(1+𝜀) moments suffice to characterise the GFF

Abstract

We show that there is "no stable free field of index α P p1, 2q", 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 p1`εq 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.

A characterisation of the continuum Gaussian free field in arbitrary dimensions

“…In [BPR20] the authors showed that for d = 2, the GFF is the only conformally invariant field satisfying a domain Markov property under some continuity and moments assumptions; these moment assumptions were then lowered in [BPR21]. Further, in [AP22], the authors proved a characterisation of the GFF in d ≥ 2, which heuristically corresponds to the stationary independent increments property.…”

“…One could get away with less, e.g. by assuming a uniform bound only for certain symmetric mollifiers; this would already very simply follow from the stronger Markov decomposition in [AP22,BPR21] combined with just the finiteness of fourth moments.…”

A characterisation of the continuum Gaussian free field in $d \geq 2$ dimensions

“…It is both the proven and the conjectured scaling limit of several natural discrete height functions [Nad97, Ken01,RV07] There are many characterisations of Brownian motion that single it out among 1d stochastic processes and it is similarly natural to also ask how the GFF in higher dimensions can be characterised. Recently, the 2d continuum GFF was characterised as the only conformally invariant field, satisfying a domain Markov property and certain minimal moment and continuity assumptions [BPR20,BPR21].…”