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2021 (1+𝜀) moments suffice to characterise the GFF
Nathanaël Berestycki, Ellen Powell, Gourab Ray
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Electron. J. Probab. 26: 1-25 (2021). DOI: 10.1214/20-EJP566


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.

Funding Statement

Nathanaël Berestycki is supported in part by EPSRC grant EP/L018896/1, the University of Vienna, and FWF grant “Scaling limits in random conformal geometry”. Gourab Ray is supported in part by NSERC 50311-57400 and University of Victoria start-up 10000-27458.


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Nathanaël Berestycki. Ellen Powell. Gourab Ray. "(1+𝜀) moments suffice to characterise the GFF." Electron. J. Probab. 26 1 - 25, 2021.


Received: 19 May 2020; Accepted: 6 December 2020; Published: 2021
First available in Project Euclid: 9 April 2021

Digital Object Identifier: 10.1214/20-EJP566

Primary: 60G05, 60G15, 60G20


Vol.26 • 2021
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