  Random field excursions is an increasingly vital topic within data analysis
in medicine, cosmology, materials science, etc. This work is the first detailed
study of their Betti numbers in the so-called `sparse' regime. Specifically, we
consider a piecewise constant Gaussian field whose covariance function is
positive and satisfies some local, boundedness, and decay rate conditions. We
model 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 level
together grow to infinity. Our results include asymptotic mean and variance
estimates, a vanishing to non-vanishing phase transition with a precise
estimate of the transition threshold, and a weak law in the non-vanishing
regime. We further obtain a Poisson approximation and a central limit theorem
close to the transition threshold. Our proofs combine extreme value theory and
combinatorial topology tools.
