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 $Rto 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.
تحميل البحث