Benjamini-Schramm convergence and spectra of random hyperbolic surfaces of high genus. (English) Zbl 1493.58015
Summary: 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 \(\mathcal{H}\) 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 proved using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of \(\mathcal{H}\) as \(g\rightarrow+\infty\) and a uniform Weyl law as \(b\rightarrow +\infty\). We deduce a bound on the number of small eigenvalues and the multiplicity of any eigenvalue.
MSC:
58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
58J65 | Diffusion processes and stochastic analysis on manifolds |
Keywords:
hyperbolic surfaces; eigenvalues of the Laplacian; Selberg trace formula; Benjamini-Schramm convergence; moduli spaces; Weil-Petersson volumeReferences:
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