Spectral convergence of high-dimensional spheres to Gaussian spaces

TL;DR

This paper demonstrates that the spectral properties of high-dimensional spheres converge to those of Gaussian spaces as the dimension increases, establishing a connection between geometric and probabilistic spectral structures.

Contribution

It provides a rigorous proof of spectral convergence from high-dimensional spheres to Gaussian spaces, including eigenvalues and eigenfunctions, as the dimension tends to infinity.

Findings

Spectral structure on high-dimensional spheres converges to Gaussian spectral structure.

First Dirichlet eigenvalue on spheres converges to that in Gaussian space.

Eigenfunctions on spheres approximate those in Gaussian space as dimension grows.

Abstract

We prove that the spectral structure on the $N$-dimensional standard sphere of radius $(N-1)^{1/2}$ compatible with a projection onto the first $n$-coordinates converges to the spectral structure on the $n$-dimensional Gaussian space with variance $1$ as $N\to \infty$. We also show the analogue for the first Dirichlet eigenvalue and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space.