Backbone probability of planar Brownian motion
Gefei Cai, Zhuoyan Xie
TL;DR
The paper addresses the backbone event for planar Brownian motion, establishing that the probability of two disjoint subpaths connecting a small ε-neighborhood of the origin to a macroscopic circle decays only via an iterated logarithm. It leverages a deep connection to SLE_8/3, conformal radii, and Liouville quantum gravity to derive exact laws for conformal radii of SLE-type loops and a layer structure for Brownian cut points, enabling precise tail estimates. The main result shows P[ Bac_ε ] is asymptotically proportional to 1/ log|log ε| as ε → 0, with extensions to alternative target radii; this reveals a fundamental difference between Brownian motion and critical percolation in backbone exponents. The work combines exact probabilistic identities with Tauberian arguments and continuum geometric tools to produce a sharp, quantifiable backbone decay and hints at broader applicability to loop-soup and half-plane variants.
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 $(\log|\log\varepsilon|)^{-1}$ up to a multiplicative constant.