Multiple points on the boundaries of Brownian loop-soup clusters

Yifan Gao, Xinyi Li, Wei Qian

Published 2022-05-23Version 1

For a Brownian loop soup with intensity $c\in(0,1]$ in the unit disk, we show that the almost-sure Hausdorff dimensions of simple and double points on the boundaries of its clusters are respectively equal to $2-\xi_c(2)$ and $2-\xi_c(4)$, where $\xi_c(k)$ is the generalized disconnection exponent computed in arxiv:1901.05436. We further show that almost surely such points are dense on every portion of boundary of every cluster, when they exist. 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 non-disconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents $\xi_c(k, \lambda)$, and show that $\xi_c(k)$ is the limit as $\lambda\searrow 0$ of $\xi_c(k, \lambda)$.