Multiplicative chaos of the Brownian loop soup

Abstract

We construct a measure on the thick points of a Brownian loop soup in a bounded domain D of the plane with given intensity $θ>0$, 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 [BBK94, AHS20, Jeg20a], or via a discrete loop soup approximation. At the critical intensity $θ= 1/2$, 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 $θ= 1/2$. 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.