Asymptotic distributions of the Betti and Hodge numbers of $\overline{\mathcal M}_{g,n}$

Abstract

We study the asymptotic distributions of the Betti and Hodge numbers of the moduli spaces $\overline{\mathcal M}_{g,n}$ of stable pointed curves and the all-light Hassett spaces $\overline{\mathcal M}_{g,0|n}$, for fixed genus $g\ge1$ as the number of markings tends to infinity. We prove central limit theorems for the full Betti distributions, whose variances grow linearly in $n$ with leading coefficients independent of the genus. For the normalized even Betti numbers, we establish refined local limit theorems with estimates uniform in degree. These local estimates yield asymptotic formulas for individual even Betti numbers and determine the sharp central ultra-log-concavity thresholds $\frac{3(e-2)}{2(3-e)}$ and $3$, respectively. For $g\ge2$, the off-diagonal Hodge components form proportions of exact orders $n^{-1/2}$ and $n^{-1}$ of the total cohomology of the two spaces, respectively. The conditional distributions of the Hodge difference $p-q$ given $p\ne q$ converge, for both spaces, to a single discrete law independent of the genus. We also prove that tautological classes account for all but a proportion of exact order $n^{-1}$ of the even cohomology of $\overline{\mathcal M}_{g,n}$. The tautological degree distribution converges to the standard Gaussian under the centering and scaling of the ambient even Betti distribution. An explicit asymptotic formula for the ratio of the ordinary Euler characteristics of the light and heavy spaces answers a question of Kannan, Serpente and Yun. The proofs combine the heavy/light substitution with a stratification of the light spaces by stable cores, singularity analysis of Hodge generating functions, and refined Fourier estimates exploiting Poincaré duality.