Asymptotic distribution of the Betti numbers of [Formula: see text]

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 [Formula: see text] marked points [Formula: see text] and the Fulton-MacPherson configuration space [Formula: see text] 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 [Formula: see text] are also asymptotically normally distributed. In contrast, we provide examples of geometric spaces that do not follow this Gaussian law.