(1+𝜀) moments suffice to characterise the GFF

Berestycki, Nathanaël; Powell, Ellen; Ray, Gourab

doi:10.1214/20-ejp566

(1+𝜀) moments suffice to characterise the GFF

Berestycki, Nathanaël; Powell, Ellen; Ray, Gourab

Authors

Nathanaël Berestycki

Dr Ellen Powell ellen.g.powell@durham.ac.uk
Associate Professor

Gourab Ray

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
Publisher	Institute of Mathematical Statistics
Peer Reviewed	Peer Reviewed
Volume	26
Issue	44
Pages	1-25
DOI	https://doi.org/10.1214/20-ejp566

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