Busemann Process: Asymptotic Geometry & Random Fields

• The Busemann process is defined as a family of boundary-indexed functions or cocycles derived from asymptotic limits along rays, capturing underlying geometric and random structures.
• It unifies various settings—from metric and Finsler geometry to directed percolation models and hyperbolic neural networks—with concrete applications in random growth and equilibrium analysis.
• Technical constructions ensure properties like cocycle additivity, ergodicity, and convergence to scaling limits, exemplified by the Stationary Horizon in KPZ models.

The term Busemann process denotes a family of Busemann functions or Busemann-type cocycles indexed by rays, boundary points, directions, or tilts. Its meaning is not uniform across the literature. In metric and differential geometry it refers to boundary-indexed limits of renormalized distance functions and, in some Finsler settings, to a smooth family of solutions of $F(\nabla b)=1$ and $\Delta b=h$ attached to rays at infinity. In directed first- and last-passage percolation and KPZ models it usually means a joint random field of Busemann functions across all directions, coupled on one probability space and used to encode semi-infinite geodesics, equilibrium measures, and interface statistics. More recent work extends the same boundary-indexed viewpoint to statistical depth on Hadamard manifolds and to hyperbolic neural-network layers (Shah et al., 2021, Fan et al., 2018, Janjigian et al., 11 Jul 2025, Jiang et al., 20 Apr 2026).

1. Variants of the term and its common structure

Across fields, the recurring structure is a family of functions obtained from asymptotic limits along rays or from cocycles that recover the underlying geometry or random environment. What changes is the index set—geodesic rays, visual boundary points, KPZ directions, or tilts in the super-differential of a limit shape—and the role played by the resulting family.

Setting	Index set	Characteristic object
Asymptotically harmonic Finsler manifolds	rays, lines, or $IM$	smooth functions $b_\gamma$; total Busemann function $B:IM\to A(M)$
Directed FPP, LPP, and KPZ models	directions $\alpha,\rho,\theta,\xi$ or tilts $h$	random cocycle field $\{B_\alpha(x,y)\}$, $\{B^\rho_{x,y}\}$, or $\{B^h(x,y)\}$
Hadamard statistics and hyperbolic learning	boundary points $\Delta b=h$0 or learned directions $\Delta b=h$1	horospherical depth, Busemann median, BMLR/BFC layers

In the Hammersley and lattice LPP literature, the term is explicit: the Busemann process is the joint object

$\Delta b=h$2

a random, direction-indexed cocycle field (Cator et al., 2010). In the planar corner growth model, the terminology shifts to a tilt-indexed Busemann process

$\Delta b=h$3

where tilts lie in a subset $\Delta b=h$4 of the super-differential of the shape function (Janjigian et al., 11 Jul 2025). In Brownian last-passage percolation, the global object is

$\Delta b=h$5

simultaneously for all initial points and directions (Seppäläinen et al., 2021).

A persistent source of ambiguity is that some papers work with unmistakably process-like families without formally adopting the name. Shah–Taha explicitly note that their Finsler paper does not use the term “Busemann process,” but that its constructions naturally yield a process-like structure: rays are organized by Busemann functions up to additive constants, and the total Busemann function

$\Delta b=h$6

assigns a differentiable potential to each direction in the unit tangent bundle (Shah et al., 2021). A similar situation appears in horospherical depth on Hadamard manifolds, where the paper does not define a Busemann process as such, but builds an entire depth theory from the family $\Delta b=h$7 (Jiang et al., 20 Apr 2026).

2. Geometric and analytic foundations

At the metric level, a Busemann function is the limit of renormalized distances along a geodesic ray. On a Hadamard manifold $\Delta b=h$8, for a ray $\Delta b=h$9,

$IM$0

and for a chosen basepoint $IM$1 and boundary point $IM$2, one writes $IM$3 for the Busemann function of the unique ray from $IM$4 to $IM$5. If two rays are asymptotic, their Busemann functions differ only by a constant. Horoballs and horospheres are then

$IM$6

and in a Hadamard manifold $IM$7 is $IM$8-Lipschitz, geodesically convex, $IM$9, and satisfies $b_\gamma$0 (Jiang et al., 20 Apr 2026).

In the general Finsler setting, for a forward complete noncompact Finsler manifold $b_\gamma$1 without conjugate points and a forward unit speed ray $b_\gamma$2, the associated Busemann function is

$b_\gamma$3

The approximants $b_\gamma$4 are monotonically decreasing, bounded below by $b_\gamma$5, and converge uniformly on compact sets. The asymmetric Lipschitz estimate

$b_\gamma$6

reflects the non-reversibility of the Finsler distance. In asymptotically harmonic Finsler manifolds, Busemann functions become smooth distance functions: $b_\gamma$7, $b_\gamma$8, their level sets are smooth horospheres, and asymptoticity of rays becomes equivalent to differing by an additive constant (Shah et al., 2021).

The analytic framework in the Finsler case depends on Shen’s Laplacian associated to a chosen smooth Finsler volume form,

$b_\gamma$9

and on the mean curvature of geodesic spheres and horospheres. An asymptotically harmonic Finsler manifold is defined by requiring the mean curvature of horospheres to be a real constant $B:IM\to A(M)$0. In the weak sense this becomes the distributional condition $B:IM\to A(M)$1 for every Busemann function, and the central regularity theorem states that in such a weak AHF-manifold every Busemann function is smooth (Shah et al., 2021).

These geometric constructions already contain the main ingredients of a process viewpoint: boundary- or direction-indexed potentials, horospheres as level sets, cocycle-type additivity along geodesics, and an ambient PDE or convexity structure that ties the entire family together.

3. Random cocycles in first- and last-passage models

In directed growth models, the Busemann process is a random field of directional cocycles built from passage-time differences to infinity. In Hammersley last-passage percolation, for $B:IM\to A(M)$2,

$B:IM\to A(M)$3

exists almost surely, and in lattice LPP with exponential weights,

$B:IM\to A(M)$4

exists almost surely as well. These fields satisfy cocycle and antisymmetry identities,

$B:IM\to A(M)$5

and inherit stationarity and ergodicity from the underlying homogeneous environment. Along the horizontal axis, the induced process $B:IM\to A(M)$6 is the unique spatially ergodic equilibrium for the associated interacting particle dynamics, with intensity $B:IM\to A(M)$7 in Hammersley and $B:IM\to A(M)$8 in the lattice model (Cator et al., 2010).

In the exactly solvable corner growth model with exponential weights, Fan–Seppäläinen construct a càdlàg process

$B:IM\to A(M)$9

simultaneously in all directions of growth. For each fixed $\alpha,\rho,\theta,\xi$0, the cocycle satisfies

$\alpha,\rho,\theta,\xi$1

recovers the environment through

$\alpha,\rho,\theta,\xi$2

is monotone in $\alpha,\rho,\theta,\xi$3, and has independent exponential increments along down-right paths. Across a single edge, the direction process admits an explicit marked point process representation: if $\alpha,\rho,\theta,\xi$4 has a point at $\alpha,\rho,\theta,\xi$5 and on $\alpha,\rho,\theta,\xi$6 is a Poisson point process with intensity $\alpha,\rho,\theta,\xi$7, with independent marks $\alpha,\rho,\theta,\xi$8, then

$\alpha,\rho,\theta,\xi$9

has the same law as $h$0 (Fan et al., 2018).

For general two-dimensional first-passage percolation, pointwise directional limits are often unavailable. Damron–Hanson therefore build a distributional framework from Busemann functions toward supporting hyperplanes $h$1, average the induced laws $h$2, and pass to subsequential weak limits $h$3. Under the limiting law one reconstructs a random Busemann-type function $h$4 with additivity, translation covariance, and a shape theorem

$h$5

for a random supporting functional $h$6. This distributional Busemann framework is sufficient to prove existence of sector-directed geodesics, coalescence, and nonexistence of infinite backward paths under minimal assumptions (Damron et al., 2012).

4. Strong existence, uniqueness, and exact joint laws

A major development in recent work is the upgrade from existence “in law” to strong, pathwise constructions. In the i.i.d. planar corner growth model, the tilt-indexed Busemann process is indexed by

$h$7

where $h$8 denotes forward measurable, shift-covariant, recovering cocycles with coalescing geodesics. The principal theorem states that on the canonical i.i.d. weight space there exists a process

$h$9

such that for each $\{B_\alpha(x,y)\}$0, $\{B_\alpha(x,y)\}$1 almost surely, each $\{B_\alpha(x,y)\}$2, $\{B_\alpha(x,y)\}$3, and monotonicity and left/right continuity in tilt hold. Any other process with these properties coincides with it almost surely. For a fixed tilt, strong uniqueness says that if $\{B_\alpha(x,y)\}$4 have the same mean tilt, then $\{B_\alpha(x,y)\}$5 almost surely (Janjigian et al., 11 Jul 2025).

In Brownian last-passage percolation, the global Busemann process is constructed simultaneously for all directions and all space-time points: $\{B_\alpha(x,y)\}$6 The process satisfies additivity, monotonicity in $\{B_\alpha(x,y)\}$7, continuity in space-time, and queueing relations

$\{B_\alpha(x,y)\}$8

and agrees with point-to-point Busemann limits for every fixed direction $\{B_\alpha(x,y)\}$9. This global object is the basis for constructing semi-infinite Busemann geodesics in every asymptotic direction on a single probability-one event (Seppäläinen et al., 2021).

Exactly solvable KPZ models exhibit additional process-level structure. Shen proves that in the corner growth model and inverse-gamma polymer, if a down-right path is partitioned into disjoint segments and each segment is assigned a different direction $\{B^\rho_{x,y}\}$0, then the corresponding collections of Busemann increments are mutually independent. Analogous theorems hold for Brownian LPP and the O’Connell–Yor polymer, and via scaling or coupling for the KPZ equation and the directed landscape (Shen, 2023).

A complementary exact description is provided by permutation invariance in exponential LPP. Bates, Emrah, Martin, Seppäläinen, and Sorensen show that the Busemann increments within a $\{B^\rho_{x,y}\}$1 grid, associated to $\{B^\rho_{x,y}\}$2 different directions, are equal in distribution to a particular collection of last-passage increments inside a $\{B^\rho_{x,y}\}$3 grid in a finite inhomogeneous environment. This yields an exact finite-dimensional sampling scheme for joint Busemann distributions over arbitrary finite edge sets, not just along a single horizontal line (Bates et al., 14 Jun 2025).

5. Geodesics, particles, interfaces, and scaling limits

One of the central uses of a Busemann process is that it encodes semi-infinite geodesics. In Brownian last-passage percolation, the global Busemann process yields, for every initial point and every $\{B^\rho_{x,y}\}$4, at least one $\{B^\rho_{x,y}\}$5-directed semi-infinite geodesic; for each fixed starting point and direction the geodesic is almost surely unique, all $\{B^\rho_{x,y}\}$6-directed geodesics coalesce, every semi-infinite geodesic has an asymptotic direction, and for fixed northeast and southwest directions there are almost surely no bi-infinite geodesics in those directions (Seppäläinen et al., 2021). In two-dimensional first-passage percolation, the distributional Busemann framework of Damron–Hanson similarly yields sector-directed geodesics, coalescing families, and evidence against bigeodesics without the curvature assumptions used in earlier work (Damron et al., 2012).

In interacting particle systems, the Busemann process acts as an equilibrium object. In TASEP and the Hammersley interacting particle process, the joint field across directions governs competition interfaces and the asymptotic speed of a second class particle. For a deterministic TASEP configuration with rarefaction fan $\{B^\rho_{x,y}\}$7, the second class particle speed has support

$\{B^\rho_{x,y}\}$8

and its law is expressed through suprema of random walks built from Busemann increments. In the Hammersley process, the asymptotic speed $\{B^\rho_{x,y}\}$9 has support $\{B^h(x,y)\}$0 and

$\{B^h(x,y)\}$1

where $\{B^h(x,y)\}$2 is an independent Poisson process. In both models, the Busemann process supplies the equilibrium laws that turn rarefaction geometry into explicit speed distributions (Cator et al., 2010).

McKeown’s directed first-passage percolation work extends this viewpoint to steep highways and competition-interface clustering. In planar directed FPP, the Busemann process exists under differentiability and strict convexity of the limit shape, and in several integrable strict-weak models its distribution can be computed explicitly. These explicit laws quantify semi-infinite geodesics passing through thin rectangles, show how branch points along a column grow logarithmically, and identify convoy phenomena for competition interface angles. In the exponential strict-weak model, for the convoy

$\{B^h(x,y)\}$3

one has

$\{B^h(x,y)\}$4

with explicit $\{B^h(x,y)\}$5. The same framework also produces multi-class invariant distributions for discrete-time TASEP with parallel updates (McKeown, 22 Oct 2025).

At the KPZ $\{B^h(x,y)\}$6 scale, the direction-indexed process itself has a nontrivial scaling limit. In exponential LPP, with

$\{B^h(x,y)\}$7

Busani studies the rescaled horizontal process

$\{B^h(x,y)\}$8

and proves convergence to a limit $\{B^h(x,y)\}$9, the Stationary Horizon. For each fixed $\Delta b=h$00, $\Delta b=h$01 is distributed as a two-sided Brownian motion with drift $\Delta b=h$02 and diffusivity $\Delta b=h$03, while on compact spatial windows the map $\Delta b=h$04 is a pure jump process with finitely many jumps on finite $\Delta b=h$05-intervals. The limit satisfies the scaling relation

$\Delta b=h$06

and is described as an ensemble of “sticky” lines of Brownian regularity. The paper further states that $\Delta b=h$07 is believed to be the universal scaling limit of Busemann processes in the KPZ universality class (Busani, 2021).

6. Extensions, applications, and open directions

Beyond random growth, Busemann processes appear wherever boundary-indexed limits or cocycles organize geometry. On a locally finite regular tree, Dumont studies the Busemann cocycle after applying a Poisson transform

$\Delta b=h$08

showing that

$\Delta b=h$09

Hence the norm is asymptotically linear with respect to the square root of the distance. In the rank-one $\Delta b=h$10-adic setting this realizes the Busemann cocycle in the Steinberg representation, providing a representation-theoretic incarnation of a Busemann process (Dumont, 2017).

On Hadamard manifolds, the boundary-indexed family $\Delta b=h$11 underlies a full statistical depth theory. The horospherical depth

$\Delta b=h$12

generalizes Tukey half-space depth by replacing linear functionals with Busemann functions and halfspaces with horospherical halfspaces. The associated Busemann median is the set of maximizers of this depth. The depth regions are nested and geodesically convex, a centerpoint of depth at least $\Delta b=h$13 exists, and hence the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique (Jiang et al., 20 Apr 2026).

In hyperbolic machine learning, Busemann functions have become explicit computational primitives. Hyperbolic Busemann Neural Networks introduce Busemann MLR

$\Delta b=h$14

and Busemann FC layers, which lift multinomial logistic regression and fully connected layers into hyperbolic space. BMLR admits a point-to-horosphere distance interpretation,

$\Delta b=h$15

uses compact parameters, is batch-efficient, and has a Euclidean limit. BFC generalizes FC and activation layers with comparable complexity. Experiments on image classification, genome sequence learning, node classification, and link prediction report improvements in effectiveness and efficiency over prior hyperbolic layers (Chen et al., 21 Feb 2026).

Several open directions recur across these literatures. The terminology itself remains nonstandard: some papers define a Busemann process formally, while others build process-like structures without naming them as such. In the Finsler setting, asymptoticity is not an equivalence relation in general and the AHF hypothesis is what restores equivalence via Busemann functions (Shah et al., 2021). In the corner growth model, the realized tilt set $\Delta b=h$16 satisfies

$\Delta b=h$17

and the possibility of gaps in $\Delta b=h$18 is left open (Janjigian et al., 11 Jul 2025). In KPZ scaling, the Stationary Horizon is proposed as a universal limit but identified rigorously only for exponential LPP so far (Busani, 2021). These unresolved points underscore a broader theme: the Busemann process is not a single object but a versatile organizing principle for asymptotic geometry, random growth, equilibrium particle systems, and boundary-based analysis.