Paper
Anomalous scaling law for the two-dimensional Gaussian free field
Authors
Pierre-François Rodriguez, Wen Zhang
Abstract
We consider the Gaussian free field on at large spatial scales and give sharp bounds on the probability that the radius of a finite cluster in the excursion set on the corresponding metric graph is macroscopic. We prove a scaling law for this probability, by which transitions from fractional logarithmic decay for near-critical parameters to polynomial decay in the off-critical regime. The transition occurs across a certain scaling window determined by a correlation length scale , which is such that for typical heights as diverges, with an explicit exponent that we identify in the process. This is in stark contrast with recent results from arXiv:2101.02200 and arXiv:2312.10030 in dimension three, where similar observables are shown to follow regular scaling laws, with polynomial decay at and near criticality, and rapid decay in away from it.