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

TL;DR

The paper addresses how random $2r$-regular Schreier graphs arising from $r$ uniformly random permutations acting on $k_n$-tuples achieve near-optimal spectral gaps for growing $k_n$, extending Friedman's theorem to polynomially large $k_n$. The authors develop a robust framework combining strong convergence of random permutation representations, word-map bounds, and an extended Weingarten calculus within Schur–Weyl–Jones duality to control stable irreducible characters. A key technical contribution is a bound proving $\mathbb{E}_w[\chi^{\lambda^{+}(n)}]=O(n^{-k})$ for nontrivial words when $\lambda$ lies in a stable regime, achieved via a polynomial-ratio formulation in $1/n$ and a novel $w$-cycle/stacking argument extending Louder–Wilton. Collectively, these results yield a near-optimal spectral gap with high probability for random Schreier graphs when $k_n \le n^{1/20-\varepsilon}$, and establish strong convergence to the left-regular representation in a broad dimension regime, with potential implications for random fixed-degree Cayley graphs of $S_n$.

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 $k_{n}\leq n^{\frac{1}{20}-ε}.$ 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 $\mathbb{E}\left[χ^μ\left(w(σ_{1},\dots,σ_{r})\right)\right]=O\left(\frac{1}{\dimχ^μ}\right)=O\left(n^{-k}\right)$, where $k$ is the number of boxes outside the first row of the Young diagram $μ,$ solving one aspect of a conjecture of Hanany and Puder. We obtain this bound using an extension of Wise's $w$--cycle conjecture.

Figures (4)

• Figure 1: Here we have shown how to begin constructing $G\left(\sigma_{f},\tau_{f},\pi_{i}\right)$ with $w=x_{1}x_{1}x_{2}x_{1}^{-1}x_{2}^{-1}$ and $k=2.$ There are $2kl(w)=20$ vertices, split in to $k=2$ distinct subsets, each containing $2l(w)=10$ vertices each. The vertices on the left (i.e. with the outer subscript "$1$") are the $1^{\mathrm{st}}$$w$--loop and the vertices on the right are the $2^{\mathrm{nd}}$$w$--loop. There are $k|w|_{x_{1}}=6$ directed $x_{1}$--edges and $k|w|_{x_{2}}=4$ directed $x_{2}$--edges.
• Figure 2: Here we continue the construction of the graph from Figure \ref{['fig: constructing G step 1 - vertices and directed edges only']} by adding in the $\sigma_{x_{1}}$--edges (in red) and the $\sigma_{x_{2}}$--edges (in purple). In this example, we have $\sigma_{x_{1}}=\{\{1,3,6\},\ \{2,4,5\}\}\in\mathrm{Part}\left([3k]\right]$ and $\sigma_{x_{2}}=\{\{1,3\},\ \{2,4\}\}\in$.
• Figure 3: We continue the construction from Figures \ref{['fig: constructing G step 1 - vertices and directed edges only']} and \ref{['fig: consturcting graph with vertices, directed edges and sigma edges']}. We have added in the $\tau_{x_{1}}$--edges (in dark blue) and the $\tau_{x_{2}}$--edges (in light blue). In this example, $\tau_{x_{1}}=\{\{1,3,5\}\ \{2,4,6\}\}$ and $\tau_{x_{2}}=\{\{1,4\}\ \{2,3\}\}$.
• Figure 5: Above we show how to construct $\Gamma\left(\sigma_{f},\tau_{f},\pi_{i}\right)$ from the graph $G\left(\sigma_{f},\tau_{f},\pi_{i}\right)$ constructed in Figure \ref{['fig: final construction of gsigma']}. We have labeled one vertex to show which vertices of $G\left(\sigma_{f},\tau_{f},\pi_{i}\right)$ have been glued together in the construction.

Theorems & Definitions (78)

• Theorem 1.1: Friedman
• Theorem 1.2: Kassabov
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Definition 1.9
• Theorem 1.10