# Polynomials and High-Dimensional Spheres

## TL;DR

This paper investigates the asymptotic behavior of orthogonal polynomials on high-dimensional spheres, revealing their convergence to Hermite polynomials as the dimension grows large, and analyzes related differential operators.

## Contribution

It establishes the limiting behavior of spherical orthogonal polynomials and the spherical Laplacian in high dimensions, connecting them to Hermite polynomials.

## Key findings

- Orthogonal polynomials on large spheres tend to Hermite polynomials as dimension increases.
- The spherical Laplacian's behavior is characterized in the large-$N$ limit.
- Zonal harmonic polynomials exhibit specific asymptotic properties in high dimensions.

## Abstract

We show that a natural class of orthogonal polynomials on large spheres in $N$ dimensions tend to Hermite polynomials in the large-$N$ limit. We determine the behavior of the spherical Laplacian as well as zonal harmonic polynomials in the large-$N$ limit.