Betti numbers, Euler characteristic and their sum for excursion sets are expressed in terms of Binomial coefficients of a topological basis, predicting Gaussian statistics except at high thresholds.
Specifically we focus on Gaussian random fields, for which the closed form analytic formula for ⟨χ⟩ is available
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On the statistical nature of Betti numbers and Euler characteristic of smooth random fields
Betti numbers, Euler characteristic and their sum for excursion sets are expressed in terms of Binomial coefficients of a topological basis, predicting Gaussian statistics except at high thresholds.