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

Jinwon Choi, Young-Hoon Kiem

TL;DR

The paper addresses whether Betti numbers of prominent moduli spaces exhibit asymptotic normality. By encoding even Betti data into normalized generating functions and applying the Quasi-Powers framework with singularity analysis of the bivariate generating function $\varphi(z,u)$, it establishes that the Betti numbers of $\overline{\mathcal{M}}_{0,n}$ and the Fulton–MacPherson configuration space $\mathbb{P}^1[n]$ are asymptotically normal, with linear-in-$n$ variance and explicit means: $m_n=\frac{n-3}{2}$ and $m_n=\frac{n}{2}$ respectively, both converging at rate $O(n^{-1/2})$. It also proves asymptotic log-concavity and extends the analysis to quotients by $\mathbb{S}_n$ conjecturally, while providing clear counterexamples (Hilbert schemes and certain GIT quotients) where Gaussian laws fail. The paper thereby links the analytic structure of generating functions to probabilistic behavior of topological invariants, offering a criterion to distinguish spaces with Gaussian Betti distributions from those without. The results deepen the understanding of statistical regularities in topology and identify natural geometric conditions influencing asymptotic normality.

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.

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

TL;DR

The paper addresses whether Betti numbers of prominent moduli spaces exhibit asymptotic normality. By encoding even Betti data into normalized generating functions and applying the Quasi-Powers framework with singularity analysis of the bivariate generating function , it establishes that the Betti numbers of and the Fulton–MacPherson configuration space are asymptotically normal, with linear-in- variance and explicit means: and respectively, both converging at rate . It also proves asymptotic log-concavity and extends the analysis to quotients by conjecturally, while providing clear counterexamples (Hilbert schemes and certain GIT quotients) where Gaussian laws fail. The paper thereby links the analytic structure of generating functions to probabilistic behavior of topological invariants, offering a criterion to distinguish spaces with Gaussian Betti distributions from those without. The results deepen the understanding of statistical regularities in topology and identify natural geometric conditions influencing asymptotic normality.

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 marked points and the Fulton-MacPherson configuration space 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 are also asymptotically normally distributed. In contrast, we provide examples of geometric spaces that do not follow this Gaussian law.
Paper Structure (8 sections, 6 theorems, 63 equations, 7 figures, 1 table)

This paper contains 8 sections, 6 theorems, 63 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moduli space of rational curves with $n$ marked points
  4. Fulton-MacPherson space of $n$ points in $\mathbb{P}^1$
  5. More examples
  6. Spaces of rational curves with unordered marked points
  7. Hilbert scheme of $n$ points on a surface
  8. GIT quotient of $(\mathbb{P}^1)^n$

Key Result

Theorem 2.3

Let $\xi_n$ be non-negative integer-valued random variables with probability generating function $p_n(u)$. Suppose that there exist analytic functions $A(u)$, $B(u)$, independent of $n$ with $A(1)=B(1)=1$ and sequences $\beta_n, \kappa_n\to \infty$ such that uniformly in a fixed neighborhood $\Omega$ of $u=1$. Assume furthermore the variability condition holds. Then, $\xi_n$ is asymptotically no

Figures (7)

  • Figure 1: Betti distribution for $\overline{{\mathcal{M}}}_{0,50}$
  • Figure 2: Betti distribution for $\mathbb{P}^1[50]$
  • Figure 3: Betti distribution for $\overline{{\mathcal{M}}}_{0,50}/\mathbb{S}_{50}$
  • Figure 4: Betti distribution for $\mathbb{P}^1[50]/\mathbb{S}_{50}$
  • Figure 5: Betti distribution for $\mathrm{Hilb}^{25}(\mathbb{P}^2)$
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3: FS, Quasi-Powers Theorem
  • Theorem 2.4: FS, Local limit theorem
  • Example 2.5
  • Theorem 3.1
  • Example 3.2
  • Definition 3.3
  • Corollary 3.4
  • proof
  • ...and 5 more