Abstract
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 $k'\neq k$.
Citation
Slim Kammoun Kammoun. Mylène Maïda. "A product of invariant random permutations has the same small cycle structure as uniform." Electron. Commun. Probab. 25 1 - 14, 2020. https://doi.org/10.1214/20-ECP334
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