Equidistribution of random waves on small balls

TL;DR

This paper studies how random waves, formed by sums of Laplacian eigenfunctions, distribute evenly on small scales of compact manifolds, providing expectation, variance, and probability decay results.

Contribution

It establishes small scale equidistribution properties of random waves on all compact manifolds, including expectation, variance, and exponential decay of non-equidistribution probabilities.

Findings

Proves expectation and variance results for small scale random waves.

Shows probability of failure to equidistribute decays exponentially with eigenvalue.

Provides estimates for larger but still small scales.

Abstract

In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation and variance results for random waves on all compact manifolds. Here, "small scale" refers to balls of radius $r(\lambda)\to 0$ such that $r/r_{\text{Planck}}\to\infty$, where $r_{\text{Planck}}$ is the Planck scale. For balls at a larger scale (although still $r(\lambda)\to 0$) we also obtain estimates showing that the probability that a random wave fails to equidistribute decays exponentially with the eigenvalue.