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March 2019 Critical radius and supremum of random spherical harmonics
Renjie Feng, Robert J. Adler
Ann. Probab. 47(2): 1162-1184 (March 2019). DOI: 10.1214/18-AOP1283


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.


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Renjie Feng. Robert J. Adler. "Critical radius and supremum of random spherical harmonics." Ann. Probab. 47 (2) 1162 - 1184, March 2019.


Received: 1 February 2017; Revised: 1 April 2018; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053567
MathSciNet: MR3916945
Digital Object Identifier: 10.1214/18-AOP1283

Primary: 33C55 , 60G15
Secondary: 60F10 , 60G60

Keywords: asymptotics , critical radius , curvature , Gaussian ensemble , large deviations , reach , spherical ensemble , Spherical harmonics

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 2 • March 2019
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