The Brownian loop soup. (English) Zbl 1049.60072

The subject of the paper is the study of Brownian measures on spaces of curves in the plane and their images under conformal invariance. Bridge-type Brownian measures are first considered. The Brownian bubble measure is a measure on loops in the upper half-plane rooted at the origin. The Brownian loop measure is a conformally invariant measure on unrooted loops. The last section is devoted to the relation between the Brownian loop soup of intensity \(\lambda>0\) which is a Poisson point process of intensity \(\lambda\) times the Brownian loop measure and the Brownian bubble soup which is a Poisson point process with intensity \(\lambda\) times the Brownian bubble measure. The point is that if we travel along a simple curve in the upper half-plane and encounter a loop in the loop soup, we can transform it into a bubble rooted at the origin. The relation of these topics with the Schramm-Loewner evolution curves is detailed in the introduction.


60J65 Brownian motion
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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