  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\to \infty $. This has been generalised to
excursion/level sets at arbitrary levels, implying the existence of functionals
$c_{ES}(\ell )$ and $c_{LS}(\ell )$ that encode the density of excursion/level
set components at the level $\ell $. 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 $c_{ES}(\ell )$ and $c_{LS}(\ell )$ are monotone.

Probability