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Precise asymptotics of longest cycles in random permutations without macroscopic cycles

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Precise asymptotics of longest cycles in random permutations without macroscopic cycles. / Betz, Volker; Mühlbauer, Julian; Schäfer, Helge; Zeindler, Dirk.

In: Bernoulli, Vol. 27, No. 3, 31.08.2021, p. 1529-1555.

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Betz, Volker ; Mühlbauer, Julian ; Schäfer, Helge ; Zeindler, Dirk. / Precise asymptotics of longest cycles in random permutations without macroscopic cycles. In: Bernoulli. 2021 ; Vol. 27, No. 3. pp. 1529-1555.

Bibtex

@article{5c3170604b364a3c9c911c5e208a873b,
title = "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 $n^\beta$ 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.",
keywords = "Random permutations, Ewens measure, long cycles, functional limit theorem, Total variation distance, cycle structure",
author = "Volker Betz and Julian M{\"u}hlbauer and Helge Sch{\"a}fer and Dirk Zeindler",
year = "2021",
month = aug,
day = "31",
doi = "10.3150/20-BEJ1282",
language = "English",
volume = "27",
pages = "1529--1555",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "International Statistical Institute",
number = "3",

}

RIS

TY - JOUR

T1 - Precise asymptotics of longest cycles in random permutations without macroscopic cycles

AU - Betz, Volker

AU - Mühlbauer, Julian

AU - Schäfer, Helge

AU - Zeindler, Dirk

PY - 2021/8/31

Y1 - 2021/8/31

N2 - 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.

AB - 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.

KW - Random permutations

KW - Ewens measure

KW - long cycles

KW - functional limit theorem

KW - Total variation distance

KW - cycle structure

U2 - 10.3150/20-BEJ1282

DO - 10.3150/20-BEJ1282

M3 - Journal article

VL - 27

SP - 1529

EP - 1555

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 3

ER -