Word Measures on Symmetric Groups

TL;DR

This paper investigates the properties of word-induced permutations in symmetric groups, revealing how subgroup ranks influence permutation statistics and eigenvalues of associated Schreier graphs.

Contribution

It extends previous results by linking subgroup rank to all stable characters of symmetric groups and analyzes spectral properties of Schreier graphs with random generators.

Findings

Average number of t-cycles approaches 1/t with error term depending on subgroup rank

Second eigenvalue of Schreier graphs is asymptotically bounded by 2√(2r-1)+ε

Subgroup rank π(w) governs estimates of stable characters in symmetric groups

Abstract

Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating $w\left(\sigma_{1},\ldots,\sigma_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a $w$-random permutation is $1+\theta\left(N^{1-\pi\left(w\right)}\right)$, where $\pi\left(w\right)$ is the smallest rank of a subgroup $H\le F$ containing $w$ as a non-primitive element. We show that $\pi\left(w\right)$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge2$, the average number of $t$-cycles is $\frac{1}{t}+O\left(N^{-\pi\left(w\right)}\right)$. As an application, we prove that for every $s$, every $\varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have second eigenvalue at most $2\sqrt{2r-1}+\varepsilon$ asymptotically almost surely. An important ingredient in this work is a systematic study of not-necessarily connected Stallings core graphs.