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

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 c_ES(ℓ) and c_LS(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 some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which c_ES(ℓ) and c_LS(ℓ) are monotone.