(1+)-moments suffice to characterise the GFF
Abstract
We show that there is "no stable free field of index α∈ (1,2)", in the following sense. It was proved in a previous work by the authors, 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+)-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 It\o excursions.
0