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

TL;DR

This paper investigates the one-arm probabilities in the level-set percolation of discrete and metric-graph Gaussian free fields, providing asymptotic estimates and bounds that highlight differences between the two cases.

Contribution

It offers new asymptotic estimates for metric-graph GFF and bounds for discrete GFF, clarifying their probabilistic behaviors and differences.

Findings

Asymptotic estimates for metric-graph one-arm probabilities

Up-to-constants bounds for discrete point-to-bulk probability

Demonstration of differences between discrete and metric-graph GFF behaviors

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.