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

TL;DR

This paper demonstrates that conditioning a Brownian loop-soup cluster on part of its boundary results in a conformal restriction property for the touched loops and an independent loop-soup for the others, revealing a phase transition at c=14/15.

Contribution

It extends previous work by showing partial boundary conditioning preserves conformal restriction and explores subcritical cases, revealing new structural insights.

Findings

Loops touching the conditioned boundary satisfy conformal restriction.

Remaining loops form an independent loop-soup.

Phase transition at c=14/15 for loop connectivity.

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 $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $l$ 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 $l$. Our results can be viewed as an extension of some of the results in our earlier paper 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 there 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.