Random permutations acting on $k$--tuples have near--optimal spectral gap for $k=\mathrm{poly}(n)$

Abstract: We extend Friedman's theorem to show that, for any fixed $r$, the random $2r$--regular Schreier graphs depicting the action of $r$ random permutations of $[n]$ on $k_{n}$--tuples of distinct elements in $[n]$ are a.a.s. weakly Ramanujan, for any $k_{n}\leq n^{\frac{1}{12}-\epsilon}.$ Previously this was known only for $k$--tuples where $k$ is fixed. In fact, we prove the stronger result of strong convergence of random matrices in irreducible representations of $S_{n}$ of quasi--exponential dimension. We also give a new bound for the expected stable irreducible character of a random permutation obtained via a word map, showing that $\mathbb{E}\left[\chi^{\mu}\left(w(\sigma_{1},\dots,\sigma_{r})\right)\right]=O\left(n^{-k}\right)$, where $k$ is the number of boxes outside the first row of the Young diagram $\mu,$ solving one aspect of a conjecture of Hanany and Puder. We obtain this bound using an extension of Wise's $w$--cycle conjecture.