Variation of the Nazarov–Sodin constant for random plane waves and arithmetic random waves

Abstract

This is a manuscript containing the full proofs of results announced in [KW], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions. 1 arXiv:1707.00766v3 [math-ph]

On the number of excursion sets of planar Gaussian fields

“…Nazarov-Sodin also obtained analogous results in higher dimensions and for Gaussian ensembles on manifolds [13]. In the case that f is not ergodic, it has been shown [10] (under the additional assumption that ρ has compact support) that the expected number of nodal components, scaled by the area, still converges, i.e.…”

Direction distribution for nodal components of random band-limited functions on surfaces

“…is the number of points on the nodal line of f normal to ζ; equivalently, N ζ (f ) is the number of nodal points of f in direction ±ξ := ζ ⊥ . The direction distribution carries a lot of information on the nodal line of f ; for example, the number of nodal components of f on T 2 essentially majorizes [25] the nodal count of f for every ζ, as every nodal component of f of trivial homology contains at least two points tangent to ξ, and it is usually easy to control the contribution of all the other components. For the N ζ (·) corresponding to toral Laplace eigenfunctions Rudnick and Wigman [32] gave the optimal upper bounds, as well as evaluated their total expected number for the associated Gaussian random model, "Arithmetic Random Waves" (1.7), precisely, appealing to the Kac-Rice method.…”

“…However, there exist [13,24] other weak- * partial limits of the sequence {ρ n } n∈S . It was established [23,36,25], that for the nodal structures of f n to exhibit a limit law, it is essential to divide S into subsequences whose corresponding ρ n obey a limiting distribution, i.e. take a subsequence {n} ⊆ S so that ρ n ⇒ τ for some probability measure τ on S 1 .…”

“…Let N (f n ) be the total number of nodal components of f n . A combination of the techniques by Nazarov-Sodin [36,31] and [25] implies that for every sequence {n} ⊆ S so that ρ n ⇒ τ for some probability measure τ on S 1 , there exists a constant c N S (τ ) ≥ 0 so that…”

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

“…More precisely, the authors relied on a result from semi‐classical analysis (see [, Theorem 2.3] in which the authors extend a result from ). Other works in this field are . All of the aforementioned works study parametric families of smooth functions on a manifold of dimension that vary at a natural scale and that possess ‘local limits’.…”