Backbone probability of planar Brownian motion

Motivated by critical planar percolation, we investigate a ``backbone’’ event of planar Brownian motion, i.e.~the existence of two disjoint subpaths on the Brownian trajectory connecting the $\epsilon$-neighborhood of the starting point to a macroscopic distance. We show that the probability of this event is $C(\log \mathrm{\mid }\log \epsilon \mathrm{\mid }{)}^{-1}(1+o(1))$ as $\epsilon \to 0$ for some constant $C\in (0,\mathrm{\infty })$.