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Abstract
We show that there is âno stable free field of index αâ(1,2)â, 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+đ) 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 âhitting densityâ of ItĂŽ excursions.
Highlights
The Gaussian free field (GFF) is a universal object believed to govern the fluctuation statistics of many natural random surface models [10, 18, 17, 12, 6, 3, 2, 7, 16]
The GFF can be defined in any dimension, this article is concerned with the planar continuum version, which satisfies two special properties; namely, conformal invariance and a domain Markov property
The former roughly entails that applying a conformal map to a GFF in any domain produces a GFF in the image domain
Summary
The Gaussian free field (GFF) is a universal object believed (and in many cases proved) to govern the fluctuation statistics of many natural random surface models [10, 18, 17, 12, 6, 3, 2, 7, 16] (see, e.g., [1, 20] for an introduction and survey of some recent developments). In [4], a fourth moment assumption on the field was used to apply Kolmogorovâs criterion, and thereby prove that the circle average process possesses an almost surely continuous modification This modification must be Brownian motion and, in particular, Gaussian. Note that this does not immediately imply joint Gaussianity of circle averages (for which significantly more work would be needed) It is enough (with a little extra work) to prove existence of fourth moments (Proposition 1.3) and given the result of [4], this concludes the proof of Theorem 1.4. Theorem 1.4 shows that conformal invariance and the domain Markov property (in the sense of Assumptions 1.1) are incompatible with these phD, Ïqs having α-stable (rather than Gaussian) distributions, for any value of the index α P p1, 2q. What are the minimal moment assumption necessary for Theorem 1.4 to hold? Do moments of order Ο for any Ο Ä 0 suffice?
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