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arXiv · 1704.04943

Two point function for critical points of a random plane wave

Dmitry Beliaev·Valentina Cammarota·Igor Wigman
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Abstract

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.

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Dmitry Beliaev, Valentina Cammarota, Igor Wigman. 2017-04-17. Two point function for critical points of a random plane wave. https://arxiv.org/abs/1704.04943

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