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

2025/07/27 by Gao, Yifan, Li, Xinyi, Liu, Runsheng +1
#60J65 (Primary) 28A80 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.20324

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''.

Citations

• Up-to-constants estimates on four-arm events for simple conformal loop ensemble
• Uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for c ∈ (0,1]
• Parity questions in critical planar Brownian loop-soups (or "where did the free planar bosons go?")
• Crossing exponent in the Brownian loop soup
• Multiple points on the boundaries of Brownian loop-soup clusters
• Minkowski content of Brownian cut points
• Fast convergence to an invariant measure for non-intersecting 3-dimensional Brownian paths

Related

• Brownian geometry
• The fractal cylinder process: existence and connectivity phase transition
• The existence phase transition for scale invariant Poisson random\n fractal models
• Multiple points on the boundaries of Brownian loop-soup clusters
• On clusters of Brownian loops in d dimensions