Multiple points on the boundaries of Brownian loop-soup clusters

For a Brownian loop soup with intensity c∈(0,1] in the unit disk, we show that almost surely, the set of simple (resp., double) points on any portion of boundary of any of its clusters has Hausdorff dimension 2−ξc(2) (resp., 2−ξc(4)), where ξc(k) is the generalized disconnection exponent computed in (Probab. Theory Related Fields 179 (2021) 117–164). As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and nondisconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents ξc(k,λ), and show that ξc(k) is the limit as λ↘0 of ξc(k,λ).