Abstract
The Nazarov–Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower bounds showing that the functional is at least bimodal for certain isotropic fields, including the important special case of the random plane wave.
Highlights
1.1 The Nazarov–Sodin constantLet f : R2 → R be a continuous stationary planar Gaussian field normalised to have zero mean and unit variance
The first contribution of this paper is to extend the results of Nazarov–Sodin and [10] to arbitrary levels
Since the densities pm+, pm− and ps are in principle known by the Kac– Rice formula, our result demonstrates that the study of the Nazarov–Sodin constant can be reduced to an analysis of the density ps−, which may be an easier quantity to handle
Summary
Let f : R2 → R be a continuous stationary planar Gaussian field normalised to have zero mean and unit variance. One of the main analytical results concerning this set, due to Nazarov and Sodin [13,15], states that the number of components of N in a large domain scales like the area of the domain. If NR denotes the number of components of N inside the centred ball of radius R > 0, provided f is ergodic, there exists a constant cLS = cLS(ρ) ≥ 0 such that. In the case that f is not ergodic, it has been shown [10] (under the additional assumption that ρ has compact support) that the expected number of nodal components, scaled by the area, still converges, i.e. E[NR]/(π R2) → cL S as R → ∞. In [10] it was shown that among fields with compactly supported spectral measures, the constant cLS varies continuously with ρ (in the weak∗ topology)
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