Cycles of free words in several independent random permutations

Florent Benaych-Georges (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this text, extending results of Nica and Neagu, 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 $A_1$,\ldots, $A_k$ non empty sets of positive integers, we consider, for all $n$ \st it is possible, an independent family $s_1(n),\ldots, s_k(n)$ of random permutations chosen uniformly among the permutations of $n$ objects which have all their cycle lengths in respectively $A_1$, \ldots, $A_k$, and for $w$ word in the letters $g_1,g_1^{-1},\ldots, g_k,g_k^{-1}$, for $l$ positive integer, we are going to give asymptotics (as $n$ goes to infinity) on the number $N_l(n)$ of cycles of length $l$ of the permutation obtained by changing any letter $g_i$ in $w$ by $s_i(n)$. We first consider the case of words in a single letter: it amounts to consider a single random permutation with restricted cycle lengths. Then, for words in several letters, we prove that the order of $w$ in a certain quotient of the free group with generators $g_1,\ldots, g_k$ as 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 $(N_1(n), N_2(n),...)$ 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.

Mots clés: random permutation; cycle; free group

Domaines: Probabilités [math.PR]

Florent Benaych-Georges. Cycles of free words in several independent random permutations. 2007. ⟨hal-00114382v2⟩