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Random field excursions is an increasingly vital topic within data analysisin medicine, cosmology, materials science, etc. This work is the first detailedstudy of their Betti numbers in the so-called `sparse' regime. Specifically, weconsider a piecewise constant Gaussian field whose covariance function ispositive and satisfies some local, boundedness, and decay rate conditions. Wemodel its excursion set via a Cech complex. For Betti numbers of this complex,we then prove various limit theorems as the window size and the excursion leveltogether grow to infinity. Our results include asymptotic mean and varianceestimates, a vanishing to non-vanishing phase transition with a preciseestimate of the transition threshold, and a weak law in the non-vanishingregime. We further obtain a Poisson approximation and a central limit theoremclose to the transition threshold. Our proofs combine extreme value theory andcombinatorial topology tools.

Betti Numbers of Gaussian Excursions in the Sparse Regime