Geodesic stars in random geometry

TL;DR

This paper characterizes the fractal dimensions of geodesic stars with up to four arms in the Brownian sphere, revealing precise dimensionalities that complement previous upper bounds.

Contribution

It establishes exact dimensions of geodesic stars with 1 to 4 arms in the Brownian sphere, advancing understanding of random geometric structures.

Findings

Dimension of 1-arm geodesic stars is 4

Dimension of 2-arm geodesic stars is 3

Dimension of 3-arm geodesic stars is 2

Abstract

A point of a metric space is called a geodesic star with $m$ arms if it is the endpoint of $m$ disjoint geodesics. For every $m\in\{1,2,3,4\}$, we prove that the set of all geodesic stars with $m$ arms in the Brownian sphere has dimension $5-m$. This complements recent results of Miller and Qian, who proved that this dimension is smaller than or equal to $5-m$.