Crossing exponent in the Brownian loop soup

We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity θ∈(0,1/2]. We show that a cluster which is targeting a given interior point forms a dynamical system, which satisfies asymptotically a fixed point equation. As a consequence, we obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii r and rs as r→0 (s>1 fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius r decays like |logr|−1+θ+o(1) as r→0. Finally, we characterise the polar sets of clusters, that is, sets that are not hit by the closure of any cluster, in terms of logα-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 θ=1 that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain.