One-arm probabilities for the two-dimensional metric-graph and discrete Gaussian free field

Abstract: We study the one-arm probability in the level-set percolation of the discrete and metric-graph Gaussian free field (GFF) defined on a box with Dirichlet boundary conditions. For the metric-graph case, we establish asymptotic estimates on two one-arm probabilities of interest. For the discrete case, we show up-to-constants bounds on the point-to-bulk probability and demonstrate its difference from the metric-graph case.