Sharp character bounds and cutoff for symmetric groups

Abstract

We develop a flexible technique to bound the characters of symmetric groups, via the Naruse hook length formula, the Larsen--Shalev character bounds, and appropriate diagram slicings. It allows us to prove a uniform exponential character bound with optimal constant $1/2$. We furthermore prove sharp character bounds for conjugacy classes having a macroscopic number of fixed points, and deduce that the random walks on the associated Cayley graphs exhibit a total variation and $L^2$ cutoff.

v2: 40 pages. Change in title. The exposition and proofs were improved. The application to cutoff profiles was moved to a follow up paper. v1: 65 pages, comments welcome!