Multi-window trace connectivity in subcritical planar Brownian loop soups

Let \(\cL_D^θ\) be a Brownian loop soup of intensity \(0<θ<1/2\) in a bounded smooth planar domain. We estimate the probability that one cluster of intersecting loop traces meets a fixed number of separated shrinking bulk discs. For every fixed \(q\ge3\), this probability is, up to an arbitrary loss in the logarithmic exponents, the product of the corresponding one-arm probabilities. The upper bound is not obtained by multiplying pairwise estimates or by a formal application of BK. We condition on the loops not confined to one target collar, charge a local arm down to their penetration radius at every target, and control the joint penetration depths by a multi-target Brownian loop-measure estimate. The latter follows from a marked Schur-complement expansion on finite killed networks, uniform planar capacity estimates, and a random-walk-loop-soup coupling. The lower bound uses winding separators, one macroscopic bridge loop, and Poisson FKG. The one-window exponent and the two-window estimate are due to Jego, Lupu and Qian.