Mathematics

Dr. Gugan Thoppe
Technion - Israel Institute of Technology, Haifa, Israel

Excursions of random fields is an increasingly important topic within data analysis in medicine, cosmology, materials science, etc. In this talk, we will discuss some detailed results concerning their Betti numbers. Specifically, we shall consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local and decay rate conditions. We will discuss a way to model its excursion set using a Cech complex. For each Betti number of this complex, we shall then prove various limit theorems in different regimes based on how fast the window size and excursion level grow to infinity. These shall include asymptotic mean 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 shall further see a Poisson and a central limit theorem close to the transition threshold. The expected vertex degree asymptotically vanishes in the regimes we shall deal with. This places all our above results in the so-called `sparse' regime. Our proofs combine tools from both extreme value theory and combinatorial topology.