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

Technical Summary: Asymptotic Distribution of the Betti Numbers of $M_{0,n}$

Problem Statement
The paper investigates the asymptotic behavior of the Betti numbers (topological invariants) of specific moduli spaces of rational curves as the number of marked points, $n$, tends to infinity. While the total cohomology of these spaces grows exponentially, making explicit computation for arbitrary $n$ combinatorially intractable, the authors seek to determine whether the distribution of these Betti numbers follows a specific limiting shape. Specifically, the study asks if the sequence of Betti numbers exhibits asymptotic normality, a phenomenon frequently observed in large combinatorial structures (e.g., cycles in permutations, prime factors of integers) but less understood in the context of geometric moduli spaces.

Methodology
The authors employ techniques from analytic combinatorics, specifically singularity analysis of generating functions, to study the limiting distribution.

1. Probabilistic Framework: The normalized Poincaré polynomial of a space $X_n$, defined as $f_n(u) = P_{X_n}(u)/\chi(X_n)$, is treated as a probability generating function (PGF) for an integer-valued random variable $\xi_n$ representing the degree of cohomology.
2. Quasi-Powers Theorem: The core analytical tool is the Quasi-Powers Theorem (Hwang, 1998). This theorem asserts that if a PGF $p_n(u)$ behaves asymptotically like $A(u)B(u)^{\beta_n}$ (where $\beta_n \to \infty$), then the associated random variable converges in distribution to a normal distribution, provided a variability condition on $B(u)$ is met.
3. Singularity Analysis: The authors analyze the bivariate generating functions $\phi(z, u)$ associated with the Poincaré polynomials. By identifying the location of singularities (specifically square-root type singularities) and performing Taylor expansions near these points, they derive the asymptotic form of the coefficients (the Betti numbers).
4. Local Limit Theorems: To establish log-concavity properties, the authors utilize local limit theorems to approximate individual coefficients with Gaussian density functions.

Key Results

1. Asymptotic Normality of $M_{0,n}$:
   The authors prove that the Betti numbers of the moduli space of stable rational curves with $n$ marked points, $M_{0,n}$, are asymptotically normally distributed.

   • Mean: $m_n \sim \frac{n-3}{2}$.
   • Variance: $\sigma_n^2 \sim \frac{3-e}{6(e-2)}n$.
   • Convergence Speed: $O(n^{-1/2})$.
   • This result implies that the topology of $M_{0,n}$ has a statistically predictable nature in the limit, mimicking the behavior of sums of independent random variables.

2. Asymptotic Normality of Fulton-MacPherson Spaces:
   The study extends to the Fulton-MacPherson configuration space $P^1[n]$ (the compactification of $n$ distinct ordered points in $\mathbb{P}^1$). Using the relation between the generating functions of $M_{0,n}$ and $P^1[n]$, the authors establish that the Betti numbers of $P^1[n]$ are also asymptotically normally distributed with a mean of $n/2$ and a variance growing linearly with the same coefficient as $M_{0,n}$.

3. Asymptotic Log-Concavity:
   Leveraging the Gaussian approximation in the central region, the authors prove that the Betti sequences of both $M_{0,n}$ and $P^1[n]$ are asymptotically 3-ultra-log-concave in the central region ($|k - \text{mean}| = O(\sqrt{n})$). This strengthens previous results which were primarily effective in the tail regions.

4. Conjectures on Quotients:
   Based on numerical evidence (up to $n=72$) and established log-concavity, the authors conjecture that the Betti numbers of the symmetric group quotients $M_{0,n}/S_n$, $M_{0,n+1}/S_n$, and $P^1[n]/S_n$ are also asymptotically normally distributed with variances growing linearly at the same rate.

5. Counter-Examples (Non-Normality):
   The paper demonstrates that asymptotic normality is not a universal feature of all moduli spaces.

   • Hilbert Schemes: The Betti numbers of the Hilbert scheme of points on a smooth projective surface ($Hilb^n(S)$) do not follow a Gaussian law. The infinite product structure of their generating functions causes singularities to accumulate, preventing the application of the Quasi-Powers Theorem. The distribution is suggested to converge to a Gumbel distribution (modeling extreme values) rather than a normal distribution.
   • GIT Quotients: The Betti numbers of the GIT quotient $(\mathbb{P}^1)^n // SL_2(\mathbb{C})$ (for odd $n$) are shown to be non-normal. These numbers correspond to partial sums of binomial coefficients, which approximate a cumulative distribution function rather than a density function, resulting in a distinct deviation from Gaussian behavior.

Significance and Claims
The paper claims to provide the first rigorous establishment of asymptotic normality for the Betti numbers of fundamental geometric moduli spaces like $M_{0,n}$. The significance lies in:

• Statistical Predictability: It reveals that despite the exponential combinatorial complexity of these spaces, their topological invariants exhibit a "statistically predictable" Gaussian nature in the limit.
• Analytic Classification: It proposes that the asymptotic normality of a moduli space's topology depends on the analytic nature of its generating function (specifically, the existence of an isolated square-root singularity suitable for the Quasi-Powers Theorem).
• Distinction of Spaces: By providing counter-examples (Hilbert schemes and GIT quotients), the paper delineates the boundary between spaces that obey this Gaussian law and those that do not, suggesting a new approach to classifying the topology of moduli spaces based on their generating function structures.

The authors remain modest regarding the quotient spaces, presenting their findings as a conjecture supported by numerical data and log-concavity, rather than a proven theorem, noting that a full proof would require arguments analogous to those used for the ordered cases.