Local Statistics of Random Permutations from Free Products

Let a and /3 be uniformly random permutations of orders 2 and 3, respectively, in S_N, and consider, say, the permutation a/Ja/J^-1. How many fxed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let r = G₁ *... * G_k be a free product of groups where each of G₁,..., G_k is either fnite, fnitely generated free, or an orientable hyperbolic surface group. For a fxed element y e r, a y-random permutation in the symmetric group S_N is the image of y through a uniformly random homomorphism r -> S_N. In this paper we study local statistics of y-random permutations and their asymptotics as N grows. We frst consider E f x N)], the expected number of fxed points in a y -random permutation in S_N. We show that unless y has fnite order, the limit of E[fx N)] as N -> oo is an integer, and is equal to the number of subgroups H <F containing y such that H = Z or H = C₂*C₂. Equivalently, this is the number of subgroups H < r containing y and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for E f x N)] and determine the limit distribution of the number of fxed points as N —>- oo. These results are then generalized to all statistics of cycles of fxed lengths.