Busemann Process in Planar FPP

• The paper demonstrates that the Busemann process rigorously constructs asymptotic passage time differences and defines geodesic uniqueness and coalescence.
• It utilizes monotonicity and geometric arguments to establish a limiting directed geodesic graph that reveals one-ended forest topology.
• The approach removes restrictive assumptions, offering a universal framework for analyzing infinite geodesics in planar directed first passage percolation.

The Busemann process in planar directed first passage percolation provides a rigorous framework for analyzing asymptotic passage time differences and the geometric structure of geodesics in random environments on $\mathbb{Z}^2$. The process is key to understanding existence, uniqueness, coalescence, and the large-scale organization of infinite geodesics—questions that have been addressed under minimal assumptions by extending and unifying methods developed by Newman and successors. The process is constructed as a distributional limit using monotonicity, geometric arguments, and ergodicity, and it induces a limiting directed geodesic graph that encodes the asymptotic geometry and topological complexity of the percolation model (Damron et al., 2012, Auffinger et al., 2013).

1. Construction of the Busemann Process

The Busemann function is defined in planar directed FPP as the limit of passage time differences to boundary points receding in a deterministic direction. Given a connected, infinite, locally finite subgraph $V \subseteq \mathbb{Z}^2$ whose complement is also infinite and connected, and a sequence of boundary vertices $\{v_n\}$ tending to infinity in a fixed direction, the Busemann function is

$$B(x, y) := \lim_{n \to \infty} \left[ \tau(x, v_n) - \tau(y, v_n) \right],$$

where $\tau(x, v_n)$ is the minimal passage time from $x$ to $v_n$. Existence of this limit for all vertex pairs $(x, y)$ is proved under the sole assumption that finite geodesics exist for each pair (and, when necessary, uniqueness of passage times). The convergence is monotone in $n$, a property deduced from geometric constructions that exploit planar topology, crossing arguments, and Jordan curve techniques (Auffinger et al., 2013).

This process yields a field of Busemann differences over all vertex pairs and induces a sequence of directed graphs $G_n$ (for geodesics to $v_n$) that converge in the product topology of edge-indicator variables to a limiting directed geodesic graph $\mathfrak{G}$.

2. Limiting Geodesic Structure and Coalescence

The limiting directed graph $\mathfrak{G}$ encapsulates the global behavior of geodesics:

• Out-degree: For domains like the half-plane, almost surely every vertex has out-degree 1 in $\mathfrak{G}$, reflecting that from each site there is a unique infinite forward path.
• Acyclicity: The undirected version of $\mathfrak{G}$ contains no cycles; the graph is a (random) forest.
• Coalescence: Any two infinite geodesics eventually merge—starting from different vertices, the forward paths coincide after finitely many steps (having finite symmetric difference).
• Backward clusters: For every $x$, the set $\{y : \text{directed path from } y \text{ to } x\}$ is finite; that is, no infinite backward paths exist.
• One-endedness: There is, almost surely, a unique “end” in $\mathfrak{G}$: all infinite geodesics coalesce to form a single topological end (Auffinger et al., 2013).

These results, obtained for general ergodic edge-weight distributions and without restrictive shape or moment conditions, solidify the geometric theory of directed FPP.

3. Methodological Innovations and Minimal Assumptions

The methodology is founded on two principles:

• Monotonicity/stability arguments: Monotonicity of passage-time differences (Busemann-type variables) allows almost-sure convergence of edge-inclusion indicators. The “paths crossing” trick and careful geometric analysis are used to control convergence and monotonicity properties.
• Geometric constructions: Planar topology (e.g., Jordan curves) ensures that comparison of geodesics can be reduced to local (finite) configurations, which in turn allows extending arguments to infinite-volume limits.

A significant aspect is the removal of restrictive moment or curvature assumptions that featured in earlier work (such as Newman's). For instance, by using edge modification arguments and the upward finite energy property, the proofs circumvent the need for finite first moments or strict convexity of the limit shape (Auffinger et al., 2013).

4. Busemann Function Properties and Mathematical Formulation

Key mathematical forms include:

• Busemann function: For any $x, y \in V$,

$$B(x, y) = \lim_{n \to \infty} [\tau(x, v_n) - \tau(y, v_n)],$$

which always exists and produces a cocycle: $B(x, y) + B(y, z) = B(x, z)$.

• Geodesic graph indicator: For each directed edge $(x, y)$,

$$\eta_n((x, y)) = \begin{cases} 1 & \text{if } \{x, y\} \text{ lies on a geodesic to } v_n \text{ and } \tau(x, v_n) \geq \tau(y, v_n), \ 0 & \text{otherwise} \end{cases}$$

The limiting graph $\mathfrak{G}$ is defined by $\eta = \lim_n \eta_n$.

• Geodesic uniqueness and coalescence: In the half-plane, the (unique) infinite geodesic from any $x$ almost surely merges with that from any other $y$. For any $x$, the backward cluster is finite, implying the absence of bigeodesics (doubly-infinite geodesics) (Auffinger et al., 2013).

5. Directional Conditions and Existence of Geodesics

The framework replaces global curvature assumptions with a purely directional condition. When the directional condition (which is strictly weaker than global curvature) holds, existence of infinite geodesics in a deterministic direction is established. Otherwise, geodesics directed within sectors still exist, and coalescence phenomena persist. The analysis characterizes the transition from sector-directed geodesics to sector-wide coalescence, showing that even minimal directional regularity suffices for robust asymptotic structure.

6. Implications and Applications

These structural results have several implications:

• Geodesic geometry: The existence and properties of Busemann functions provide a canonical tool for constructing and analyzing infinite geodesics, making it possible to rigorously define their direction, uniqueness, and merging behavior.
• Random graph topology: The limiting geodesic graph describes the asymptotic topology as a random forest with one end, capturing key conjectures concerning absence of bigeodesics and global tree-like structure.
• Minimal requirement universality: The arguments demonstrate that such geometric organization is a generic feature of planar directed FPP, not an artifact of special assumptions, thus justifying the naturalness of these objects for broader application.
• Direction-sensitive phenomena: The methodology delineates the boundary between scenarios where geodesics are deterministic-direction-directed and sector-directed, clarifying the geometric consequences of subtle underlying regularity properties (Auffinger et al., 2013).

7. Connections to Broader Disordered Media Theory

The organization and convergence of geodesic graphs, as controlled by the Busemann process, reflect a general principle in disordered systems: local rules and structural monotonicity can create global order. The Busemann function acts as a microscopic gradient encoding macroscopic shape and directionality. These findings bridge first passage percolation with ergodic random geometry, coalescence theory, and related interacting stochastic processes, and provide a foundational perspective for further development in stochastic growth and random metric space theory.

Table: Key Mathematical Structures

Object	Definition	Role in Busemann Process
Busemann Function $B(x, y)$	$\lim_{n \to \infty} [\tau(x, v_n) - \tau(y, v_n)]$	Encodes asymptotic passage gradients
Limiting Geodesic Graph $\mathfrak{G}$	Edge indicator limits: $\eta((x, y)) = \lim_n \eta_n((x, y))$	Captures asymptotic geodesic structure
Out-degree Property	Each $x$ has out-degree $1$ in half-plane	Ensures uniqueness of infinite paths
Coalescence Property	Any two infinite geodesics eventually merge	Topological one-endedness
Backward Cluster	$\{y :$ path from $y$ to $x$ in $\mathfrak{G}\}$ is finite	No backward- or doubly-infinite geodesics

All results are established without moment, shape, or curvature assumptions except minimal requirements for finite geodesic existence and uniqueness. This abstention from restrictive assumptions demonstrates the universality and robustness of the Busemann process as an organizing principle in planar directed first passage percolation (Damron et al., 2012, Auffinger et al., 2013).