Expected number of nodal components for cut-off fractional Gaussian fields

Alejandro Rivera (1)

1 IF - Institut Fourier

Let (X , g) be a closed Riemmanian manifold of dimension n > 0. Let ∆ be the Laplacian on X , with eigenfunctions (resp. eigenvalues) (e k) k (resp. (λ k) k). We assume that (λ k) k is increasing and that the e k are real-valued and L 2-normalized. Let (ξ k) k be a sequence of iid N (0, 1) random variables. For each L > 0 and s ∈ R, set f s L = 0<λj ≤L λ − s 2 j ξ j e j. Then, f s L is almost surely regular on its zero set. Let N L be the number of connected components of its zero set. If s < n 2 , then we deduce from previous results that there exists ν = ν(n, s) > 0 such that N L ∼ νVol g (X)L n/2 in L 1 and almost surely. In particular, E[N L ] ≍ L n/2. On the other hand, we prove that if s = n 2 then E[N L ] ≍ L n/2 ln L 1/2. In the latter case, we also obtain an asymptotic formula for the expected Euler characteristic of the zero set of f s L as well as upper bounds for its Betti numbers.

Mots clés: Log-correlated Gaussian fields; Random geometry; Nodal set; Gaussian field

Domaines: Probabilités [math.PR]; Géométrie différentielle [math.DG]

Alejandro Rivera. Expected number of nodal components for cut-off fractional Gaussian fields. 2018. ⟨hal-01689153v1⟩