Journal article

Smoothness and monotonicity of the excursion set density of planar gaussian fields

D Beliaev, M McAuley, S Muirhead

Electronic Journal of Probability | Institute of Mathematical Statistics | Published : 2020


Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius R, normalised by area, converges to a constant as R → ∞. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals cES(ℓ) and cLS(ℓ) that encode the density of excursion/level set components at the level ℓ. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result, which derives differentiability of the functionals from the decay of the probability of ‘four-arm events’ for the field conditioned to have a saddle point at the origin. For so..

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University of Melbourne Researchers


Awarded by Engineering & Physical Sciences Research Council (EPSRC) Fellowship

Awarded by EPSRC

Funding Acknowledgements

The authors thank an anonymous referee for their careful reading of the manuscript, for making us aware of [23] and, in particular, for pointing out an error in Lemma 4.4. The authors also thank Igor Wigman and Ben Hambly for helpful comments on a slightly different version of this work. The first author was supported by the Engineering & Physical Sciences Research Council (EPSRC) Fellowship EP/M002896/1.