On the radius of Gaussian free field excursion clusters

Abstract

We consider the Gaussian free field $φ$ on $\mathbb{Z}^d$, for $d\geq3$, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set $\{φ\geq h\}$ exceeds a large value $N$, for any height $h \neq h_*$, where $h_*$ refers to the corresponding percolation critical parameter. In dimension $d=3$, we prove that this probability is sub-exponential in $N$ and decays as $\exp\{-\fracπ{6}(h-h_*)^2 \frac{N}{\log N} \}$ as $N \to \infty$ to principal exponential order. When $d\geq 4$, 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.