Critical radius and supremum of random spherical harmonics (II)

TL;DR

This paper extends previous work on the critical radius of embeddings of spheres using spherical harmonics, now including mixed degrees, leading to improved lower bounds and insights into the distribution of random spherical harmonics' suprema.

Contribution

It introduces new results on mixed degree spherical harmonics, providing larger lower bounds on critical radii compared to prior work focused on common degrees.

Findings

Larger lower bounds on critical radii for mixed degree spherical harmonics

Extended the analysis to include mixed degrees, not just common degree harmonics

Enhanced understanding of the distribution of suprema of random spherical harmonics

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.