
Arm events in critical planar loop soups
We establish up-to-constants estimates for arm events in the Brownian loop soup on the 2D metric graph associated with the square lattice. More specifically, we consider two natural geometric events: first, ``bulk'' four-arm events, corresponding to two large connected components of loops getting close to each other; and then, two-arm events in the half-plane, used to estimate the probability that a cluster of loops approaches the boundary. Our proof relies on an estimate by Lupu-Werner [Probab…