Spectral gap with polynomial rate for Weil-Petersson random surfaces (2508.14874v1)

Abstract: We show that there is a constant $c>0$ such that a genus $g$ closed hyperbolic surface, sampled at random from the moduli space $\mathcal{M}_{g}$ with respect to the Weil-Petersson probability measure, has Laplacian spectral gap at least $\frac{1}{4}-O\left(\frac{1}{g^{c}}\right)$ with probability tending to $1$ as $g\to\infty$. This extends and gives a new proof of a recent result of Anantharaman and Monk proved in the series of works [2,3,5,4,6]. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel [19], and its generalization to the strong convergence of surface groups by Magee, Puder and van Handel [41], to the Laplacian on Weil-Petersson random hyperbolic surfaces.

• The paper proves that the spectral gap on Weil-Petersson random hyperbolic surfaces is at least 1/4 minus an O(1/g^c) term as genus increases.
• The approach adapts the polynomial method from random matrix theory to approximate the Laplacian operator and control error terms.
• The results set a benchmark in spectral analysis for geometric probability, influencing the study of prime geodesic theorems and mixing rates.

Spectral Gap with Polynomial Rate for Weil-Petersson Random Surfaces

Overview of the Paper

The paper investigates the spectral gap of the Laplacian on random Weil-Petersson hyperbolic surfaces. It proves that for a genus $g$ closed hyperbolic surface, sampled randomly from the moduli space $\mathcal{M}_g$ using the Weil-Petersson probability measure, the spectral gap is at least $\frac{1}{4} - O\left(\frac{1}{g^c}\right)$ with high probability as $g \to \infty$. This is an improvement of previous results which achieved spectral gap bounds with $o(1)$ error terms. The approach adapts the polynomial method for strong convergence of random matrices to the Laplacian operator on Weil-Petersson random surfaces.

Implementation and Application of Results

Polynomial Method and Random Matrices

The polynomial method, originally introduced for random matrix analysis, is adapted in this paper to analyze the Laplacian spectra on random hyperbolic surfaces. This method involves approximating operators using power series and expanding functions to bound traces. For practical implementation:

1. Choice of Function: Select $f$ such that $f(\sqrt{\Delta - \frac{1}{4}})$ is a bounded operator. Use functions like $f_0 \in C_c^\infty$ which are non-negative and even.

import numpy as np def bounded_operator_sqrt_laplacian(laplacian_eigenvalues, f0_func): sqrt_laplacian = np.sqrt(laplacian_eigenvalues - 0.25) return f0_func(sqrt_laplacian)

1. Convolution Method: The convolution theorem is used for transforming functions involved in spectral gap estimations.
2. Spectral Analysis: Analyze discrete eigenvalues and ensure the measure of spectral gaps is $f(\sqrt{\Delta_X - 0.25})$. This governs error terms in prime geodesic theorems and mixing rates of geodesic flows.

Estimation Techniques

1. Drift Estimation: Integrate over hyperbolic surfaces and compute intersection numbers and Weil-Petersson volumes for surfaces.

# Example code for estimating drift in Python def drift_estimate(num_surfaces, wp_volume, intersections): drift = 0 for i in range(num_surfaces): drift += np.random.normal() * wp_volume * intersections[i] return drift / num_surfaces

1. Error Term Management: Utilize polynomial expansions to manage error terms in spectral gap computations. This is reflected in estimating Weil-Petersson volume expansions.

Predictions and Further Research

• Higher Order Terms in Expansion: The improvements suggest that polynomials with higher degree accuracy can lead to better spectral gap predictions.
• Possible Extensions: The methods can be further extended to analyze other types of spectral gaps beyond hyperbolic surfaces, possibly incorporating other forms of random manifolds.

Conclusion

This paper extends earlier results on the spectral gap phenomena, proving the existence of polynomial rates of convergence for spectral gaps in Weil-Petersson sampled hyperbolic surfaces. The techniques developed can serve as a benchmark for further research in spectral analysis of random manifolds, providing a robust framework for computational analysis and probabilistic estimations in geometric contexts. By adapting the polynomial method used for random matrices to geometric settings, this paper opens avenues for advanced spectral estimations in geometric probability spacings, enabling further refinements in the paper of hyperbolic manifolds.