(1+𝜀) moments suffice to characterise the GFF

Abstract

We show that there is “no stable free field of index $\mathit{\alpha }\in (1,2)$”, 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 $(1+\mathrm{𝜀})$ 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.