Cycles of free words in several independent random permutations with restricted cycle lengths

In this text, we consider random permutations which can be written as free words in several independent random permutations: firstly, we fix a non trivial word $w$ in letters $g_1,g_1^{-1},..., g_k,g_k^{-1}$, secondly, for all $n$, we introduce a $k$-tuple $s_1(n),..., s_k(n)$ of independent random permutations of $\{1,..., n\}$, and the random permutation $\sigma_n$ we are going to consider is the one obtained by replacing each letter $g_i$ in $w$ by $s_i(n)$. For example, for $w=g_1g_2g_3g_2^{-1}$, $\sigma_n=s_1(n)\circ s_2(n)\circ s_3(n)\circ s_2(n)^{-1}$. Moreover, we restrict the set of possible lengths of the cycles of the $s_i(n)$'s: we fix sets $A_1,..., A_k$ of positive integers and suppose that for all $n$, for all $i$, $s_i(n)$ is uniformly distributed on the set of permutations of $\{1,..., n\}$ which have all their cycle lengths in $A_i$. For all positive integer $l$, we are going to give asymptotics, as $n$ goes to infinity, on the number $N_l(\sigma_n)$ of cycles of length $l$ of $\sigma_n$. We shall also consider the joint distribution of the random vectors $(N_1(\sigma_n),..., N_l(\sigma_n))$. We first prove that the order of $w$ in a certain quotient of the free group with generators $g_1,..., g_k$ determines the rate of growth of the random variables $N_l(\sigma_n)$ as $n$ goes to infinity. We also prove that in many cases, the distribution of $N_l(\sigma_n)$ converges to a Poisson law with parameter $1/l$ and that the random variables $N_1(\sigma_n),N_2(\sigma_n), ...$ are asymptotically independent. We notice the surprising fact that from this point of view, many things happen as if $\sigma_n$ were uniformly distributed on the $n$-th symmetric group.