Word Measures on Symmetric Groups

TL;DR

This work studies word-induced probability measures on symmetric groups via w-random permutations, connecting their statistics to free-group invariants such as the primitivity rank π(w) and the Crit(w) set. By embedding the problem in the MOCC framework and using multi core graphs, the authors develop a general Φ-function counting lifts to random covers and apply Möbius inversions (both basis-dependent and algebraic) to extract precise leading-term asymptotics for stable class functions, including ξ_k-based statistics. The main result shows that, for non-power words w, E_w[ξ1^{α1}…ξk^{αk}] equals the uniform expectation plus a correction proportional to |Crit(w)| and ⟨ξ1^{α1}…ξk^{αk}, ξ1−1⟩, with a first-order decay governed by N^{π(w)−1}. This approach yields applications to expansion properties of random Schreier graphs, providing near-Ramanujan-type eigenvalue bounds, and suggests deep connections between word measures, stability phenomena, and free-group morphisms with potential universality across group families.

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 $σ_{1},\ldots,σ_{r}\in S_{N}$ and evaluating $w\left(σ_{1},\ldots,σ_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a $w$-random permutation is $1+θ\left(N^{1-π\left(w\right)}\right)$, where $π\left(w\right)$ is the smallest rank of a subgroup $H\le F$ containing $w$ as a non-primitive element. We show that $π\left(w\right)$ 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 $\frac{1}{t}+O\left(N^{-π\left(w\right)}\right)$. 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.

Figures (3)

• Figure 3.1: Let $\mathbb{\mathbb{\mathbf{F}}}_{2}$ have basis $B=\left\{ x,y\right\}$, and let $w=yxyxy^{-2}\in\mathbb{\mathbb{\mathbf{F}}}_{2}$. The multi core graph in the top part of the figure is $\Gamma=\Gamma_{B}\left({\cal H}\right)$ where ${\cal H}=\left\{ \left\langle w\right\rangle ^{\mathbb{\mathbb{\mathbf{F}}}_{2}},\left\langle w\right\rangle ^{\mathbb{\mathbb{\mathbf{F}}}_{2}},\left\langle w^{2}\right\rangle ^{\mathbb{\mathbb{\mathbf{F}}}_{2}},\left\langle w^{3}\right\rangle ^{\mathbb{\mathbb{\mathbf{F}}}_{2}}\right\}$. It is denoted $\Gamma_{2,1,1}^{w}$ in the notation from Example \ref{['exa:main thm in terms of Phi']}. The bottom part shows the bouquet $X_{B}$. There is a single morphism of multi core graphs between these two, and we denote it by $\eta_{2,1,1}^{w}$. We have $\Phi_{\eta_{2,1,1}^{w}}=\mathbb{E}_{w}\left[\xi_{1}^{~2}\xi_{2}\xi_{3}\right]$, where $\Phi_{\eta_{2,1,1}^{w}}$ is defined in Definition \ref{['def:Phi']}.
• Figure 7.1: The graph of groups in Louder's Theorem \ref{['thm:Louder']}.
• Figure C.1: In every pair of figures, the one on the left shows the path $p$ of length $h_{i}>0$ which corresponds to the immediate morphism $\beta_{i}$, and two vertices whose merging corresponds to the immediate morphism $\beta_{i+1}$ (with $h_{i+1}=0$). The right figure in every pair shows how the same final result can be obtained by first performing a step which is equivalent to merging the two vertices and only then performing a step equivalent to $\beta_{i}$. Making this change results in a lexicographically smaller pair of integers.

Theorems & Definitions (112)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Corollary 1.4
• Corollary 1.5
• Corollary 1.6
• Corollary 1.7
• Conjecture 1.8
• Definition 1.9