Stability of geodesics in the Brownian map

TL;DR

The paper investigates the geodesic geometry of the Brownian map, proving stability and confluence properties of geodesics and cut loci. It introduces precise notions such as the weak and strong cut loci $S(x)$ and $C(x)$, and the geodesic nets $G(x)$, establishing continuity and local stability results for typical points and showing that the geodesic framework is typically sparse (first Baire category). A central achievement is the classification of dense geodesic networks into six topological types with explicit Hausdorff-dimension counts, together with a detailed analysis of convergence and confluence near typical points. The results deepen understanding of the random geometry of the Brownian map, with implications for related random planar-map limits such as the Brownian plane and higher-genus analogues, and provide a robust basis for future exploration of geodesic networks in random metric spaces.

Abstract

The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure. Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a closed, nowhere dense set of zero measure. We obtain similar stability results for the set of points inside geodesics to a fixed point. Furthermore, we show that the set of points inside geodesics of the map is of first Baire category. Hence, most points in the Brownian map are endpoints. Finally, we classify the types of geodesic networks which are dense. For each $k\in\{1,2,3,4,6,9\}$, there is a dense set of pairs of points which are joined by networks of exactly $k$ geodesics and of a specific topological form. We find the Hausdorff dimension of the set of pairs joined by each type of network. All other geodesic networks are nowhere dense.

Figures (7)

• Figure 1: As depicted, $(x,y)\in N(2,3)$. Note that $(u,x)$ does not induce a normal $(j,k)$-network.
• Figure 2: Theorem \ref{['T_normal']}: Classification of networks which are dense in the Brownian map (up to symmetries and homeomorphisms of the sphere).
• Figure 3: Proposition \ref{['P_conf-point']}: All geodesics from points in $N'$ to points in the complement of $N\supset N'$ pass through a confluence point $x_0$.
• Figure 4: Lemma \ref{['L_u']}: $[u_\ell,v_\ell]-\gamma$ is contained in $(B(u,\delta)\cup B(v,\delta)) \cap L$ (as viewed through the homeomorphism $\psi$).
• Figure 5: Given $[x',y']\subset \gamma$ we find a geodesic $\gamma_\ell =[u_\ell,v_\ell]$ which intersects $\gamma$ in $[u"_\ell,v"_\ell]$, which is almost all of $[x',y']$, and similarly $[u_r,v_r]$. These are used to define the sets $V_\eta$ (shaded), and subsets $H_\ell$ and $H_r$ (dark gray). For large $n$, the geodesics $\gamma_n$ are included in $V_\eta$ and cannot enter $H_\ell\cup H_r$, leading to strong convergence. The points $u,v,u'_r,u"_r,v'_r,v"_r,u",v"$ are not shown. For clarity, we omitted $\psi(\cdot)$ from all points (besides $\psi(x)=0$ and $\psi(y)=1$) named in the figure.

Theorems & Definitions (67)

• Definition
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