Multiple points on the boundaries of Brownian loop-soup clusters

TL;DR

This paper analyzes the boundary points of Brownian loop-soup clusters, determining their Hausdorff dimensions and density properties, and introduces a separation lemma to estimate non-intersection probabilities.

Contribution

It provides the first precise Hausdorff dimension results for boundary points of Brownian loop-soup clusters and introduces a separation lemma for sharp probability estimates.

Findings

Hausdorff dimension of simple boundary points is 2 - ξ_c(2)

Hausdorff dimension of double boundary points is 2 - ξ_c(4)

No triple points on cluster boundaries almost surely

Abstract

For a Brownian loop soup with intensity $c\in(0,1]$ 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 $2-\xi_c(2)$ (resp. $2-\xi_c(4)$), where $\xi_c(k)$ is the generalized disconnection exponent computed in arxiv:1901.05436. 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 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 $\xi_c(k, \lambda)$.