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

Florent Benaych-Georges (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this text, extending results of A.Nica and M. Neagu, we consider random permutations which can be written as free words in several independent random permutations: we first 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 family $s_1(n),..., s_k(n)$ of independent random permutations of $\{1,..., n\}$, and the random permutation $\sigma_n$ we are going to consider is the one obtained by replacing any letter $g_i$ in $w$ by $s_i(n)$. As an example, for $w=g_1g_2g_3g_2^{-1}$, $\sigma_n=s_1(n)s_2(n)s_3(n)s_2(n)^{-1}$. Moreover, we generalize the problem by allowing to 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 even 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 $g_1,..., g_k$ has an influence on the asymptotics of the random variables $N_l(n)$ as $n$ goes to infinity. We also prove that in many cases, the distribution of $N_l(n)$ converges to a Poisson law with parameter $1/l$ and that the family of random variables $(N_1(n),..., N_l(\sigma_n))$ is asymptotically independent. We notice the pretty surprising fact that from this point of view, many things happen as if $\sigma_n$ were uniformly distributed on the $n$-th symmetric group.

Mots clés: random permutation; cycle; free group

Domaines: Probabilités [math.PR]

Florent Benaych-Georges. Cycles of free words in several independent random permutations with restricted cycle lengths. 2007. ⟨hal-00114382v3⟩