Small scale equidistribution of random eigenbases

TL;DR

This paper proves that, under certain conditions, random eigenbases on compact manifolds are almost surely equidistributed at small scales, with the scale depending on eigenfrequency multiplicity growth, including spheres and tori.

Contribution

It establishes small scale equidistribution of random eigenbases on manifolds with high eigenfrequency multiplicity, extending understanding of eigenfunction distribution at fine scales.

Findings

Almost surely, random eigenbases are equidistributed at small scales.

Equidistribution scales depend on the multiplicity growth rate.

Results apply to spheres (n>=2) and tori (n>=5).

Abstract

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity of eigenfrequency tends to infinity at least logarithmically. We prove that, with respect to the natural probability measure on the space of eigenbases, almost surely a random eigenbasis is equidistributed at small scales; furthermore, the scales depend on the growth rate of multiplicity. In particular, this implies that almost surely random eigenbases on the n-dimensional sphere (n>=2) and the n-dimensional tori (n>=5) are equidistributed at polynomial scales.