# A central limit theorem for integrals of random waves

## TL;DR

This paper establishes a central limit theorem for the mean-square of high-frequency random waves on compact Riemannian manifolds, utilizing advanced integral estimates and the local Weyl law.

## Contribution

It introduces a universal CLT for random waves on any compact Riemannian manifold, extending previous results to arbitrary dimensions and shrinking sets.

## Key findings

- Proves a CLT for the mean-square of random waves in high-frequency limit.
- Uses a novel estimate involving Bessel functions and Gegenbauer's addition formula.
- Demonstrates universality across different manifold geometries.

## Abstract

We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer's addition formula.