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We use moment method to understand the cycle structure of the composition ofindependent invariant permutations. We prove that under a good control on fixedpoints and cycles of length 2, the limiting joint distribution of the number ofsmall cycles is the same as in the uniform case i.e. for any positive integerk, the number of cycles of length k converges to the Poisson distribution withparameter 1/k and is asymptotically independent of the number of cycles oflength k' different from k.

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