On the distribution of the critical values of random spherical harmonics
Mathematical Physics
2018-01-09 v1 math.MP
Probability
Abstract
We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed.
Keywords
Cite
@article{arxiv.1409.1364,
title = {On the distribution of the critical values of random spherical harmonics},
author = {Valentina Cammarota and Domenico Marinucci and Igor Wigman},
journal= {arXiv preprint arXiv:1409.1364},
year = {2018}
}