ON THE RADIUS OF GAUSSIAN FREE FIELD EXCURSION CLUSTERS

We consider the Gaussian free field phi on Z(d), for d >= 3, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set {phi >= h} exceeds a large value N for any height h not equal 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{-pi/6 (h - h(*))(2) N/logN} as N -> infinity 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.

Automated summary

This paper considers the Gaussian free field phi on Z(d), and provides sharp bounds on the probability of the radius of a finite cluster exceeding a given value for any height. The results show the decay rate of this probability in different dimensions.