Berestycki, NathanaëlPowell, EllenRay, Gourab2025-02-192025-02-192021Berestycki, 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-ejp566https://doi.org/10.1214/20-ejp566https://hdl.handle.net/1828/21221We 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.enAttribution 4.0 InternationalGaussian free fieldcharacterisationharnessexcursion measuremomentsArticle