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We study the clusters of loops in a Brownian loop soup in some boundedtwo-dimensional domain with subcritical intensity $\theta \in (0,1/2]$. Weobtain an exact expression for the asymptotic probability of the existence of acluster crossing a given annulus of radii $r$ and $r^s$ as $r \to 0$ ($s >1$fixed). Relying on this result, we then show that the probability for amacroscopic cluster to hit a given disc of radius $r$ decays like $|\logr|^{-1+\theta+ o(1)}$ as $r \to 0$. Finally, we characterise the polar sets ofclusters, i.e. sets that are not hit by the closure of any cluster, in terms of$\log^\alpha$-capacity.  This paper reveals a connection between the 1D and 2D Brownian loop soups.This connection in turn implies the existence of a second critical intensity$\theta = 1$ that describes a phase transition in the percolative behaviour oflarge loops on a logarithmic scale targeting an interior point of the domain.

Crossing exponent in the Brownian loop soup