Spectral gap with polynomial rate for random covering surfaces

TL;DR

This work addresses the spectral gap for random coverings of closed hyperbolic surfaces by studying the first new Laplacian eigenvalue $\lambda_{1}^{\mathrm{new}}(X_n)$ on a uniformly random degree-$n$ cover $X_n$ of a fixed surface $X$. It combines Selberg-transform-based operator methods with the recent strong convergence results of Magee–Puder–van Handel for random permutation representations to obtain a polynomial-rate lower bound: $\lambda_{1}^{\mathrm{new}}(X_n) \ge \tfrac{1}{4} - c\,n^{-b}$ with probability tending to $1$ as $n\to\infty$, where $b,c>0$ depend only on the genus of $X$. The approach uses a kernel $k_t$ and its Selberg transform $h_t$ to relate gaps to operator norms, then approximates non-finite-rank parts by finite-rank operators and leverages strong convergence to compare with the regular representation. This yields the first polynomial-error-rate result for the spectral gap in the uniform random covering model, marking a significant improvement over prior subpolynomial and polylogarithmic rates and aligning with expectations from random-matrix-type spectral statistics for chaotic hyperbolic surfaces.

Abstract

In this note we show that the recent work of Magee, Puder and van Handel [MPvH25] can be applied to obtain an optimal spectral gap result with polynomial error rate for uniformly random covers of closed hyperbolic surfaces. Let $X$ be a closed hyperbolic surface. We show there exists $b,c>0$ such that a uniformly random degree-$n$ cover $X_{n}$ of $X$ has no new Laplacian eigenvalues below $\frac{1}{4}-cn^{-b}$ with probability tending to $1$ as $n\to\infty$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 2.1: Ma.Pu.vH2025
• Theorem 2.2: Ma.Pu.vH2025
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['thm:main-thm']}