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Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves

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This is joint work with Par Kurlberg. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for “arithmetic random waves”, i.e. random toral Laplace eigenfunctions.

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