There are no geodesic hubs in the Brownian sphere

TL;DR

This work studies exceptional geodesic structures in the Brownian sphere and proves that almost surely there are no $3^+$-hubs (hubs with three or more disjoint geodesic arms). The authors develop a Brownian-snake framework, introduce Brownian slices, and use spine decompositions and Palm calculus to reduce hub existence to a triple-point event, then show that in the relevant slice no geodesic can pass through the apex. The result strengthens the understanding of geodesic networks in scaling limits of random planar maps and contrasts with related models by showing rigidity of geodesic connections. The techniques—spine decompositions, Poisson point measures, and 0–1 laws—provide a robust toolkit for analyzing geodesic intersections in random geometry and may inform future investigations of higher-hub questions and related fractal structures.

Abstract

A point of a metric space is called a $k$-hub if it is the endpoint of exactly $k$ disjoint geodesics, and that the concatenation of any two of these paths is still a geodesic. We prove that in the Brownian sphere, there is no $k$-hub for $k\geq 3$.

Figures (4)

• Figure 1: Illustration where $x$ is a $4$-hub, and $(x_1,x_2,x_3,x_4)$ border a $4$-hub. Each of the black paths between $x_i$ and $x_j$ for $i\neq j$ in $\{1,2,3,4\}$ is a geodesic.
• Figure 2: Representation of $T_0,\,\widehat{T}_0,\,T_1$ and $\widehat{T}_1$ in the random tree $\mathcal{T}_1$. Note that the two blue (respectively pink, green) portions correspond to the same path in $\mathcal{S}_1$.
• Figure 3: Illustration of the path $\gamma_{0,1}$. We need to determine whether it can be a geodesic.
• Figure 4: The same illustration as Figure \ref{['Arbre']}, but viewed in $\mathcal{S}_1$. The black path corresponds to the projection of the spine of $\mathcal{T}_1$, and the yellow (respectively gray) area is the projection of the atom containing $u_*$ (respectively $u_{**}$).

Theorems & Definitions (29)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Proposition 2.2
• Definition 2.3
• Lemma 3.1
• proof
• Proposition 3.2
• proof