We show that there is “no stable free field of index ”, in the following sense. It was proved in  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 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.
Nathanaël Berestycki is supported in part by EPSRC grant EP/L018896/1, the University of Vienna, and FWF grant “Scaling limits in random conformal geometry”. Gourab Ray is supported in part by NSERC 50311-57400 and University of Victoria start-up 10000-27458.
" moments suffice to characterise the GFF." Electron. J. Probab. 26 1 - 25, 2021. https://doi.org/10.1214/20-EJP566