Non-existence of several random fractals in Brownian motion and Brownian loop soup

TL;DR

This paper proves that certain types of random fractals related to Brownian motion and loop soups do not exist, based on their intersection properties and Hausdorff dimension.

Contribution

It introduces a unified method to show the non-existence of specific fractals in Brownian motion and loop soups, resolving open questions.

Findings

Proves pioneer triple points of planar Brownian motion do not exist.

Shows double cut points of Brownian motions are absent.

Establishes no double points on loop soup cluster boundaries at critical intensity.

Abstract

We develop a unified approach to establish the non-existence of three types of random fractals: (1) the pioneer triple points of the planar Brownian motion, answering an open question in [7], (2) the pioneer double cut points of the planar and three-dimensional Brownian motions, and (3) the double points on the boundaries of the clusters of the planar Brownian loop soup at the critical intensity, answering an open question in [39]. These fractals have the common feature that they are associated with an intersection or disconnection exponent which yields a Hausdorff dimension ``exactly zero''.