Excursion decomposition of the 2D continuum GFF
Abstract
In this note we show that the 2D continuum Gaussian free field (GFF) admits an excursion decomposition that is on the one hand similar to the classical excursion decomposition of the Brownian motion, and on the other hand can be seen as an FK representation of the continuum GFF. In particular, 2D continuum GFF can be written as an infinite sum of disjoint positive and negative sign excursions, which are given by Minkowski content measures of clusters of a critical 2D Brownian loop soup with i.i.d. signs. Although the 2D continuum GFF is not even a signed measure, we show that the decomposition to positive and negative parts is unique under natural conditions.
Cite
@article{arxiv.2304.03150,
title = {Excursion decomposition of the 2D continuum GFF},
author = {Juhan Aru and Titus Lupu and Avelio Sepúlveda},
journal= {arXiv preprint arXiv:2304.03150},
year = {2023}
}
Comments
28 pages, a negligible number of figures, second version contains some new results, viewpoints and a conjecture