On the radius of Gaussian free field excursion clusters

Abstract

We consider the Gaussian free field φ on Zd, for d≥3, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set {φ≥h} exceeds a large value N for any height h≠h∗, where h∗ refers to the corresponding percolation critical parameter. In dimension 3, we prove that this probability is subexponential in N and decays as exp{−π6(h−h∗)2NlogN} as N→∞ to principal exponential order. When d≥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.

Topics & Concepts