Cycles of free words in several independent random permutations with restricted cycle lengths Florent Benaych-Georges 20B3060B1520P0560C05random permutation$A$-permutationsfree group 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 $g_1, g_1^{-1}, \dots, g_k, g_k^{-1}$, secondly, for all $n$, we introduce a $k$-tuple $s_1(n), \dots, s_k(n)$ of independent random permutations of $\{1, \dots, n\}$, 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 $w=g_1 g_2 g_3 g_2^{-1}$, $\sigma_n = s_1(n) \circ s_2(n) \circ s_3(n) \circ s_2(n)^{-1}$. Moreover, we restrict the set of possible lengths of the cycles of the $s_i(n)$'s: we fix sets $A_1, \dots, A_k$ of positive integers and suppose that for all $n$, for all $i$, $s_i(n)$ is uniformly distributed on the set of permutations of $\{1, \dots, n\}$ which have all their cycle lengths in $A_i$. For all positive integers $\ell$, we are going to give asymptotics, as $n$ goes to infinity, on the number $N_{\ell}(\sigma_n)$ of cycles of length $\ell$ of $\sigma_n$. We shall also consider the joint distribution of the random vectors $(N_1(\sigma_n), \dots, N_{\ell}(\sigma_n))$. We first prove that the representant of $w$ in a certain quotient of the free group with generators $g_1, \dots, g_k$ determines the rate of growth of the random variables $N_{\ell}(\sigma_n)$ as $n$ goes to infinity. We also prove that in many cases, the distribution of $N_{\ell}(\sigma_n)$ converges to a Poisson law with parameter $1/\ell$ and that the random variables $N_1(\sigma_n), N_2(\sigma_n), \dots$ 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. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4119 10.1512/iumj.2010.59.4119 en Indiana Univ. Math. J. 59 (2010) 1547 - 1586 state-of-the-art mathematics http://iumj.org/access/