Asymptotic topology of excursion and nodal sets of Gaussian random fields

Let M be a compact smooth manifold of dimension n with or without boundary, or an affine polytope, and let f : M -> R be a smooth Gaussian random field. It is very natural to suppose that for a large positive real u, the random excursion set {f >= u} is mostly composed of a union of disjoint topological n -balls. Using the constructive part of (stratified) Morse theory, we prove that in average, this intuition is true, and provide for large u the asymptotic of the expected number of such balls, and so of connected components of {f >= u). We similarly show that in average, the high nodal sets {f = u} are mostly composed of spheres, with the same asymptotic than the one for excursion set. A quantitative refinement of these results using the average of the Euler characteristic proved in former works by Adler and Taylor provides a striking asymptotic of the constant defined by Nanrov and Sodin, again for large u. This new Morse theoretical approach of random topology also applies to spherical spin glasses with large dimension.