Characters of symmetric groups: Sharp bounds and applications

We provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible. It follows from our general bound that if a permutation σ S _n has at most n ^o(1) cycles of length <m, then |χ(σ)|≤ χ(1) ^1/m+o(1) for all irreducible characters χ of S _n . This is a far reaching generalization of a result of Fomin and Lulov. We then use our various character bounds to solve a wide range of open problems regarding mixing times of random walks, covering by powers of conjugacy classes, as well as probabilistic and combinatorial properties of word maps. In particular we prove a conjecture of Rudvalis and of Vishne on covering numbers, and a conjecture of Lulov and Pak on mixing times of certain random walks on S _n . Our character-theoretic methods also yield best possible solutions to Waring type problems for alternating groups A _n , showing that if w is a non-trivial word, and n ≥ 0, then every element of A _n is a product of two values of w.