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. We first prove a quantitative estimate of the speed of Benjamini-Schramm convergence to the hyperbolic plane as the genus g goes to infinity. We then use the Selberg pretrace formula to find an estimate for the number of eigenvalues in an interval [a, b], as b and/or g go to infinity. This result proves that, as g → +∞, the spectrum approaches the continuous spectrum of the Laplacian on the hyperbolic plane, with the density appearing in the Selberg trace formula. It also leads to a probabilistic uniform Weyl law, when b → +∞. We also prove a probabilistic upper bound on the number of eigenvalues in the interval [a, b], and noticeably that the number of eigenvalues below 1/4 is O(g (log g)^(−3/4)).

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. 2020. ⟨hal-02462994v1⟩