(1+𝜀) moments suffice to characterise the GFF

Berestycki, Nathanaël; Powell, Ellen; Ray, Gourab

Abstract

We show that there is “no stable free field of index α ∈ ( 1 , 2 ) ”, in the following sense. It was proved in [4] that subject to a fourth moment assumption, any random generalised function on a domain D of the plane, satisfying conformal invariance and a natural domain Markov property, must be a constant multiple of the Gaussian free field. In this article we show that the existence of ( 1 + ε ) moments is sufficient for the same conclusion. A key idea is a new way of exploring the field, where (instead of looking at the more standard circle averages) we start from the boundary and discover averages of the field with respect to a certain “hitting density” of Itô excursions.

Citation

Berestycki, N., Powell, E., & Ray, G. (2021). (1+𝜀) moments suffice to characterise the GFF. Electronic Journal of Probability, 26(44), 1-25. https://doi.org/10.1214/20-ejp566

• Journal Article Type: Article
• Acceptance Date: Dec 6, 2020
• Online Publication Date: Apr 9, 2021
• Publication Date: 2021
• Deposit Date: Jul 19, 2021
• Publicly Available Date: Aug 19, 2021
• Journal: Electronic Journal of Probability
• Electronic ISSN: 1083-6489
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 44
• Pages: 1-25
• DOI: https://doi.org/10.1214/20-ejp566
• Public URL: https://durham-repository.worktribe.com/output/1239287