Poisson Horoball Process in Hyperbolic Space

• Poisson horoball process is a stochastic model generating random horoballs in hyperbolic space that respects isometry invariance and induces detailed horosphere boundaries.
• The model employs a one-dimensional Poisson reduction to derive explicit variance bounds and non-standard central limit theorem error estimates for surface-area functionals.
• It unifies analytic and combinatorial approaches, blending hyperbolic geometry with propagation limits on product groups to aid in low-cost graphing and density benchmarking.

Poisson horoball process denotes a random family of horoballs, or equivalently of oriented horosphere boundaries, generated in a manner compatible with the ambient geometry. In the cited literature, the term does not designate a single universal object. One rigorous realization is a stationary Poisson process of horospheres in $d$-dimensional hyperbolic space $H^d$, where the signed-distance parametrization and the choice of convex side identify each horosphere with the boundary of a unique horoball (Kabluchko et al., 2023). A second realization is a Poisson point process of pointed horoballs on a product group $G\times G'$ equipped with a weighted $l_1$ metric, obtained as a propagation limit and used as a weak factor of i.i.d. construction in measured group theory (Khezeli, 10 Sep 2025). Related deterministic work on horoball packings in $\mathbb H^3$ does not define a Poisson process, but supplies explicit horoball geometry, admissibility constraints, and density benchmarks that are directly reusable in stochastic settings (Yahya et al., 2021).

1. Fundamental objects and terminological scope

A horosphere in $H^d$ is described as “a sphere of infinite radius” and, more formally, as a complete totally umbilic hypersurface of constant normal curvature $1$. A horoball is the convex domain bounded by a horosphere. In the Poincaré ball model, horospheres are realized as Euclidean spheres tangent to the boundary of the ball model. Because a horosphere becomes a horoball boundary once its convex side is specified, any oriented horosphere process naturally induces a horoball-boundary interpretation (Kabluchko et al., 2023).

The group-theoretic formulation uses a broader notion of horoball defined through horofunctions. For a locally finite graph $H$ with boundedly finite metric $d$, one considers shifted distance functions

$d_x(\cdot):=d(x,\cdot)-d(x,o),$

and defines the horoboundary $H^d$0 as the closure of $H^d$1 in the space of $H^d$2-Lipschitz functions vanishing at $H^d$3, under pointwise convergence. For $H^d$4 and $H^d$5,

$H^d$6

is a horoball with center $H^d$7 and delay $H^d$8, and $H^d$9 is a pointed horoball. The extended case $G\times G'$0 is also allowed formally (Khezeli, 10 Sep 2025).

A persistent source of ambiguity is that some papers treat horospheres as primary objects and study only their induced boundary content, whereas others construct random horoballs directly as limiting random sets. Another common ambiguity is geometric rather than probabilistic: deterministic horoball packing papers analyze admissible placements and density optimization, but not Poisson processes. The distinction is explicit in the available literature.

2. Stationary hyperbolic model as a Poisson horoball boundary process

In the hyperbolic stochastic-geometry model, the ambient space is $G\times G'$1, the $G\times G'$2-dimensional hyperbolic space. Let $G\times G'$3 denote the space of all horospheres in $G\times G'$4. The group of hyperbolic isometries acts transitively on $G\times G'$5, and there is an invariant measure on $G\times G'$6, unique up to multiplicative constant, denoted by $G\times G'$7. Fixing an origin $G\times G'$8, a horosphere is parameterized by

$G\times G'$9

where $l_1$0 is the signed distance from the horosphere to $l_1$1, $l_1$2 is the unit tangent direction at $l_1$3 of the geodesic that intersects the horosphere orthogonally, $l_1$4 if $l_1$5 lies on the convex side, and $l_1$6 points outside of the convex side. In these coordinates,

$l_1$7

The random family of horospheres is a Poisson point process $l_1$8 on $l_1$9 with intensity measure $\mathbb H^3$0. Stationarity means invariance under the full group of hyperbolic isometries, rather than Euclidean translation invariance. Since the parametrization explicitly records the convex side, the model can be read as an oriented horosphere process or, equivalently, as a Poisson horoball boundary process. What is studied, however, is not the Boolean union of horoballs, nor coverage, connectivity, or percolation, but a surface-content functional (Kabluchko et al., 2023).

For a hyperbolic ball $\mathbb H^3$1 of radius $\mathbb H^3$2 centered at $\mathbb H^3$3, the main observable is

$\mathbb H^3$4

the total $\mathbb H^3$5-dimensional hyperbolic surface area induced by the random horospheres inside the observation window. By rotational symmetry, if $\mathbb H^3$6 is a horosphere at signed distance $\mathbb H^3$7 from $\mathbb H^3$8, then $\mathbb H^3$9 is empty for $H^d$0, and for $H^d$1,

$H^d$2

This yields the one-dimensional Poisson representation

$H^d$3

where $H^d$4 is an inhomogeneous Poisson process on $H^d$5 with density $H^d$6, and

$H^d$7

This reduction is the main structural simplification of the model.

3. Quantitative fluctuation theory in high dimension

The principal theorem for the hyperbolic surface-content functional is a quantitative non-standard central limit theorem. The random variable

$H^d$8

converges not to a centered Gaussian of variance $H^d$9, but to a centered Gaussian of variance $1$0. The theorem gives explicit error bounds in both Kolmogorov and Wasserstein distance: for $1$1 and $1$2, the approximation error is bounded by a universal constant times $1$3 when $1$4, and by

$1$5

when $1$6. In particular, if $1$7 and $1$8, then the centered and normalized $1$9 is asymptotically Gaussian with variance $H$0; for $H$1, the bound becomes $H$2 for $H$3 and $H$4 for $H$5 (Kabluchko et al., 2023).

The mechanism behind the variance $H$6 is the decomposition

$H$7

where the two terms correspond to $H$8 and $H$9. These contributions are independent, and the variance splits evenly: $d$0 The asymptotic analysis shows that the negative part becomes Gaussian after normalization, while the positive part becomes negligible in the relevant sense. This produces a non-standard limit law despite the fact that the total functional is itself a Poisson sum.

The variance admits the explicit expression

$d$1

The analysis is organized through auxiliary integrals $d$2, $d$3, $d$4, and the normalized geometric integral

$d$5

Lower bounds on $d$6 are the key high-dimensional input: $d$7

Methodologically, the proof relies on hyperbolic isometry invariance, the reduction to a one-dimensional inhomogeneous Poisson process, the Mecke formula, and a fourth-cumulant bound for Poisson functionals. The paper explicitly does not develop stabilization, Wiener–Itô chaos expansions, or Malliavin–Stein in explicit form. It also emphasizes that horospheres correspond to the case $d$8 in a broader family of $d$9-geodesic hyperplanes: horospheres are intrinsically Euclidean, whereas the $d_x(\cdot):=d(x,\cdot)-d(x,o),$0 analogues have intrinsic hyperbolic geometry, and this difference is reflected in the limiting fluctuation behavior.

4. Propagation-limit Poisson horoball process on product groups

A distinct formulation arises for

$d_x(\cdot):=d(x,\cdot)-d(x,o),$1

where $d_x(\cdot):=d(x,\cdot)-d(x,o),$2 and $d_x(\cdot):=d(x,\cdot)-d(x,o),$3 are infinite finitely generated groups equipped with Cayley graphs and graph metrics $d_x(\cdot):=d(x,\cdot)-d(x,o),$4 and $d_x(\cdot):=d(x,\cdot)-d(x,o),$5. The product is endowed with the weighted $d_x(\cdot):=d(x,\cdot)-d(x,o),$6 metric

$d_x(\cdot):=d(x,\cdot)-d(x,o),$7

If $d_x(\cdot):=d(x,\cdot)-d(x,o),$8 and $d_x(\cdot):=d(x,\cdot)-d(x,o),$9 denote the metric compactifications, then

$H^d$00

and the horofunction associated to $H^d$01 is

$H^d$02

Accordingly, a product horoball has the form

$H^d$03

The relevant horoballs are classified into three types: type I when $H^d$04 and $H^d$05, type I' when $H^d$06 and $H^d$07, and type II when $H^d$08 and $H^d$09. The process space is $H^d$10, the Polish space of tuples $H^d$11, and its marked version $H^d$12, whose elements are $H^d$13 with a mark $H^d$14 (Khezeli, 10 Sep 2025).

The Poisson horoball process is not postulated axiomatically by an explicit intensity formula. It is obtained as a weak limit via the propagation method. One begins with a sequence of radii $H^d$15 and ratios $H^d$16, subject to the growth assumption

$H^d$17

For each $H^d$18, define a Bernoulli process of centers $H^d$19 with parameter

$H^d$20

Each selected center $H^d$21 is replaced by the $H^d$22-ball $H^d$23, called a diamond, together with delay

$H^d$24

This yields the point process of pointed diamonds

$H^d$25

Tightness follows from the fact that, for each fixed $H^d$26, the number of diamonds containing $H^d$27 is binomial with parameters $H^d$28. After subsequence extraction,

$H^d$29

The unmarked limit $H^d$30 is stated to be a Poisson point process on $H^d$31 with a suitable intensity measure. The limiting objects are pointed horoballs because large balls whose centers escape to the horoboundary converge, in the Fell topology, to horoballs satisfying $H^d$32.

A decisive geometric fact is that almost surely every pointed horoball in $H^d$33 is of type II. The proof excludes type I, type I', and extended horoballs by combining the growth assumption, counting arguments for diamonds containing the root, and a mass transport argument. Thus the propagation limit is supported on genuine type-II horoballs

$H^d$34

5. Markings, independence, and graph-theoretic use

The marked Poisson horoball process $H^d$35 is obtained by equipping each pointed horoball in $H^d$36 with an independent vertically replicated i.i.d. marking. Concretely, if

$H^d$37

then $H^d$38 receives an independent i.i.d. marking $H^d$39 of $H^d$40, and the induced marking on the horoball is

$H^d$41

Thus the mark depends only on the first coordinate. Different horoballs receive independent such markings. The process may be non-simple if the intensity measure has atoms, but repeated copies of the same pointed horoball carry different marks almost surely (Khezeli, 10 Sep 2025).

This vertically replicated structure is essential in the later graphing argument. For a marked horoball

$H^d$42

and a point $H^d$43, one chooses a neighbor $H^d$44 of $H^d$45 with

$H^d$46

The existence of such a neighbor follows from the geodesic lemma asserting that for every $H^d$47 and $H^d$48 there is an infinite path $H^d$49 starting at $H^d$50 such that

$H^d$51

Because the marking is vertically replicated, $H^d$52 depends only on $H^d$53, not on $H^d$54. This yields parallel motion along vertical fibers and a directed forest $H^d$55 on the union of marked horoballs, with exactly one outgoing edge from every vertex.

Connectivity is supplied by adding an invariant bond percolation of small intensity. One fixes a symmetric, positive, equivariant kernel

$H^d$56

and then inserts percolation with intensity $H^d$57. Inside a single horoball, parallel paths stay at bounded distance forever, and positivity of $H^d$58 ensures infinitely many possible connecting edges. Between distinct horoballs, the type-II geometry allows the construction of comparison paths $H^d$59 and $H^d$60 with uniformly bounded $H^d$61-distance, again yielding infinitely many opportunities for percolative connection. After resolving overlaps with a second independent i.i.d. marking and compressing the resulting graphing, the expected degree can be made arbitrarily close to $H^d$62, so the graphing cost is arbitrarily close to $H^d$63. This is the step that produces fixed price one for products satisfying the growth assumption.

The process therefore serves two functions simultaneously: it is a weak factor-of-i.i.d. limit object with explicit horoball geometry, and it is the combinatorial substrate for a low-cost connected graphing.

6. Deterministic horoball geometry and packing benchmarks

Deterministic horoball packing results in $H^d$64 provide a complementary geometric background. The study of simply truncated Coxeter orthoschemes with parallel faces does not involve Poisson point processes, Palm theory, or random fields. It analyzes ball and horoball packings generated by tilings of Schläfli type

$H^d$65

with horoballs centered at ideal vertices and, in the two-vertex case, possibly of different types. Here “different types” means that the horoballs are globally congruent in $H^d$66 but intersect the chosen fundamental domain in sectors of different volumes and hence have different local densities (Yahya et al., 2021).

Several formulas from this deterministic setting are directly reusable in stochastic horoball modeling. In the projective Lorentz model, with ideal center

$H^d$67

and a horosphere passing through $H^d$68, the horosphere equation is

$H^d$69

Because horospheres are intrinsically Euclidean, horospherical arc length for hyperbolic chord length $H^d$70 is

$H^d$71

and the volume of a horoball sector over a domain $H^d$72 on a horosphere is

$H^d$73

For a horoball packing in a fundamental domain $H^d$74, the local density is

$H^d$75

The optimization problem is constrained by non-overlap and by the condition that no horoball extend beyond the facet opposite its center. In the two-horoball case, the volume-exchange lemma states that if the tangency point moves by hyperbolic distance $H^d$76 along the connecting edge, then

$H^d$77

for $H^d$78, so the total sector volume strictly increases as $H^d$79. This pushes optima to boundary points of the admissible interval.

Within the family studied, the densest horoball packings are realized by the tilings

$H^d$80

with density

$H^d$81

The best one-horoball packing occurs for $H^d$82 with density $H^d$83. These values are deterministic benchmarks rather than probabilistic ones, but they supply explicit local admissibility conditions, sector-volume formulas, and cell-wise density calculations that can serve as input to a Poisson horoball model on a Coxeter-tiling background.

7. Conceptual synthesis and boundaries of the notion

The available literature supports two precise interpretations of a Poisson horoball process. In hyperbolic stochastic geometry, it is most naturally understood as an isometry-invariant Poisson process of oriented horospheres, hence as a Poisson horoball boundary process, with detailed fluctuation theory for the total induced surface area in growing hyperbolic balls (Kabluchko et al., 2023). In propagation-based constructions on product groups, it is a Poisson point process of pointed type-II horoballs arising as the weak limit of sparse Bernoulli seeds expanded to large metric balls, with independent vertically replicated markings attached to each horoball (Khezeli, 10 Sep 2025).

These interpretations are related but not interchangeable. The first is analytic and integral-geometric, centered on the observable

$H^d$84

The second is combinatorial and ergodic-theoretic, centered on weak factor-of-i.i.d. realizability, multiset-valued random configurations, and low-cost graphings. Neither paper studies the Boolean union of horoballs, coverage probabilities, overlap graphs, or percolation of horoballs as primary objects. Conversely, deterministic horoball packing papers supply exact local geometry but do not define random horoball fields (Yahya et al., 2021).

A precise use of the term therefore depends on context. In one context, the essential structure is hyperbolic isometry invariance and the measure

$H^d$85

on oriented horospheres. In the other, the essential structure is the propagation limit on $H^d$86, the horofunction representation

$H^d$87

and the fact that almost surely every limiting horoball is of type II. Taken together, these constructions show that “Poisson horoball process” is not a single canonical model but a family of rigorously defined objects organized around horoballs, horoboundaries, and Poissonian randomness in non-Euclidean geometry.