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

TL;DR

This paper demonstrates that the cycle structure of compositions of invariant permutations converges to the same distribution as uniform permutations, under certain conditions on fixed points and small cycles.

Contribution

It establishes the limiting joint distribution of small cycles for composed invariant permutations, matching the uniform case under specific conditions.

Findings

Number of small cycles converges to Poisson(1/k) distribution.

Cycle counts are asymptotically independent across different lengths.

Results extend understanding of cycle structures in permutation compositions.

Abstract

We use moment method to understand the cycle structure of the composition of 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 1/k and is asymptotically independent of the number of cycles of length k' different from k.