Topology and Nesting of the Zero Set Components of Monochromatic Random Waves

Abstract

This paper is dedicated to the study of the topologies and nesting configurations of the components of the zero set of monochromatic random waves. We prove that the probability of observing any diffeomorphism type and any nesting arrangement among the zero set components is strictly positive for waves of large enough frequencies. Our results are a consequence of building Laplace eigenfunctions in euclidean space whose zero sets have a component with prescribed topological type or an arrangement of components with prescribed nesting configuration. © 2018 Wiley Periodicals, Inc.

Topologies of Nodal Sets of Random Band‐Limited Functions

“…The most difficult case of part (2) of Theorem 1.1 is the monochromatic case˛D 1. The proof of this for n > 2 is given in the companion paper [11]. Remark 1.3.…”

Universal Components of Random Nodal Sets

“…Since on the boundary of the ball W , we have either x 2 ≥ 7/2 or y 2 ≥ 3/2, the values of the function q i are greater than 1 2 e −5/2 and we get the result.…”

“…If |q i (x, y)| < δe −5/2 , then |Q i (x, y)| < δ, so that 1 − δ < ( x 2 − 2) 2 + y 2 < 1 + δ. This implies that 1 2 < 2− √ 1 + δ < x 2 and that either 1 2 < x 2 −2 or 1 4 < y 2 since δ ≤ 1 2 . Moreover, for every j ∈ {1, .…”

“…Let us finally mention [14], where Sarnak and Wigman announce a convergence in probability for N in the case of Laplace-Beltrami operators (see also [1]). …”

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’.…”