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Nazarov and Sodin have shown that the number of connected components of thenodal set of a planar Gaussian field in a ball of radius $R$, normalised byarea, converges to a constant as $R\to \infty $. This has been generalised toexcursion/level sets at arbitrary levels, implying the existence of functionals$c_{ES}(\ell )$ and $c_{LS}(\ell )$ that encode the density of excursion/levelset components at the level $\ell $. We prove that these functionals arecontinuously differentiable for a wide class of fields. This follows from amore general result, which derives differentiability of the functionals fromthe decay of the probability of `four-arm events' for the field conditioned tohave a saddle point at the origin. For some fields, including the importantspecial cases of the Random Plane Wave and the Bargmann-Fock field, we alsoderive stochastic monotonicity of the conditioned field, which allows us todeduce regions on which $c_{ES}(\ell )$ and $c_{LS}(\ell )$ are monotone.

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