Cycles of Free Words in Several Independent Random Permutations with Restricted Cycle Lengths

摘要

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 ktuple s_1(n),...,s_k(n) of independent random permutations of {1,...,n}, and the random permutation σ_n we are going to consider is the one obtained by replacing each letter gi in w by s_i(n). For example, for w = g_1g_2g_3g_2~(1) , σ_n = s_1(n) · s_2(n) · s_3(n) · 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 Ai. For all positive integersl, we are going to give asymptotics, as n goes to infinity, on the number Nl(σ_n) of cycles of length l of σ_n. We shall also consider the joint distribution of the random vectors (N_1(σ_n),..., Nl(σ_n)). We first prove that the representant 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(σ_n) as n goes to infinity. We also prove that in many cases, the distribution of N_l(σ_n) converges to a Poisson law with parameter 1/l and that the ran-dom variables N1_(σ_n),N_2(σ_n),... are symptotically independent. We notice the surprising fact that from this point of view, many things happen as if zn were uniformly distributed on then-th symmetric group.