Conditioning a Brownian loop-soup cluster on a portion of its boundary

Abstract

We show that if one conditions a cluster in a Brownian loop-soup L (of any intensity) in a two-dimensional domain by a portion ∂ of its outer boundary, then in the remaining domain, the union of all the loops of L that touch ∂ satisfies the conformal restriction property while the other loops in L form an independent loop-soup. This result holds when one discovers ∂ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch ∂. Our results can be viewed as an extension of some of the results in our paper (J. Eur. Math. Soc. (2019) to appear) in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained in (J. Eur. Math. Soc. (2019) to appear) that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds.