2024/10/02 by David Vernotte, Vernotte, David
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2410.01453
openalex publication_date 2024/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℓ=ℓc=0. We prove that there exists s1>1 such that with high probability, macroscopic nodal lines in a box of size λ are of length at least λs1. As an application, on the event that a box is crossed by a nodal line, then the shortest crossing is of length at least λs1. We also prove that there exists s2<2 such that with high probability, the shortest crossing is non degenerated, that is, its length is at most λs2. 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].