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June 2020 Random permutations without macroscopic cycles
Volker Betz, Helge Schäfer, Dirk Zeindler
Ann. Appl. Probab. 30(3): 1484-1505 (June 2020). DOI: 10.1214/19-AAP1538

Abstract

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^{\beta }$ with $0<\beta <1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of macroscopic length, this is a singular conditioning in the limit. Nevertheless, we obtain a fairly complete picture about the cycle number distribution at various lengths. Depending on the scale at which cycle numbers are studied, our results include Poisson convergence, a central limit theorem, a shape theorem and two different functional central limit theorems.

Citation

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Volker Betz. Helge Schäfer. Dirk Zeindler. "Random permutations without macroscopic cycles." Ann. Appl. Probab. 30 (3) 1484 - 1505, June 2020. https://doi.org/10.1214/19-AAP1538

Information

Received: 1 December 2017; Revised: 1 August 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

MathSciNet: MR4133379
Digital Object Identifier: 10.1214/19-AAP1538

Subjects:
Primary: 60C05, 60F05, 60F17

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 3 • June 2020
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