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

## TL;DR

This paper proves the smoothness and monotonicity properties of the density functionals for excursion and level set components of planar Gaussian fields, extending understanding of their geometric and probabilistic structure.

## Contribution

It establishes the continuous differentiability of the excursion and level set density functionals for a broad class of Gaussian fields and derives their monotonicity in specific cases.

## Key findings

- Functionals are continuously differentiable for many Gaussian fields.
- Monotonicity of the density functionals is shown for the Random Plane Wave and Bargmann-Fock fields.
- Differentiability follows from decay of four-arm event probabilities conditioned on saddle points.

## Abstract

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.