Asymptotic distribution of the Betti numbers of $\overline{\mathcal{M}}_{0,n}$

TL;DR

This paper proves that the Betti numbers of certain moduli spaces and configuration spaces become normally distributed as the number of points increases, extending known asymptotic normality to topological invariants.

Contribution

It establishes the asymptotic normality of Betti numbers for $ar{ ext{M}}_{0,n}$ and $ ext{P}^1[n]$, and conjectures similar behavior for their symmetric group quotients.

Findings

Betti numbers of $ar{ ext{M}}_{0,n}$ are asymptotically normal.

Betti numbers of $ ext{P}^1[n]$ are asymptotically normal.

Betti numbers of quotients by symmetric groups are conjectured to be normal.

Abstract

Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various parameters of random graphs. In this paper, we investigate whether this normal limit behavior extends to the topological invariants of geometric spaces. We show that the Betti numbers of the moduli space of rational curves with $n$ marked points $\overline{\mathcal{M}}_{0,n}$ and the Fulton-MacPherson configuration space $\mathbb{P}^1[n]$ are asymptotically normally distributed. Based on numerical evidence and established log-concavity, we conjecture that the Betti numbers of the quotients of these spaces by the symmetric group $\mathbb{S}_n$ are also asymptotically normally distributed. In contrast, we provide examples of geometric spaces that do not follow this Gaussian law.