Critical radius and supremum of random spherical harmonics (II)

Renjie Feng,Robert J Adler,Xingcheng Xu

doi:10.1214/18-ecp156

Abstract

We continue the study, begun in \cite{FA}, of the critical radius of embeddings, via deterministic spherical harmonics, of fixed dimensional spheres into higher dimensional ones, along with the associated problem of the distribution of the suprema of random spherical harmonics. Whereas \cite{FA} concentrated on spherical harmonics of a common degree, here we extend the results to mixed degrees, obtaining larger lower bounds on critical radii than we found previously.

Highlights

The spherical harmonics of degree ≥ 0 on the d-dimensional unit sphere Sd are the collection of eigenfunctions {φj,d}kj=d 1 of the Laplacian ∆gSd on Sd, satisfying∆gSd φj,d(x) = − ( + d − 1)φj,d(x), (1.1)where kd is kd =∆ 2 + d − 1 +d−1 . +d−1 d−1 (1.2)In [6], we studied the mapid : Sd → Skd−1, x→sd kd φ1,d(x), · · ·, φk,dd(x) (1.3)defined by the spherical harmonics of degree, where, sd denotes the surface area of the unit sphere Sd
Whereas [6] concentrated on spherical harmonics of a common degree, here we extend the results to mixed degrees, en passant improving on the lower bounds on critical radii that we found previously
The aim of the present paper is to extend the analysis of [6] to a related, but somewhat different embedding, given by the deterministic map idL : Sd → SπLd −1, x→

Summary

Introduction

The spherical harmonics of degree ≥ 0 on the d-dimensional unit sphere Sd are the collection of (real) eigenfunctions {φj,d}kj=d 1 of the Laplacian ∆gSd on Sd, satisfying. Following the ideas and proofs in [6], we will prove the existence of a lower bound for the critical radius of idL(Sd) in SπLd −1 This will allow us to derive an exact formula for the distribution of the suprema of the family of random spherical harmonics under the spherical ensemble, viz. In terms of the original motivation for studying random spherical harmonics (the ‘Berry conjecture’ of [4]), mixed spectra processes play a more central role than pure spectra ones. This is one of the main motivations behind the current paper

Spherical harmonics and the deterministic embedding

Random spherical harmonics and exceedence probabilities

Spectral projection kernels