Affiliation:
1. School of Mathematical Sciences Fudan University Shanghai China
2. Faculty of Mathematics University of Vienna Vienna Austria
3. Ecole Polytechnique Fédérale de Lausanne EPFL EPFL SB MATH RGM Lausanne Switzerland
4. Laboratoire de Probabilités Statistique et Modélisation Sorbonne Université and Université Paris Cité CNRS Paris France
Abstract
AbstractWe construct a measure on the thick points of a Brownian loop soup in a bounded domain of the plane with given intensity , which is formally obtained by exponentiating the square root of its occupation field. The measure is constructed via a regularisation procedure, in which loops are killed at a fix rate, allowing us to make use of the Brownian multiplicative chaos measures previously considered in Aïdékon et al. (Ann. Probab. 48 (2020), no. 4, 1785–1825), Bass et al. (Ann. Probab. 22 (1994), no. 2, 566–625) and Jego (Ann. Probab. 48 (2020), no. 4, 1597–1643), or via a discrete loop soup approximation. At the critical intensity , it is shown that this measure coincides with the hyperbolic cosine of the Gaussian free field, which is closely related to Liouville measure. This allows us to draw several conclusions which elucidate connections between Brownian multiplicative chaos, Gaussian free field and Liouville measure. For instance, it is shown that Liouville‐typical points are of infinite loop multiplicity, with the relative contribution of each loop to the overall thickness of the point being described by the Poisson–Dirichlet distribution with parameter . Conversely, the Brownian chaos associated to each loop describes its microscopic contribution to Liouville measure. Along the way, our proof reveals a surprising exact integrability of the multiplicative chaos associated to a killed Brownian loop soup. We also obtain some estimates on the discrete and continuous loop soups which may be of independent interest.
Funder
Agence Nationale de la Recherche
Österreichischen Akademie der Wissenschaften
Engineering and Physical Sciences Research Council
Universität Wien
Austrian Science Fund