#### Vol. 15, No. 3, 2022

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Benjamini–Schramm convergence and spectra of random hyperbolic surfaces of high genus

### Laura Monk

Vol. 15 (2022), No. 3, 727–752
##### Abstract

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil–Petersson probability measure. We first prove Benjamini–Schramm convergence to the hyperbolic plane $\mathsc{ℋ}$ as the genus $g$ goes to infinity. An estimate for the number of eigenvalues in an interval $\left[a,b\right]$ in terms of $a$, $b$ and $g$ is then proved using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of $\mathsc{ℋ}$ as $g\to +\infty$ and a uniform Weyl law as $b\to +\infty$. We deduce a bound on the number of small eigenvalues and the multiplicity of any eigenvalue.

##### Keywords
hyperbolic surfaces, eigenvalues of the Laplacian, Selberg trace formula, Benjamini–Schramm convergence, moduli spaces, Weil–Petersson volume
##### Mathematical Subject Classification 2010
Primary: 58J50, 32G15