Nodal length fluctuations for arithmetic random waves

Abstract

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (“arithmetic random waves”). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.

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      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58J51 (33C55)},
      MRNUMBER = {2670928},
      MRREVIEWER = {Nelia Charalambous},
      DOI = {10.1007/s00220-010-1078-8},
      ZBLNUMBER = {1213.33019},
      }
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    @incollection{WDartmouth,
      author={Wigman, Igor},
      TITLE={On the nodal lines of random and deterministic {L}aplace eigenfunctions},
      BOOKTITLE = {Spectral Geometry},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {84},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2012},
      VENUE={Proc. Internatl. Conf., Dartmouth College, July 19--23, 2010},
      ARXIV={1103.0150} ,}
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    @incollection {Z, MRKEY = {1500155},
      AUTHOR = {Zelditch, Steve},
      TITLE = {Real and complex zeros of {R}iemannian random waves},
      BOOKTITLE = {Spectral Analysis in Geometry and Number Theory},
      SERIES = {Contemp. Math.},
      VOLUME = {484},
      PAGES = {321--342},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      MRCLASS = {58J51 (35J05)},
      MRNUMBER = {1500155},
      MRREVIEWER = {Julie Marie Rowlett},
      DOI = {10.1090/conm/484/09482},
      ZBLNUMBER = {1176.58021},
      VENUE={International conference on the occasion of Toshikazu Sunada's 60th birthday},
      }

Authors

Manjunath Krishnapur

Indian Institute of Science
Bangalore
India

Pär Kurlberg

Royal Institute of Technology
Stockholm
Sweden

Igor Wigman

Cardiff University
Wales
UK

Current address:

King's College London
London
UK