Spectral gap with polynomial rate for Weil-Petersson random surfaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912711814676480 |
|---|---|
| author | Hide, Will Macera, Davide Thomas, Joe |
| author_facet | Hide, Will Macera, Davide Thomas, Joe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14874 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral gap with polynomial rate for Weil-Petersson random surfaces Hide, Will Macera, Davide Thomas, Joe Spectral Theory Algebraic Geometry Differential Geometry Geometric Topology Probability 58J50 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. |
| title | Spectral gap with polynomial rate for Weil-Petersson random surfaces |
| topic | Spectral Theory Algebraic Geometry Differential Geometry Geometric Topology Probability 58J50 |
| url | https://arxiv.org/abs/2508.14874 |