Fractal behavior for nodal lines of smooth planar Gaussian fields at criticality

Abstract

This paper is devoted to the study of the large scale geometry of the excursion set and nodal set of a planar smooth Gaussian field at criticality $\ell ={\ell }_c=0$. We prove that there exists $s_1>1$ such that with high probability, macroscopic nodal lines in a box of size $\lambda$ are of length at least ${\lambda }^{s_1}$. As an application, on the event that a box is crossed by a nodal line, then the shortest crossing is of length at least ${\lambda }^{s_1}$. We also prove that there exists $s_2<2$ such that with high probability, the shortest crossing is non degenerated, that is, its length is at most ${\lambda }^{s_2}$. The argument for the lower bound is based on a celebrated paper of Aizenman and Burchard [1] that provides a general argument to show that random curves present a fractal behavior. For the upper bound, our proof relies on the polynomial decay of the probability of one-arm events which was proven in [4].