Parity questions in critical planar Brownian loop-soups (or “where did the free planar bosons go?”)

Abstract The critical two-dimensional Brownian loop-soup is an infinite collection of non-interacting Brownian loops in a planar domain that possesses some combinatorial features related to the notion of “indistinguishability” of bosons. The properly renormalized occupation time field of this collection of loops is known to be distributed like the properly defined square of a Gaussian free field. In the present paper, we study how much information these fields provide about the loop-soup. Among other things, we show that the exact set of points that are actually visited by some loops in the loop-soup is not determined by these fields. We further prove that given the fields, a dense family of special points will each have a conditional probability 1/2 of being part of the loop-soup. We also exhibit another instance where the possible decompositions (given the field) into individual loops and excursions can be grouped into two clearly different groups, each having a conditional probability 1/2 of occurring.