Crossing estimates from metric graph and discrete GFF
Abstract
We compare levelset percolation for Gaussian free fields (GFFs) defined on a rectangular subset of $\delta \mathbb{Z}^2$ to levelset percolation for GFFs defined on the corresponding metric graph as the mesh size $\delta$ 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 $\delta$. In the discrete graph case, we compute the limit of the probability of a horizontal crossing for appropriate boundary conditions.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.06447
 Bibcode:
 2020arXiv200106447D
 Keywords:

 Mathematics  Probability;
 60G15;
 60G60 (Primary) 60J65;
 60J67 (Secondary)
 EPrint:
 35 pages