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(1+𝜀) moments suffice to characterise the GFF

Berestycki, Nathanaël; Powell, Ellen; Ray, Gourab

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Nathanaël Berestycki

Gourab Ray


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.


Berestycki, N., Powell, E., & Ray, G. (2021). (1+𝜀) moments suffice to characterise the GFF. Electronic Journal of Probability, 26(44), 1-25.

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
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Published Journal Article (535 Kb)

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