Critical one-arm probability on the metric graph Gaussian free field

Abstract: The Gaussian free field (GFF) on metric graphs was introduced by Titus Lupu (2016) as a natural extension of the discrete GFF. Since then, due to its favorable properties and close connection to other statistical physics models, it has been extensively studied for years. This talk will introduce our recent progress in estimating the critical one-arm probability (i.e., the probability that the origin is connected to the boundary of the box centered at the origin with side length N). Specifically, on the metric graphs of integer lattices Z^d (d>2), we derive the exact orders of one-arm probabilities for all dimensions except the critical dimension d=6, as well as the exponent at d=6. This is a joint work with Jian Ding (Peking University).