Electronic Communications in Probability

A product of invariant random permutations has the same small cycle structure as uniform

Slim Kammoun Kammoun and Mylène Maïda

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We use moment method to understand the cycle structure of the composition of two independent invariant permutations. We prove that under a good control on fixed points and cycles of length $2$, the limiting joint distribution of the number of small cycles is the same as in the uniform case i.e. for any positive integer $k$, the number of cycles of length $k$ converges to the Poisson distribution with parameter $\frac {1}{k}$ and is asymptotically independent of the number of cycles of length $k'\neq k$.

Article information

Electron. Commun. Probab., Volume 25 (2020), paper no. 57, 14 pp.

Received: 21 October 2019
Accepted: 5 July 2020
First available in Project Euclid: 8 August 2020

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Primary: 60C05: Combinatorial probability 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems 05A16: Asymptotic enumeration 05A05: Permutations, words, matrices

random permutations moment method universality results

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Kammoun, Slim Kammoun; Maïda, Mylène. A product of invariant random permutations has the same small cycle structure as uniform. Electron. Commun. Probab. 25 (2020), paper no. 57, 14 pp. doi:10.1214/20-ECP334. https://projecteuclid.org/euclid.ecp/1596852018

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