Spectral gap of random hyperbolic surfaces

Abstract

Let $X$ be a closed, connected, oriented surface of genus $g$, with a hyperbolic metric chosen at random according to the Weil--Petersson measure on the moduli space of Riemannian metrics. Let $\lambda_1=\lambda_1(X)$ bethe first non-zero eigenvalue of the Laplacian on $X$ or, in other words, the spectral gap.In this paper we give a full road-map to prove that for arbitrarily small~$\alpha>0$,\begin{align*} \Pwp{\lambda_1 \leq \frac{1}{4} - \alpha^2 } \Lim_{g\To +\infty} 0.\end{align*}The full proofs are deferred to separate papers.