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..View full abstract
Awarded by Engineering & Physical Sciences Research Council (EPSRC) Fellowship
Awarded by EPSRC
The authors thank an anonymous referee for their careful reading of the manuscript, for making us aware of  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.