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

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 1 k and is asymptotically independent of the number of cycles of length k = k.

Small cycle structure for words in conjugation invariant random permutations

“…In several cases, in particular the commutator, We can claim that these conditions are sharp. In [10] we already discussed the case of the product.…”

“…Similar results have already been proved in the case when the permutations are uniformly chosen among permutations with restrictions on cycle lengths in [3,Theorem 3.7] (the latter being an extension of [15], in which only the fixed points of the words in permutations were studied). They can also be seen as an extension of our previous work [10] for the product of permutations to any non-trivial word in the permutations. In the case of the product, we could get optimal assumptions, requiring the permutations to have only few fixed points and cycles of size 2.…”

“…Let us introduce some definitions. Some of them have already been used in [10] but we will need colored versions of the graphs used therein.…”

“…This will provide the vertices and edges of the k graphs corresponding to each permutation and each vertex is colored red if it is the entering or exiting point of the walks (see the definitions and the proof of Lemma 2.1 below for more details). In comparison to [10], considering graphs colored this way will allow more accurate control to show that the graphs containing loops do not contribute to the limiting distribution.…”

“…The technique of proof is inspired by [10]. It is based on a graphical representation of the image of several points by the permutations composing the word $$$ w. $$$ The strategy of the proof will be to identify the class of graphs giving the main contribution in the uniform case and then to show that, under our assumptions, the non vanishing contributions are the same in the conjugation invariant case.…”