Shrinking Scale Equidistribution for Monochromatic Random Waves on Compact Manifolds

Abstract

We prove equidistribution at shrinking scales for the monochromatic ensemble on a compact Riemannian manifold of any dimension. This ensemble on an arbitrary manifold takes a slowly growing spectral window in order to synthesize a random function. With high probability, equidistribution takes place close to the optimal wave scale and simultaneously over the whole manifold. The proof uses Weyl's law to approximate the two-point correlation function of the ensemble, and a Chernoff bound to deduce concentration.

Filament structure of random waves

“…Proof. We use similar argument to that used in [9] and [7] to obtain uniform equidistribution results on small balls. Namely we will approximate…”

“…G 2 α,β,µ (x, ξ) is quadratic in u so similar to [7] we will scale and perform transformations so that G 2 α,β,µ (x, ξ) can be written as y T Dy where D is a diagonal matrix and y T is a standard N -dimensional Gaussian variable (with variance one).…”

“…There are a number of interesting probability distributions from which to draw the c j . For example we may choose to look at cases where each c j is chosen as an independent random variable such as Gaussian or Rademacher (see for instance [3], [16], [7]). Alternatively in situations where it is preferable to be able to fix the 2 norm of c = [c 1 , .…”

Small scale equidistribution of random waves generated by an unfair coin flip

“…where the coefficients c j are chosen according to a some probability distribution and Λ ⊂ S n−1 . Common choices of coefficients include independent random variables such as Gaussian or Rademacher random variables (see for instance [2], [11] and [4]) and uniform probability density on high dimensional unit spheres (see for instance [10], [3], [7], [12] and [5]). Usually Λ is chosen so that the directions ξ j are equally spaced with spacing less than one wavelength (λ −1 ).…”

“…For convenience we will consider the ball about the origin however none of our analysis is dependent on this centre point so the results hold for balls centred around general points p ∈ R 2 . In the setting of manifolds the question of equidistribution on small balls where the coefficients are uniformly distributed on the sphere or Gaussian are resolved in [6] and [4] respectively. While Rademacher coefficients have not been explicitly studied, most of the results of [4] rely on properties of Gaussian random variables that are shared by Rademacher coefficients.…”