We consider the Gaussian free field φ on , for , and give sharp bounds on the probability that the radius of a finite cluster in the excursion set exceeds a large value N for any height , where refers to the corresponding percolation critical parameter. In dimension 3, we prove that this probability is subexponential in N and decays as as to principal exponential order. When , 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.
Part of this research was supported by the ERC Grant CriBLaM and an IDEX grant from Paris-Saclay. S.G.’s research was supported by the SERB grant SRG/2021/000032 and carried out in part as a member of the Infosys-Chandrasekharan virtual center for Random Geometry, supported by a grant from the Infosys Foundation. F.S.’s work was partially supported by the Swiss FNS.
We thank Jian Ding, Alexis Prévost and Mateo Wirth for discussions at various stages of this project. We are grateful to an anonymous referee for her/his numerous and valuable suggestions on a previous version of this manuscript.
"On the radius of Gaussian free field excursion clusters." Ann. Probab. 50 (5) 1675 - 1724, September 2022. https://doi.org/10.1214/22-AOP1569