## The Annals of Probability

### Critical radius and supremum of random spherical harmonics

#### Abstract

We first consider deterministic immersions of the $d$-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level $n$. The main result of the article is the, a priori unexpected, fact that there is a uniform lower bound to the critical radius of the immersions as $n\to\infty$. This fact has immediate implications for random spherical harmonics with fixed $L^{2}$-norm. In particular, it leads to an exact and explicit formulae for the tail probability of their (large deviation) suprema by the tube formula, and also relates this to the expected Euler characteristic of their upper level sets.

#### Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 1162-1184.

Dates
Revised: April 2018
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.aop/1551171648

Digital Object Identifier
doi:10.1214/18-AOP1283

Mathematical Reviews number (MathSciNet)
MR3916945

Zentralblatt MATH identifier
07053567

Subjects
Primary: 33C55: Spherical harmonics 60G15: Gaussian processes
Secondary: 60F10: Large deviations 60G60: Random fields

#### Citation

Feng, Renjie; Adler, Robert J. Critical radius and supremum of random spherical harmonics. Ann. Probab. 47 (2019), no. 2, 1162--1184. doi:10.1214/18-AOP1283. https://projecteuclid.org/euclid.aop/1551171648

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