A refinement of the asymptotic expansion of Weil-Petersson volumes

Abstract

Over the past decade, the study of the asymptotic growth of Weil-Petersson volumes of the moduli space of hyperbolic surfaces has yielded numerous results on the length spectrum and on the spectrum of the Laplacian of typical large genus surfaces. We compute the exact asymptotic value of the volume polynomials $V_{g,n}(x_1,\ldots x_n)$ for $\mathbf{x}=(x_1,\ldots x_n)$ the lengths of the boundary components such that $x_i=\mathcal{O}(\sqrt{g})$: $$\prod_{j=1}^{n}\frac{x_j}{2}\cdot\frac{V_{g,n}(x_1,\ldots x_n)}{V_{g,n}}\!=\!\frac{1}{2^n}\exp\left({\frac{|\mathbf{x}|}{2}\!-\!\frac{1}{8\pi^{2}g}\left(\frac{|\mathbf{x}|}{2}\right)^{2}}\right)\!\left(1\!+\!\mathcal{O}_{n}\left(\frac{1}{\min x_j}\right)\right).$$ This result relies on the analysis of the expansion of Witten-Kontsevitch intersection numbers, for which we obtain an analogous explicit result. We also refine the bound over the coefficients of the expansion in terms of $s$ the degree of the expansion. From the expansion of the volumes, we deduce an exact estimate of the average number of non-separating simple geodesics of length of order $\sqrt{g}$. Our result therefore explains the behavior of counting functions at the cutoff $\sqrt{g}$, at which simple geodesics become negligible with respect to non-simple ones. The existence of this cut-off was conjectured by Lipnowski and Wright and proven by Wu and Xue.