

JOURNAL ARTICLE
27 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.27 • No. 3 • August 2021
Back to Top