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 g1,g1−1,...,gk,gk−1, secondly, for all n, we
introduce a k-tuple s1(n),...,sk(n) of independent random permutations
of {1,...,n}, and the random permutation σn we are going to
consider is the one obtained by replacing each letter gi in w by si(n).
For example, for w=g1g2g3g2−1, σn=s1(n)∘s2(n)∘s3(n)∘s2(n)−1. Moreover, we restrict the set of possible lengths of
the cycles of the si(n)'s: we fix sets A1,...,Ak of positive integers
and suppose that for all n, for all i, si(n) is uniformly distributed on
the set of permutations of {1,...,n} which have all their cycle lengths in
Ai. For all positive integer l, we are going to give asymptotics, as n
goes to infinity, on the number Nl(σn) of cycles of length l of
σn. We shall also consider the joint distribution of the random vectors
(N1(σn),...,Nl(σn)). We first prove that the order of w in a
certain quotient of the free group with generators g1,...,gk determines
the rate of growth of the random variables Nl(σn) as n goes to
infinity. We also prove that in many cases, the distribution of Nl(σn)
converges to a Poisson law with parameter 1/l and that the random variables
N1(σn),N2(σn),... are asymptotically independent. We notice
the surprising fact that from this point of view, many things happen as if
σn were uniformly distributed on the n-th symmetric group.Comment: 28 page