Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform spectral gaps for random hyperbolic surfaces with not many cusps

Yuxin He, Yunhui Wu, Yuhao Xue

TL;DR

The paper establishes a uniform spectral gap for Weil-Petersson random cusped hyperbolic surfaces X in \mathcal{M}_{g,n} when the cusp count grows as n = O(g^{\alpha}) with \alpha \in [0, \tfrac{1}{2}). It achieves the bound \mathop{SpG}(X) > \tfrac{1}{4}-\left(\tfrac{1}{6(1-\alpha)}\right)^2-\varepsilon, and in the regime \alpha \to \tfrac{1}{2} this yields a new lower bound of \tfrac{5}{36}-\varepsilon, highlighting a second-order cancellation phenomenon. The method combines Mirzakhani’s integration formula, refined WP-volume asymptotics, geodesic counting in subsurfaces, and a pre-trace inequality with an inclusion-exclusion framework to isolate and cancel leading terms. A key innovation is the demonstration of a second-order cancellation between geometric counting contributions and spectral terms, enabling tight uniform bounds in the cusped setting. This advances the understanding of spectral gaps for random hyperbolic surfaces beyond closed cases, with implications for trace-theoretic analyses in moduli spaces and geometric analyses of random surfaces.

Abstract

In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if $n=O(g^α)$ where $α\in \left[0,\frac{1}{2}\right)$, then for any $ε>0$, a random cusped hyperbolic surface in $\mathcal{M}_{g,n}$ has no eigenvalues in $\left(0,\frac{1}{4}-\left(\frac{1}{6(1-α)}\right)^2-ε\right)$. If $α$ is close to $\frac{1}{2}$, this gives a new uniform lower bound $\frac{5}{36}-ε$ for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of ``second order cancellation".

Uniform spectral gaps for random hyperbolic surfaces with not many cusps

TL;DR

The paper establishes a uniform spectral gap for Weil-Petersson random cusped hyperbolic surfaces X in \mathcal{M}_{g,n} when the cusp count grows as n = O(g^{\alpha}) with \alpha \in [0, \tfrac{1}{2}). It achieves the bound \mathop{SpG}(X) > \tfrac{1}{4}-\left(\tfrac{1}{6(1-\alpha)}\right)^2-\varepsilon, and in the regime \alpha \to \tfrac{1}{2} this yields a new lower bound of \tfrac{5}{36}-\varepsilon, highlighting a second-order cancellation phenomenon. The method combines Mirzakhani’s integration formula, refined WP-volume asymptotics, geodesic counting in subsurfaces, and a pre-trace inequality with an inclusion-exclusion framework to isolate and cancel leading terms. A key innovation is the demonstration of a second-order cancellation between geometric counting contributions and spectral terms, enabling tight uniform bounds in the cusped setting. This advances the understanding of spectral gaps for random hyperbolic surfaces beyond closed cases, with implications for trace-theoretic analyses in moduli spaces and geometric analyses of random surfaces.

Abstract

In this paper, we investigate uniform spectral gaps for Weil-Petersson random hyperbolic surfaces with not many cusps. We show that if where , then for any , a random cusped hyperbolic surface in has no eigenvalues in . If is close to , this gives a new uniform lower bound for the spectral gaps of Weil-Petersson random hyperbolic surfaces. The major contribution of this work is to reveal a critical phenomenon of ``second order cancellation".
Paper Structure (39 sections, 103 theorems, 504 equations, 11 figures)

This paper contains 39 sections, 103 theorems, 504 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moduli space of hyperbolic Riemann surfaces
  4. The Weil-Petersson metric
  5. Mirzakhani’s Integration Formula
  6. Counting geodesics in subsurfaces
  7. Weil-Petersson volumes
  8. Pre-trace inequality
  9. Remainder estimate
  10. The expansion of the characteristic function
  11. First order cancellation between $\mathrm{Int}_{nsep}$ and $\mathrm{Int}_{\lambda_0}$
  12. Second order cancellation with $\mathrm{Int}_{ns}$
  13. Type $1$ in $\mathrm{Int}_{ns}$
  14. Type $2$ in $\mathrm{Int}_{ns}$
  15. Type $3$ in $\mathrm{Int}_{ns}$
  16. ...and 24 more sections

Key Result

Theorem 1.2

If $n=O(g^\alpha)$ and $\alpha\in[0,\frac{1}{2})$, then for any $\epsilon>0$,

Figures (11)

  • Figure 1: Known results and the conjecture on spectral gaps for random $X\in \mathcal{M}_{g,n}$ with $n=O(g^\alpha)$
  • Figure 2: A filling geodesic in a pants
  • Figure 3: figure-eight and one-sided iterated eight closed geodesics
  • Figure 4: $X(l,s)=\textrm{Int}(X(l,s))\cup HC_1\cup HC_2$
  • Figure 5: A geodesic arc that spirals $3$ times
  • ...and 6 more figures

Theorems & Definitions (180)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Remark
  • Proposition 3.1
  • Definition
  • Theorem 3.2
  • Remark
  • Definition
  • ...and 170 more