# Shrinking scale equidistribution for monochromatic random waves on compact manifolds

## TL;DR

This paper proves that monochromatic random waves on compact manifolds become uniformly distributed at very small scales, nearly matching the optimal wave scale, with high probability, using spectral and probabilistic tools.

## Contribution

It establishes shrinking scale equidistribution for monochromatic random waves on any compact manifold, extending previous results to more general settings.

## Key findings

- Equidistribution occurs at near-optimal wave scales.
- High probability of uniform distribution across the manifold.
- Uses Weyl's law and Chernoff bounds for proof.

## 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.