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

TL;DR

This paper demonstrates that as the genus of hyperbolic surfaces increases, most surfaces have their first eigenvalue exceeding a specific threshold and their diameter grows logarithmically with genus.

Contribution

It proves that for large genus, a generic hyperbolic surface has a first eigenvalue greater than 3/16 minus epsilon, and its diameter is bounded by a logarithmic function of genus.

Findings

First eigenvalues exceed 3/16 - epsilon for generic large genus surfaces.

Diameters grow at most logarithmically with genus.

Results hold for generic surfaces in the moduli space.

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.