On the radius of Gaussian free field excursion clusters (2101.02200v2)

Abstract: We consider the Gaussian free field $\varphi$ 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 ${\varphi \geq h}$ 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 $\exp{-\frac{\pi}{6}(h-h_*)^2 \frac{N}{\log N} }$ 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.