Crossing estimates from metric graph and discrete GFF

We compare level-set percolation for Gaussian free fields (GFFs) defined on a rectangular subset of \(\delta \mathbb{Z}^2\) to level-set 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 pat...

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Bibliographic Details
Published in:arXiv.org 2020-01
Main Authors: Ding, Jian, Wirth, Mateo, Wu, Hao
Format: Article
Language:English
Subjects:
Boundary conditions
Horizontal orientation
Percolation
Online Access:Get full text
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Summary:We compare level-set percolation for Gaussian free fields (GFFs) defined on a rectangular subset of \(\delta \mathbb{Z}^2\) to level-set 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.
ISSN:2331-8422