(1 + ε) moments suffice to characterise the GFF
Date
2021
Authors
Berestycki, Nathanaël
Powell, Ellen
Ray, Gourab
Journal Title
Journal ISSN
Volume Title
Publisher
Electronic Journal of Probability
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.
Description
Keywords
Gaussian free field, characterisation, harness, excursion measure, moments
Citation
Berestycki, N., Powell, E., & Ray, G. (2021). (1+𝜀) moments suffice to characterise the GFF. Electronic Journal of Probability, 26. https://doi.org/10.1214/20-ejp566