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

TL;DR

This paper extends Friedman's theorem to show that random Schreier graphs from permutations acting on large tuples have near-optimal spectral gaps, with new bounds on character expectations and convergence results.

Contribution

It introduces a new bound for expected irreducible characters of random permutations and proves strong convergence of permutations in irreducible representations for large tuple sizes.

Findings

Random Schreier graphs have near-optimal spectral gap for large tuples.

New bound on expected stable irreducible character of random permutations.

Proved strong convergence of permutations in irreducible representations.

Abstract

We extend Friedman's theorem to show that, for any fixed $r>1$, a random $2r$--regular Schreier graph associated with the action of $r$ uniformly random permutations of $[n]$ on $k_{n}$--tuples of distinct elements in $[n]$ has a near--optimal spectral gap with high probability, provided $k_{n}\leq n^{\frac{1}{20}-\epsilon}.$ Previously this was known only for $k$--tuples where $k$ is fixed. In fact, we prove the stronger result of strong convergence of random permutations in irreducible representations of quasi--exponential dimension. Along the way, we give a new bound for the expected stable irreducible character of a random permutation obtained via a word map, showing that $\mathbb{E}\left[\chi^{\mu}\left(w(\sigma_{1},\dots,\sigma_{r})\right)\right]=O\left(\frac{1}{\dim\chi^{\mu}}\right)=O\left(n^{-k}\right)$, 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.