Topologies of Nodal Sets of Random Band‐Limited Functions

Abstract

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band‐limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular, the results apply to random monochromatic waves and to random real algebraic hypersurfaces in projective space. © 2018 Wiley Periodicals, Inc.

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

“…that it is measurable on the given sample space of Gaussian fields F ). A detailed verification of this fact for the related random variable N(F, R, T ), that is the number of connected components contained in the ball B(R) with topological type T , was carried out in the Appendix of [35] and is sufficiently robust to prove measurability in our case. We leave the details to the interested reader.…”

“…Here we consider a sequence ("ensemble") {f l } l∈L , of smooth random Gaussian fields f l : M → R, satisfying a natural scaling property, with the scaling parameter l lying in some countable set L , and our objective is to study the distribution of the total number of nodal components of f l as l → ∞, their typical topology, geometry, relative positions, and other important properties. A particularly important such ensemble, motivating the work [35], is the ensemble of band-limited functions, depending on a fixed number α ∈ [0, 1]. This includes the important ensembles of random degree-l spherical harmonics (see §1.2.1 below), and Arithmetic Random Waves ( §1.2.2 below).…”

“…, satisfying t j → ∞. For a "band" α ∈ [0, 1) and spectral parameter T > 0 (with the intention of taking the limit T → ∞), we define [35] the random band-limited functions to be…”

“…However, for k > 0 even it is possible to construct a simple curve with number of ξ-tangencies being precisely k, that occurs as a nodal component of the scaled random field with positive density (the most subtle or delicate case being α = 1, cf. [35,Proposition 5.3] and [12]). We refer the reader to §2.1 for more details on proofs and intuitions as of why these peculiarities hold.…”

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’.…”