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Sharp character bounds and cutoff for symmetric groups
Sam Olesker-Taylor, Lucas Teyssier, Paul Thévenin · 2025-03-17 · via math.PR updates on arXiv.org

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.

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