A lower bound for the Bogomolny–Schmit constant for random monochromatic plane waves

Let $f$ be the the Planar Monochromatic Wave, i.e, the unique a.s. continuous stationary centred Gaussian field on $\mathbb{R}^2$ with covariance $\mathbb{E} [f(x)f(y)] = J_0 (\lvert x - y \rvert)$. Its average number of nodal domains per unit area is given by the Bogomolny–Schmit (or Nazarov–Sodin) constant. In this paper, we prove that $\nu_{BS} \geq 1.39 \times 10^{-4}$.

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Received 19 April 2019

Accepted 26 May 2019

Published 25 October 2019