Asymptotics for the Nodal Components of Non-Identically Distributed Monochromatic Random Waves

Abstract

Abstract We study monochromatic random waves on ${\mathbb{R}}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure on the sphere with a suitably smooth density, which connects the problem with the scattering regime of monochromatic waves. In this setting, we compute the asymptotic distribution of the nodal components of random monochromatic waves, showing that the number of nodal components contained in a large ball $B_R$ grows asymptotically like $R/\pi $ with probability $p_n>0$ and is bounded uniformly in $R$ with probability $1-p_n$ (which is positive if and only if $n\geqslant 3$). In the latter case, we show the existence of a unique noncompact nodal component. We also provide an explicit sufficient stability criterion to ascertain when a more general Gaussian probability distribution has the same asymptotic nodal distribution law.