September 2022 On the radius of Gaussian free field excursion clusters
Subhajit Goswami, Pierre-François Rodriguez, Franco Severo
Author Affiliations +
Ann. Probab. 50(5): 1675-1724 (September 2022). DOI: 10.1214/22-AOP1569


We consider the Gaussian free field φ on Zd, for d3, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set {φh} exceeds a large value N for any height hh, 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(hh)2NlogN} as N to principal exponential order. When d4, 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.

Funding Statement

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.


Download Citation

Subhajit Goswami. Pierre-François Rodriguez. Franco Severo. "On the radius of Gaussian free field excursion clusters." Ann. Probab. 50 (5) 1675 - 1724, September 2022.


Received: 1 January 2021; Revised: 1 November 2021; Published: September 2022
First available in Project Euclid: 24 August 2022

MathSciNet: MR4474499
zbMATH: 1512.60023
Digital Object Identifier: 10.1214/22-AOP1569

Primary: 60G15 , 60G60 , 60J45 , 60K35 , 82B43

Keywords: capacity , Gaussian free field , percolation , Random walk

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 5 • September 2022
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