Sharp character bounds and cutoff for symmetric groups

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 $1\mathrm{/}2$. 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.