On the Number of Nodal Domains of Random Spherical Harmonics

Let N(f) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N(f)/n² tends to a positive constant a, and that N(f)/n² exponentially concentrates around a. This result is consistent with predictions made by Bogomolny an...

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Veröffentlicht in:	American journal of mathematics 2009-10, Vol.131 (5), p.1337-1357
Hauptverfasser:	Nazarov, Fedor, Sodin, Mikhail
Format:	Artikel
Sprache:	eng
Schlagworte:	Covariance Exact sciences and technology General mathematics General, history and biography Harmonic analysis Hypersurfaces Inference from stochastic processes time series analysis Mathematical analysis Mathematical constants Mathematical functions Mathematical problems Mathematics Modeling Numbers Polynomials Probability and statistics Probability theory and stochastic processes Radius of a sphere Random variables Sciences and techniques of general use Special functions Spheres Spherical harmonics Statistics Stochastic processes Theorems Zero
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Beschreibung
Zusammenfassung:	Let N(f) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N(f)/n² tends to a positive constant a, and that N(f)/n² exponentially concentrates around a. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.
ISSN:	0002-9327 1080-6377 1080-6377
DOI:	10.1353/ajm.0.0070