Cycles of Free Words in Several Independent Random Permutations

In this text, extending results of [Ni94] and [Ne05], we study the asymptotics of the number of cycles of a given length of a word in several independent random permutations with restricted cycle lengths. Specifically, for A1,. . . , Ak non empty sets of positive integers and for w word in the letters g1, g −1 1 , . . . , gk, g −1 k , we consider, for all n such that it is possible, an independent family s1(n), . . . , sk(n) of random permutations chosen uniformly among the permutations of n objects which have all their cycle lengths in respectively A1, . . . , Ak, and for l positive integer, we are going to give asymptotics (as n goes to infinity) on the number Nl(n) of cycles of length l of the permutation obtained by changing any letter gi in w by si(n). In many cases, we prove that the distribution of Nl(n) converges to a Poisson law with parameter 1/l and that the family of random variables (N1(n), N2(n), ...) is asymptotically independent. We notice the pretty surprising fact that from this point of view, many things happen like if we considered the number of cycles of given lengths of a single permutation with uniform distribution on the n-th symmetric group.