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Theory of Probability and Mathematical Statistics



On the correlation between critical points and critical values for random spherical harmonics

V. Cammarota and A. P. Todino

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Abstract: We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I⊂ℝ. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L2-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics.

Keywords: Critical points, spherical harmonics, partial correlation, Wiener–Chaos expansion

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