2020/01/17 by Jian Ding, Mateo Wirth, Ding, Jian +3 · 1 citation
Computer Science · #60G15 #60G60 (Primary) 60J65 #60J67 (Secondary) #Advanced Graph Theory Research #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2001.06447
openalex publication_date 2020/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We compare level-set percolation for Gaussian free fields (GFFs) defined on a rectangular subset of δℤ2 to level-set percolation for GFFs defined on the corresponding metric graph as the mesh size δ goes to 0. In particular, we look at the probability that there is a path that crosses the rectangle in the horizontal direction on which the field is positive. We show this probability is strictly larger in the discrete graph. In the metric graph case, we show that for appropriate boundary conditions the probability that there exists a closed pivotal edge for the horizontal crossing event decays logarithmically in δ. In the discrete graph case, we compute the limit of the probability of a horizontal crossing for appropriate boundary conditions.