Word measures on the symmetric group

Given a word w in the free group on k generators, the word measure induced by w on the symmetric group S_n is the distribution obtained when substituting k independent uniform permutations into the word w.  Puder and Parzanchevski (2015) estimated the expected number of fixed points of a permutation sampled according to this distribution. 

We generalize this result to other statistics of the permutation, such as the expected number of cycles of a constant length, or the expected number of pairs of fixed points.  As an application of these results, we prove results regarding the expansion of random Schreier graphs of the symmetric group.

Based on joint work with Doron Puder.