AI 中文总结
研究双曲曲面模空间的Weil - Petersson体积渐近展开，通过分析Witten - Kontsevitch相交数展开计算体积多项式渐近值，改进展开系数界，推导出长度为\(\sqrt{g}\)量级非分离简单测地线平均数量的精确估计。
AI 中文摘要
在过去十年中，对双曲曲面模空间的Weil - Petersson体积渐近增长的研究在典型大属曲面的长度谱和拉普拉斯谱方面取得了许多成果。我们计算了体积多项式\(V_{g,n}(x_1,\ldots x_n)\)的精确渐近值，其中\(\mathbf{x}=(x_1,\ldots x_n)\)是边界分量的长度且\(x_i=\mathcal{O}(\sqrt{g})\)。此结果依赖于对Witten - Kontsevitch相交数展开的分析，我们也得到了类似的明确结果。我们还根据展开的阶数\(s\)改进了展开系数的界。从体积展开中，我们推导出了长度为\(\sqrt{g}\)量级的非分离简单测地线平均数量的精确估计。我们的结果解释了计数函数在\(\sqrt{g}\)截止点处的行为，在该点简单测地线相对于非简单测地线可忽略不计。Lipnowski和Wright曾猜想此截止点的存在，Wu和Xue证明了这一点。
英文摘要
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 $x_i=\mathcal{O}(\sqrt{g})$: $$\prod_{j=1}^{n}\frac{x_j}{2}\cdot\frac{V_{g,n}(x_1,\ldots x_n)}{V_{g,n}}\!=\!\frac{1}{2^n}\exp\left({\frac{|\mathbf{x}|}{2}\!-\!\frac{1}{8π^{2}g}\left(\frac{|\mathbf{x}|}{2}\right)^{2}}\right)\!\left(1\!+\!\mathcal{O}_{n}\left(\frac{1}{\min x_j}\right)\right).$$ 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.
