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Arm events in critical planar loop soups
Yijie Bi, Yifan Gao, Pierre Nolin, Wei Qian
https://arxiv.org/abs/2509.25024 https://arxiv.org/pdf/2509.25024

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…