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Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -