Crossing estimates from metric graph and discrete GFF

Understand

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 $\delta$.