We study the fine-scale (Formula presented.) -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.
Bibliographical noteFunding Information:
The authors of this paper wish to express their gratitude to J. Benatar, A. Granville, P. Kurlberg, Z. Rudnick, P. Sarnak and M. Sodin for numerous stimulating and fruitful discussions concerning various aspects of our work, and their interest in our research. It is a pleasure to thank the anonymous referee for his comments on an earlier version of the paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013), ERC grant agreement no. 335141.
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- 35P20 (primary)
ASJC Scopus subject areas
- Mathematics (all)