Bigeodesic Brownian Plane

• Bigeodesic Brownian plane is a random, non-compact, locally compact length space featuring a unique bi-infinite geodesic and emerging as a tangent limit of Brownian surfaces.
• Its construction uses continuum random trees, Bessel processes, and Poisson point processes to glue excursions along a spine, rigorously defining its metric structure.
• Its invariance properties and scaling limits unify discrete random surface models with continuous Brownian geometries, offering insight into geodesic structures in random geometry.

The bigeodesic Brownian plane is a random, non-compact, locally compact length space characterized by the presence of a unique bi-infinite geodesic. It emerges as a tangent-plane limit of the Brownian sphere at points on its simple geodesic, and as a local limit of the classical Brownian plane rerooted along its unique infinite geodesic. These constructions equip it with a rich metric, topological, and probabilistic structure, closely linked to both continuous and discrete models of random surfaces.

1. Construction and Formal Definition

The bigeodesic Brownian plane, denoted $(\mathcal{P},d,\rho,\mu)$, is constructed from a gluing of continuum random trees along a real line (“spine”) with stochastic label functions. The process is as follows:

1. Underlying random tree and labels: Start with a Bessel(7) process $R=(R_t)_{t\geq0}$ initiated from zero. Its last hitting time at level $L>0$ is $S_L=\sup\{t\geq0:R_t=L\}$. Conditioned on $R$, consider two independent Poisson point processes $\mathcal{N}^1, \mathcal{N}^2$ on $$[0,\infty)\times\mathcal{C}(\mathbb{R}_+,\mathcal{W})$$ with time-dependent intensities $2\,\mathbf{1}_{[0,S_L]}(t)\,N_{R_t}(d\omega)$ and $2\,dt\,N_{R_t}(d\omega)$, where $N_x$ is the Brownian snake excursion measure from label $x$.
2. Gluing into the bigeodesic tree: The resulting real tree $\mathcal{T}$ is formed by gluing the continuum random trees rooted at each Poisson atom into the spine $[0,S_L]$, providing a continuous label function:

$$Z(u)= \begin{cases} R_t & \text{if } u=t\in[0,S_L] \ \omega_i(s) & \text{if }u \text{ is the point at time }s\text{ of excursion }(t_i,\omega_i) \end{cases}$$

1. Metric and measure definition: On $\mathcal{T}$, define a pseudo-metric:

$$d^\circ(u,v) = Z(u) + Z(v) - 2\min_{w\in[u,v]} Z(w),$$

where $[u,v]$ is the unique geodesic in the tree. The intrinsic metric $d$ is the largest pseudo-metric $\leq d^\circ$ satisfying the triangle inequality:

$$d(u, v) = \inf_{u_0 = u, ..., u_n = v} \sum_{i=1}^n d^\circ(u_{i-1}, u_i).$$

Quotienting by zero distance yields the measured metric space $(\mathcal{P},d,\rho,\mu)$, with $\rho = \pi(0)$ the root and $\mu$ the pushforward of Lebesgue measure along the spine and excursions.

1. Local Gromov–Hausdorff–Prokhorov–Uniform (GHPU) topology: The structure is considered up to isometry as a pointed, locally compact, complete length-measure-curve space $(X,d,\rho,\mu,\eta)$, with a continuous curve $\eta:\mathbb{R}\to X$ (the bigeodesic). Convergence is defined via truncations of balls of increasing radius around the root.

2. Main Limiting Properties

The bigeodesic Brownian plane is characterized as the deep tangent and scaling limit of classical Brownian surfaces.

1. Tangent-plane limit from the Brownian sphere: For the Brownian sphere $(\mathcal{S},D,\rho,\mu)$ (with minimal label $W^*<-b$), and its distinguished geodesic $\Gamma$, the rescaled neighborhoods around $\Gamma(b)$ converge:

$$\text{Law}\big(\mathcal{S}, \varepsilon^{-1}D, \Gamma(b), \varepsilon^{-4}\mu\big) \to \text{Law}(\mathcal{P})$$

in the local GHPU topology as $\varepsilon\to 0$. This describes $\mathcal{P}$ as the “tangent plane” in distribution at a typical geodesic point.

1. Local limit along Brownian plane’s infinite geodesic: For the classical Brownian plane $(\mathrm{BP}, D_\infty, \rho_\infty)$ with unique infinite geodesic $\gamma$, rerooting at $\gamma(b)$ and sending $b\to\infty$ yields:

$$\text{Law}(\mathrm{BP}, D_\infty, \gamma(b)) \to \text{Law}(\mathcal{P})$$

in the local GHPU topology.

A plausible implication is that these local convergence results situate the bigeodesic Brownian plane as a universal local object capturing the geometry near marked geodesic pairs on random metric 2-spheres and their planar limits (Mourichoux, 1 Oct 2024, Mourichoux, 11 Nov 2025).

3. Topology, Geodesic Structure, and Coding

1. Topology: Almost surely, $\mathcal{P}$ is connected, locally compact, complete, and homeomorphic to the Euclidean plane. Each metric ball $B(\rho,r)$ is embedded in a topological disk.
2. Bigeodesic structure: There exists a unique bi-infinite geodesic $\Gamma^\circ:\mathbb{R}\to\mathcal{P}$ — the image of the spine under the quotient — which divides $\mathcal{P}$ into two “half-planes”. Two canonical geodesic rays $\Gamma^+(t)=\Gamma^\circ(t)$ and $\Gamma^-(t)=\Gamma^\circ(-t)$ for $t\geq 0$ emanate from the root. No other infinite geodesic rays exist from $\rho$, with all geodesics from any point coalescing with one of these rays at finite distance.
3. Continuum random unicycle and infinite volume limit: The coding of $\mathcal{P}$ admits an explicit “unicycle” representation, replacing the finite excursion in the CRLU construction by a two-sided Bessel(3) process $X$ on $\mathbb{R}$, with two independent Poisson clouds $M, M'$ of Brownian-snake excursions. The quotient pseudo-metric is derived from the labels along the constructed tree-of-trees. The images under this quotient of hitting times of $X$ at $\pm t$ parametrize the bi-infinite geodesic, and the remainder of the space is divided into two Brownian “half-planes” (Mourichoux, 11 Nov 2025).

4. Invariance and Symmetry Properties

The bigeodesic Brownian plane possesses notable invariance and symmetry attributes:

• Scale invariance:

For any $\lambda>0$:

$$\text{Law}(\mathcal{P},d,\rho,\mu) = \text{Law}(\mathcal{P}, \lambda d, \rho, \lambda^4\mu)$$

• Infinite-geodesic shift invariance:

Rerooting the space at $\Gamma^\circ(s)$, for any $s\in\mathbb{R}$, does not change its law; the associated labeled tree rooted at $s$ is isomorphic in distribution.

• Geodesic reversal invariance:

The process $t \mapsto -\Gamma^\circ(-t)$ coincides in law with $\Gamma^\circ(t)$.

• Symmetry by cutting along the bigeodesic:

Slicing along $\Gamma^\circ$ yields two independent Brownian half-planes, which can be glued together in either order to recover the original space (Mourichoux, 11 Nov 2025).

5. Relations to Other Brownian Surfaces and Discrete Models

Model	Compactness	Geodesic Structure
Brownian map/sphere	Compact, Hausdorff dim. 4	Typical points: unique geodesic to root; points on geodesics are rare
Brownian plane	Non-compact limit (“generic”)	Unique infinite geodesic from root
Bigeodesic Brownian plane	Non-compact “non-generic”	Unique bi-infinite geodesic dividing space into two half-planes

• Tangent-cone perspectives:

The classical Brownian plane arises as the tangent-cone at a typical point of the Brownian sphere, whereas the bigeodesic Brownian plane arises as the tangent-cone at points on simple geodesics.

• Liouville quantum gravity (LQG) connections:

For $\gamma = \sqrt{8/3}$, certain LQG surfaces coincide with these Brownian spaces; analogous geodesic limits exist in the LQG framework (Mourichoux, 1 Oct 2024).

• Discrete bijections and scaling limits:

The Miermont bijection for quadrangulations and its continuum analog, the CRLU, allow explicit discrete-to-continuum constructions. The bigeodesic Brownian plane arises as a scaling limit of uniform quadrangulations with two marked vertices at prescribed graph distance, with the local limit taken around a point on the geodesic between them (Mourichoux, 11 Nov 2025).

6. Voronoï Cells, Perimeter Laws, and Metric Structure

• Voronoï cells in the Brownian sphere:

The decomposition of the bi-pointed Brownian sphere into Voronoï cells (regions closer to one marked point, given a delay) yields, under scaling, a structure locally equivalent to the “half-planes” defined by the bigeodesic in the bigeodesic Brownian plane. The joint law of cell volumes is Dirichlet$(\tfrac{1}{4},\tfrac{1}{4})$, and the process of boundary distances evolves as a Brownian bridge (Section 3.15, (Mourichoux, 11 Nov 2025)).

• Metric and geodesic properties:

For parametric points $\gamma(s), \gamma(t)$ along the spine, the induced distance is $|s-t|$ for $s, t \geq 0$ or $s, t \leq 0$, confirming the geodesic isometric structure along the spine.

• Perimeter and volume distribution:

The volume and perimeter laws of Voronoï cells at the scaling limit involve Beta and Bessel-type distributions.

A plausible implication is that the bigeodesic Brownian plane gives a canonical local model for phenomena in which a random geometry is “cut” along a typical geodesic, providing both metric and boundary information in limiting regimes.

7. Proof Techniques and Coupling Arguments

• Spinal decomposition:

The principal approach for constructing the bigeodesic Brownian plane is a spinal decomposition under the Brownian snake excursion measure, associating the geodesic spine with a Bessel(7) process and the attached subtrees with Poisson clouds.

• Tightness and coupling:

Tightness and coupling arguments, especially for balls around the root in truncated and limiting spaces, establish convergence in the local GHPU topology.

• Localization and continuity lemmas:

Through explicit localization lemmas and showing equivalence between finite excursions and Bessel process tails, one ensures control of local topology during approximation by finite Brownian spheres or finite unicycle codings (Mourichoux, 11 Nov 2025).

These construction and convergence techniques unite discrete combinatorial models with their continuum limits, reinforcing the significance of the bigeodesic Brownian plane in the landscape of random geometry.