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Excursion decomposition of the 2D continuum GFF
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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.
Forward citations
Cited by 2 Pith papers
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A switching identity for cable-graph loop soups and Gaussian free fields
Conditioning two cable-graph points to lie in the same Brownian loop-soup cluster adds an odd-numbered Poisson cloud of Brownian excursions between them, yielding an exact law for the conditional cluster and its GFF analogue.
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Relation between Wick powers and excursion clusters of the 2D GFF
Restricted odd Wick powers of the 2D GFF appear as coefficients of the half-integer logarithmic asymptotics of conformal-radius neighborhoods of first passage sets and sign clusters; even powers appear only after comp...
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