Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$

Abstract

Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon >0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue ${\lambda }_1(X)>\frac{3}{16}-\epsilon$. As an application, we also show that a generic surface $X\in M_g$ satisfies that the diameter $\mathrm{diam}(X)<(4+\epsilon )\ln (g)$ for large genus.

arXiv commentGeometric and Functional Analysis (GAFA), to appear

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• A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\epsilon$
• A high-genus asymptotic expansion of Weil-Petersson volume polynomials
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• Bootstrap Bounds on Closed Hyperbolic Manifolds
• Extension of Alon's and Friedman's conjectures to Schottky surfaces
• Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus
• Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps
• Bose-Einstein condensation on hyperbolic spaces
• Spectral convergence of the Dirac operator on typical hyperbolic surfaces of high genus
• The Cheeger constants of random Belyi surfaces
• Averages of determinants of Laplacians over moduli spaces for large genus
• Degenerating hyperbolic surfaces and spectral gaps for large genus
• Length spectrum of large genus random metric maps