Backbone probability of planar Brownian motion

TL;DR

This paper studies the probability of a specific connectivity event in planar Brownian motion, revealing it scales inversely with a double logarithmic function as the neighborhood size shrinks.

Contribution

It provides a precise asymptotic estimate for the backbone event probability in planar Brownian motion, inspired by critical percolation models.

Findings

Probability scales as C/(log|log ε|) as ε→0

Identifies the constant C in the asymptotic expression

Connects Brownian motion behavior to percolation theory

Abstract

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 $\varepsilon$-neighborhood of the starting point to a macroscopic distance. We show that the probability of this event is $C(\log|\log\varepsilon|)^{-1}(1+o(1))$ as $\varepsilon\to0$ for some constant $C\in(0,\infty)$.