Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus

Laura Monk (1)

1 IRMA - Institut de Recherche Mathématique Avancée

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 H as the genus g goes to infinity. An estimate for the number of eigenvalues in an interval [a,b] in terms of a, b and g is then proven using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of H as g →+∞, and a uniform Weyl law as b →+∞. We deduce a bound on the number of small eigenvalues, and the multiplicity of any eigenvalue.

Mots clés: hyperbolic surfaces; eigenvalues of the Laplacian; Selberg trace formula; Benjamini-Schramm convergence; moduli spaces; Weil-Petersson volume

Domaines: Théorie spectrale [math.SP]; Probabilités [math.PR]

Laura Monk. Benjamini-Schramm convergence and spectrum of random hyperbolic surfaces of high genus. Analysis & PDE, In press. ⟨hal-02462994v2⟩