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CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

  • Igor Wigman (a1) and Nadav Yesha (a2)

Abstract

We study the fine-scale $L^{2}$ -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

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Current address: Department of Mathematics, University of Haifa, 3498838 Haifa, Israel email nyesha@univ.haifa.ac.il

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CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

  • Igor Wigman (a1) and Nadav Yesha (a2)

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