Word Measures on Symmetric Groups

Fix a word w in a free group Fr on r generators. A w-random permutation in the symmetric group Sn is obtained by sampling r independent uniformly random permutations σ1,...,σr in Sn and evaluating w(σ1,...,σr). In (Puder 2014, Puder-Parzanchevski 2015) it was shown that the average number of fixed points in a w-random permutation is 1+θ(n¹-^π(w)), where π(w) is the smallest rank of a subgroup H <= Fr containing w as a non-primitive element. We show that π(w) plays a role in estimates of other natural families of characters. In particular, we show that for all s>=2, the average number of s-cycles is (1/s)+O(n-^π(w)).