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Fluctuations of the nodal length of random spherical harmonics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2010 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 298, no 3, 787-831 p.Article in journal (Refereed) Published
Abstract [en]

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order n. It is natural to conjecture that the variance should be of order n, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order log n. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.

Place, publisher, year, edition, pages
2010. Vol. 298, no 3, 787-831 p.
Keyword [en]
Complex-Manifolds, Zeros, Eigenfunctions, Sets, Lines
National Category
URN: urn:nbn:se:kth:diva-149626DOI: 10.1007/s00220-010-1078-8ISI: 000280259000008ScopusID: 2-s2.0-77954953234OAI: diva2:740751
Knut and Alice Wallenberg Foundation

QC 20140826

Available from: 2014-08-26 Created: 2014-08-25 Last updated: 2014-08-26Bibliographically approved

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Wigman, Igor
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