On the radius of Gaussian free field excursion clusters

Abstract

We consider the Gaussian free field φ on ${\mathbb{Z}}^{\mathit{d}}$, for $\mathit{d}\ge 3$, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set $\{\mathit{\phi }\ge \mathit{h}\}$ exceeds a large value N for any height $\mathit{h}\ne {\mathit{h}}_{\ast }$, where ${\mathit{h}}_{\ast }$ refers to the corresponding percolation critical parameter. In dimension 3, we prove that this probability is subexponential in N and decays as $exp\{-\frac{\mathit{\pi }}{6}{(\mathit{h}-{\mathit{h}}_{\ast })}^2\frac{\mathit{N}}{log\mathit{N}}\}$ as $\mathit{N}\to \infty$ to principal exponential order. When $\mathit{d}\ge 4$, we prove that these tails decay exponentially in N. Our results extend to other quantities of interest, such as truncated two-point functions and the two-arms probability for annuli crossings at scale N.