Asymptotic independence for random permutations from surface groups

TL;DR

This work analyzes random n-sheeted covers of a fixed hyperbolic surface X through Hom(Γ,S_n). It proves that, for distinct primitive conjugacy classes, products of fixed-point counts F_n(γ^a) become asymptotically independent as n grows, and characterizes their limiting distributions in terms of independent Poisson sums. The authors develop a robust framework of subcovers, resolutions, and Euler-analytic tools to decompose lift counts and relate them to BR/SBR subcovers, enabling precise asymptotic factorization via a key lemma. They further extend the analysis to short cycle statistics and establish joint convergence to Poisson-derived limits, with a multivariate method of moments ensuring distributional convergence. The results deepen the connection between random covers, fixed-point statistics in random permutations, and combinatorial-geometric structures on surface groups, with implications for high-energy spectral statistics in random surface models.

Abstract

Let $X$ be an orientable hyperbolic surface of genus $g\geq 2$ with a marked point $o$, and let $Γ$ be an orientable hyperbolic surface group isomorphic to $π_{1}(X,o)$. Consider the space $\text{Hom}(Γ,S_{n})$ which corresponds to $n$-sheeted covers of $X$ with labeled fiber. Given $γ\inΓ$ and a uniformly random $φ\in\text{Hom}(Γ,S_{n})$, what is the expected number of fixed points of $φ(γ)$? Formally, let $F_{n}(γ)$ denote the number of fixed points of $φ(γ)$ for a uniformly random $φ\in\text{Hom}(Γ,S_{n})$. We think of $F_{n}(γ)$ as a random variable on the space $\text{Hom}(Γ,S_{n})$. We show that an arbitrary fixed number of products of the variables $F_{n}(γ)$ are asymptotically independent as $n\to\infty$ when there are no obvious obstructions. We also determine the limiting distribution of such products. Additionally, we examine short cycle statistics in random permutations of the form $φ(γ)$ for a uniformly random $φ\in\text{Hom}(Γ,S_{n})$. We show a similar asymptotic independence result and determine the limiting distribution.

Figures (2)

• Figure 1: On the right we have a genus-2 surface with its CW-structure (from MP1). On the left, we have a subcover $Y$ of a genus-2 surface. The subcover $Y$ is the subcover $Y_{\gamma}$ (see Definition \ref{['def Y gamma']}) of the word $\gamma = a^{2}bca^{-1}d$.
• Figure 2: On the left we have a subcover $Y$ of the genus-2 surface with CW-structure as in Figure \ref{['fig 1']}. On the right, we have its natural resolution. Both subcovers in the top right have $\pi_{1}^{\text{lab}}\cong F_{3}$ and so they have $\chi^{\text{grp}}=-2$. The copy of $Y$ and both subcovers on the bottom have $\pi_{1}^{\text{lab}}\cong F_{2}$ which implies $\chi^{\text{grp}}=-1$ for them.

Theorems & Definitions (54)

• Theorem 1.3
• Theorem 1.4: Corollary 1.7 of PZ
• Theorem 1.5: Theorem 1.11 in MPN
• Theorem 1.6: Theorem 1.15 in PZ
• Theorem 1.7: Theorem 1.14 in PZ
• Theorem 1.8
• Corollary 1.9
• Corollary 1.10
• Theorem 1.11
• Theorem 1.12