On the radius of Gaussian free field excursion clusters

TL;DR

This paper derives sharp bounds on the probability that the radius of Gaussian free field excursion clusters exceeds a large value, revealing different decay rates depending on the dimension, and extends results to related percolation quantities.

Contribution

It provides the first sharp bounds on the tail probabilities of excursion cluster radii for the Gaussian free field in dimensions three and higher, including precise decay rates.

Findings

In 3D, the tail probability decays sub-exponentially as ^{-rac{\u03c0}{6}(h-h_*)^2 rac{N}{\u03bb N}}.

In dimensions  and higher, the tail probability decays exponentially in N.

Results extend to truncated two-point functions and two-arms probabilities for annuli crossings.

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.