Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions

Abstract

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.

On the Number of Nodal Domains of Toral Eigenfunctions

“…The proof of this lemma is essentially the same as that of the so-called 'integral-geometric sandwich' (see [10,Lemma 1]). Since we are using boxes rather than balls, we give a direct proof for the reader's convenience.…”

“…where c 0 > 0 is some positive constant borrowed from theory of random fields ("the universal Nazarov-Sodin constant" [10], see Sect. 1.3 below).…”

“…(Equivalently, μ is the spectral measure of h μ .) Now for R > 0 let N hμ (R) be the number of nodal domains of h μ lying entirely inside the disc B(R) centred at the origin of radius R. Nazarov and Sodin [10,Theorem 1] proved that there exists a non-negative number c μ = c NS (μ) ≥ 0 such that as R → ∞ the expected number of nodal domains lying in B(R) satisfies the asymptotic E[N hμ (R)] = c μ · πR 2 (1 + o(1)), (1.9) and gave some very mild conditions on μ that guarantee that c μ is strictly positive; they also proved the almost sure convergence and convergence in mean of N hμ (R) πR 2 to c μ , under some extra regularity assumptions on μ. The constant c 0 mentioned in Sect.…”

“…1.2. Nazarov and Sodin [10] showed that it is quite easy to check whether for a given symmetric measure μ ∈ P(S 1 ) the corresponding NazarovSodin constant c NS (μ) = 0, and hence whether for a given set K ⊆ P(S 1 ) the image c NS (K) is bounded away from zero. …”

“…Our final ingredient is an application of the work of Nazarov and Sodin [10], who have comprehensively studied nodal domains of Gaussian fields. We do this in two steps, the details of which are given in Sect.…”