Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes

Seeing Through Hyperbolic Space: V isibility for λ -Geodesic Hyperplanes Zakhar Kabluchko a V anessa Mattutat b Christoph Thäle c Abstract W e study visibility from a fixed point in the presence of a P oisson process of λ – geodesic hyperplanes in a d -dimensional hyperbolic space. The family of λ –geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fun- damental visibility properties are invariant with respect to the parameter λ ∈ [0 , 1] . Namely , there is a critical intensity γ crit > 0 such that the visible region is unbounded with positive probability for γ < γ crit and almost surely bounded for γ > γ crit . F or d = 2 we establish almost sure boundedness also at criticality . The value for γ crit is explicit and does not depend on λ . In the bounded phase, we show that the mean vis- ible volume is identical with the known formula for λ = 0 . The key integral-geometric step is an explicit computation showing that the measure of λ -geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, indepen- dent of λ . Keywords. Crofton formula, hyperbolic space, integral geometry , λ -geodesic hyper- plane, percolation, phase transition, P oisson process, universality , visibility MSC 2020. 51M10, 52A22, 53C65, 60D05 1 Introduction Random tessellations generated by P oisson hyperplane processes in Euclidean space form a classical and well-studied topic in stochastic geometry . In R d , a stationary Pois- son hyperplane process gives rise to a mosaic of convex cells, and many aspects of its geometry have been analyzed in detail, including the cell containing the origin, see [16] and Chapter 10 of [24]. When passing from Euclidean space to hyperbolic space H d of constant negative cur- vature − 1 , the situation becomes richer , since hyperbolic geometry admits several nat- ural analogues of Euclidean hyperplanes. Besides totally geodesic hyperplanes, which are isometric copies of H d − 1 , one encounters equidistant hypersurfaces, which are de- fined as the set of points at fixed distance from a given geodesic hyperplane. A further limiting case is provided by horospheres, which arise as limits of equidistant hypersur- faces whose base hyperplane recedes towards the ideal boundary of H d . These three families can be unified by the notion of λ –geodesic hyperplanes as introduced in [26]. F or λ ∈ [0 , 1] , a λ –geodesic hyperplane is a complete totally umbilical hypersurface in H d with normal curvature λ . Thus, λ = 0 corresponds to totally geodesic hyperplanes, which have vanishing intrinsic curvature, 0 < λ < 1 yields equidistant hypersurfaces, whose curvature depends continuously on λ , and λ = 1 gives horospheres. a University of Münster , Germany . Email: zakhar .kabluchko@uni-muenster .de b Ruhr University Bochum, Germany . Email: vanessa.mattutat@rub.de c Ruhr University Bochum, Germany . Email: christoph.thaele@rub.de 1 Every λ –geodesic hyperplane separates H d into two connected components. As we are not interested in all such hypersurfaces, we introduce the space Hyp o λ of all λ – geodesic hyperplanes for which the distinguished point o ∈ H d , which we refer to as the origin, does not lie on the convex side. Let ν λ denote the natural isometry-invariant measure on the space of all λ -geodesic hyperplanes, as introduced in Section 2.2 below , and ν o λ its restriction to Hyp o λ . F or an intensity parameter γ > 0 , we define η γ ,λ to be a P oisson point process on Hyp o λ with intensity measure γ ν o λ . Each hyperplane in η γ ,λ acts as a random obstacle to visibility from the reference point o . F or a point x ∈ H d , we say that x is visible from o if the geodesic segment [ o, x ] connecting o with x does not intersect any H ∈ η γ ,λ . This leads to the random closed set Z γ ,λ,d := { x ∈ H d : [ o, x ] ∩ H = ∅ for all H ∈ η γ ,λ } , (1.1) which we call the visibility region of o . Equivalently , for λ = 0 , Z γ , 0 ,d is the connected component of o in the complement of the union of all hyperplanes in η γ , 0 . Simulations of Poisson processes of λ -geodesic hyperplanes for different values of λ together with the corresponding visibility regions are shown in Figure 1. The case λ = 0 , corresponding to totally geodesic hyperplanes, has been studied in detail in previous works [3, 7, 8, 12, 23]. These papers show that the visibility region Z γ , 0 ,d may be unbounded, and that its behaviour undergoes a sharp phase transition as the intensity γ varies. More precisely , there exists the critical value γ crit := √ π ( d − 1) 2 Γ( d − 1 2 ) Γ( d 2 ) (1.2) such that • if γ < γ crit , then Z γ , 0 ,d is unbounded with strictly positive probability , • if γ ≥ γ crit , then Z γ , 0 ,d is almost surely bounded. The critical case γ = γ crit was treated in [12, 23] for d = 2 and in [7] for general dimen- sions. Thus, for small intensities, there remains a positive chance of seeing arbitrar- ily far from the origin, whereas for large intensities the random hypersurfaces almost surely form a finite “cocoon” around o . In the latter regime, the expected hyperbolic volume of the bounded visibility region has been computed in [8, Theorem 7.1], yielding an explicit expression in terms of the dimension d and the intensity parameter γ . In the present work we investigate the same questions for general λ -geodesic hyper- planes. One might expect the behaviour of the visibility region to depend sensitively on the parameter λ , since varying λ interpolates between different geometric objects. Sur- prisingly , our results show that the situation is completely rigid with respect to λ . The critical intensity at which the transition from unbounded to bounded visibility occurs is identical for all λ ∈ [0 , 1] , and in the bounded regime the expected volume of the visibil- ity region Z γ ,λ,d agrees exactly with the expression known from the case λ = 0 . In other words, neither the existence of infinite sightliness nor the size of the bounded visibility region depends in any way on the geometric nature of the obstructing hypersurfaces. Our main findings are summarized in the following result. Theorem 1.1. Fix d ≥ 2 , γ > 0 and 0 ≤ λ ≤ 1 . Consider the visibility region Z γ ,λ,d of o in a P oisson process of λ -geodesic hyperplanes in H d with intensity measure γ ν o λ . (a) Let γ crit be as in (1.2) . If γ < γ crit , then Z γ ,λ,d is unbounded with strictly positive probability , whereas if γ > γ crit , then Z γ ,λ,d is almost surely bounded. Moreover , if d = 2 , then Z γ crit ,λ, 2 is almost surely bounded. 2 (a) λ = 0 , γ = 2 < γ crit (b) λ = 0 . 5 , γ = 2 < γ crit (c) λ = 1 , γ = 2 < γ crit (d) λ = 0 , γ = π = γ crit (e) λ = 0 . 5 , γ = π = γ crit (f) λ = 1 , γ = π = γ crit (g) λ = 0 , γ = 7 > γ crit (h) λ = 0 . 5 , γ = 7 > γ crit (i) λ = 1 , γ = 7 > γ crit Figure 1: Simulation of a P oisson process of λ -geodesic hyperplanes in Hyp o λ with the corresponding visibility set in red in the Poincaré ball model. Note that for d = 2 , γ crit = π . 3 (b) If γ > γ crit then E v ol d ( Z γ ,λ,d ) = E v ol d ( Z γ , 0 ,d ) = π d − 1 2 Γ  d + 1 2  Γ  γ ∗ d − d +1 2  Γ  γ ∗ d + d +1 2  (1.3) with γ ∗ d := γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) . On the other hand, E v ol d ( Z γ ,λ,d ) = + ∞ for all γ ≤ γ crit . The proofs of the results in (a) and (b) rely on two complementary ingredients. F or the existence of a phase transition, we adapt a covering criterion due to Hoffmann– Jørgensen [14]. This method parallels the approach in [12] used in the case λ = 0 , and it shows that the critical threshold does not depend on the geometric nature of the hyperplanes, i.e., on λ . The concrete computations rely on a subtle cancellation, which eliminates the λ -dependence from the relevant density terms. The computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure ν o λ of λ –geodesic hyperplanes that intersect a fixed geodesic segment of length h > 0 . F or λ = 0 , this quantity can be determined using the classical Crofton formula in hyperbolic space. In fact, if x h is a point with hyperbolic distance h from o we have Z Hyp o 0 1 { H ∩ [ o, x h ]  = ∅ } ν o 0 (d H ) = Z Hyp o 0 χ ( H ∩ [ o, x h ]) ν o 0 (d H ) = Γ( d 2 ) 2 √ π Γ( d +1 2 ) h, (1.4) where χ denotes the Euler characteristic, see [8, Section 7]. However , the Crofton argument breaks down for λ > 0 , since a λ –geodesic hyperplane may intersect a geodesic segment either once or twice. This implies that for all such λ the indicator function cannot be replaced by the Euler characteristic and it seems that no direct integral-geometric relation is available. W e overcome this by introducing an explicit parametrization of λ –geodesic hyperplanes and carrying out the computation directly . The resulting expression turns out to be independent of λ and again proportional to the length h . This invariance is a surprising integral-geometric fact of independent in- terest and is ultimately responsible for the λ –independence of the expected volume as described in Theorem 1.1. Our findings are placed within the broader context of stochastic geometry in hyper- bolic space, a field that has seen significant recent activity . This area encompasses a rich variety of random structures adapted to negative curvature. A selection of recent results includes the asymptotic geometry of random polytopes [4, 9], the investigation of the Boolean model [3, 8, 15, 27] and the study of hyperbolic random graphs, which connects geometric probability with network science [5, 10, 20]. The investigation of P oisson point processes and their associated V oronoi tessellations remained a central theme, with foundational works including [17]. A particular focus has recently devel- oped around ideal Poisson–V oronoi tessellations and their applications, where the gen- erating points reside on the boundary at infinity [1, 2, 6, 11]. Our work complements this diverse landscape by focusing on the geometry induced by P oisson processes of λ -geodesic hyperplanes, uncovering a surprising universality in visibility properties. The paper is organized as follows. Section 2 collects the necessary background on hyperbolic geometry to make the paper self-contained. In Section 3, we establish the visibility phase transition, proving Theorem 1.1(a). Section 4 is dedicated to the core integral-geometric mechanism: we prove that the Crofton-type identity (1.4) holds universally for all λ ∈ [0 , 1] . Finally , we apply this result in Section 5 to derive the mean visible volume formula in Theorem 1.1(b). 4 2 Background material 2.1 Hyperbolic space and the Poincaré ball model F or d ∈ N with d ≥ 2 , let H d denote the d –dimensional hyperbolic space of constant curvature − 1 . One convenient realisation of H d is given by the Poincaré ball model, in which the underlying space is the open unit ball B d := { y ∈ R d : ∥ y ∥ < 1 } , equipped with the Riemannian metric 4 (1 −∥ y ∥ 2 ) 2 d y 2 . In this model, the geodesics are precisely the intersections with B d of Euclidean circles and lines that meet the boundary ∂ B d = S d − 1 orthogonally . The hyperbolic distance between the origin and a point y ∈ B d is given explicitly by d P (0 , y ) = 2 artanh( ∥ y ∥ ) . More generally , for x, y ∈ B d the hyperbolic distance satisfies cosh  d P ( x, y )  = 1 + 2 ∥ x − y ∥ 2 (1 − ∥ x ∥ 2 )(1 − ∥ y ∥ 2 ) . A totally geodesic hyperplane in H d is a ( d − 1) -dimensional submanifold such that any geodesic contained in the submanifold is also a geodesic of the ambient space. In the P oincaré ball model, these hyperplanes are represented by the intersection of B d with Euclidean ( d − 1) -spheres or Euclidean hyperplanes that intersect the boundary S d − 1 orthogonally . Another important class of hypersurfaces consists of horospheres. Intuitively , these can be viewed as the limits of hyperbolic spheres as their radii tend to infinity . In the P oincaré ball model, a horosphere is realized as a Euclidean sphere contained in the closure of B d that is tangent to the boundary S d − 1 at exactly one point (excluding the point of tangency itself). The P oincaré ball model for hyperbolic space will be our main tool for the geometric computations appearing later in the paper . 2.2 λ -geodesic hyperplanes Let 0 ≤ λ ≤ 1 . F ollowing [26], a λ –geodesic hyperplane is a complete totally umbilical hypersurface in H d with normal curvature λ . In the Poincaré ball model, λ -geodesic hyperplanes are of the form S ∩ B d , where S is a ( d − 1) -dimensional sphere in R d intersecting S d − 1 at angle θ with cos( θ ) = λ . In particular , λ = 0 corresponds to the case of totally geodesic hyperbolic hyperplanes and λ = 1 to the case of horospheres. Let Hyp λ denote the space of λ -geodesic hyperplanes. F or a fixed origin o ∈ H d , an element H ∈ Hyp λ can be parametrised by a pair ( s, u ) ∈ R × S d − 1 , where s ∈ R is the signed distance from the origin o to H and u is the unit vector in the tangent space of H d at o along the geodesic through o intersecting H orthogonally , pointing outside of the convex side. In the P oincaré ball model, if o is the centre of B d , then u ∈ S d − 1 is the direction which is orthogonal to H . The sign of s is chosen in such a way that s < 0 if o lies on the convex side of H . The parametrised version of H is denoted by H h ( s, u ) . As argued in [26], there exists an isometry-invariant measure ν λ on Hyp λ , which is unique up to normalization. F ollowing [18], we choose the normalization in such a way that Z Hyp λ f ( H ) ν λ (d H ) = Z R Z S d − 1 f ( H h ( s, u ))(cosh( s ) + λ sinh( s )) d − 1 σ d − 1 (d u ) d s, (2.1) for all measurable functions f : Hyp λ → R ≥ 0 , where σ d − 1 denotes the normalised sur- face measure on S d − 1 and d s refers to the integration with respect to the Lebesgue 5 measure on R . Note that in contrast to [18], we choose the sign of s differently so that the invariant measure is of a slightly different form here. As we are not interested in all λ -geodesic hyperplanes, we further introduce the space Hyp o λ of all λ -geodesic hyperplanes for which o does not lie on their convex side, i.e., for which s > 0 in the previously mentioned parametrization. In this paper , Hyp o λ will be equipped with the measure ν o λ defined as the restriction of ν λ to Hyp o λ , i.e., ν o λ ( · ) = Z ∞ 0 Z S d − 1 1 { H h ( s, u ) ∈ · } (cosh( s ) + λ sinh( s )) d − 1 σ d − 1 (d u ) d s. 2.3 A Poisson process on the space of λ -geodesic hyperplanes F or γ > 0 let η γ ,λ be a P oisson process on Hyp o λ with intensity measure µ γ ,λ = γ ν o λ , see [24, Chapter 3] for the definition of P oisson processes on general state spaces. W e want to represent this process in the P oincaré ball model. T o pass to this model, we set o to be the centre of the ball B d and r = tanh  s 2  for s ≥ 0 . In other words, if a point has hyperbolic distance s to o , then its Euclidean distance to o is r . Accordingly , we denote a λ -geodesic hyperplane H h ( s, u ) as defined in the previous subsection by H ( r, u ) in this model. This notation refers to the same geometric object, but the first coordinate is now the Euclidean distance r ∈ [0 , 1) given by r = tanh( s/ 2) . Then, we use that for x ∈ R , sinh(2 x ) = 2 sinh( x ) cosh( x ) and cosh(2 x ) = 2 cosh 2 ( x ) − 1 . T ogether with the relations cosh(artanh( x )) = 1 √ 1 − x 2 and sinh(artanh( x )) = x √ 1 − x 2 for x ∈ (0 , 1) , we get cosh(2artanh( r )) = 2 cosh 2 (artanh( r )) − 1 = 2 1 − r 2 − 1 = 1 + r 2 1 − r 2 and sinh(2artanh( r )) = 2 sinh(artanh( r )) cosh(artanh( r )) = 2 r 1 − r 2 . for r ∈ (0 , 1) . Since d(2artanh( r )) = 2 d r 1 − r 2 , it follows that the intensity measure µ γ ,λ can be written as µ γ ,λ ( · ) = γ Z 1 0 Z S d − 1 1 { H ( r , u ) ∈ · }  1 + r 2 1 − r 2 + λ · 2 r 1 − r 2  d − 1 · 2 1 − r 2 σ d − 1 (d u )d r = γ Z 1 0 Z S d − 1 1 { H ( r , u ) ∈ · } 2(1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d σ d − 1 (d u )d r . (2.2) 3 Visibility to infinity In the following we study the question whether , with positive probability , there exists an infinite geodesic ray emanating from o , which does not intersect any of the λ -geodesic hyperplanes of η γ ,λ . Note that, looking from o in the direction of some λ -geodesic hyperplane, this λ -geodesic hyperplane covers a part of S d − 1 . Thus, the question raised above is equivalent to the question whether the shadows of the λ -geodesic hyperplanes of η γ ,λ on S d − 1 , the boundary of the Poincaré ball B d , cover the whole sphere. In what follows, we choose o to be the centre of B d . 6 1 R α θ r Figure 2: W e consider a λ -geodesic hyperplane of Euclidean distance r to o realized by the intersection of B 2 with the black sphere of radius R , where R depends on r and λ = cos( θ ) . This λ -geodesic hyperplane covers a part of S 1 illustrated in green. In order to describe this green cap we aim to determine φ ( r ) = cos( α ) . T o this end, we use the red and the blue triangle. The red triangle provides sin( α ) = R R + r . Applying the law of cosine to the blue triangle leads to 2 R cos( θ ) = R 2 + 1 − ( R + r ) 2 . 3.1 Covering with spherical caps W e start with analysing which part of S d − 1 is covered by a fixed λ -geodesic hyperplane in the P oincaré ball model. Recall at first that every λ -geodesic hyperplane H ( r, u ) in the P oincaré model is of the form S ∩ B d , where S is a ( d − 1) -dimensional sphere intersecting S d − 1 at angle θ = arccos( λ ) , u ∈ S d − 1 is the unit vector intersecting S orthogonally and r is the Euclidean distance from o to S . Thus, H ( r, u ) covers a closed spherical cap. The corresponding open spherical cap, which we aim to analyze, is of the form S ◦ ( u, φ ( r )) := { x ∈ S d − 1 : ⟨ x, u ⟩ > φ ( r ) } , where φ : (0 , 1) → (0 , 1) is a suitable function depending only on r and λ . W e start by making φ explicit and refer to Figure 2 for a geometric illustration in dimension 2 . Let R denote the radius of S . Then, φ ( r ) = cos  arcsin  R R + r  , compare with the red triangle in Figure 2. Hence, it remains to express R in terms of r and θ . Using the law of cosine for a triangle that has 0 , the centre of S and an arbitrary intersection point of S and B d as vertices leads to λ = cos( θ ) = R 2 +1 − ( R + r ) 2 2 R , see the blue triangle in Figure 2. Thus, R = 1 − r 2 2( λ + r ) . Altogether , we conclude that φ ( r ) = cos arcsin 1 − r 2 2( λ + r ) 1 − r 2 2( λ + r ) + r !! = cos  arcsin  1 − r 2 1 + r 2 + 2 rλ  = r 1 −  1 − r 2 1 + r 2 + 2 rλ  2 = 2 p r ( λ + r )(1 + λr ) 1 + 2 λr + r 2 . (3.1) In the following lemma we derive the asymptotic behaviour of φ ( r ) as r ↑ 1 . 7 Lemma 3.1. Let φ : (0 , 1) → (0 , 1) be defined as in (3.1) . Then, as r ↑ 1 , 1 − φ ( r ) = (1 − r ) 2 2(1 + λ ) 2 + (1 − r ) 3 2(1 + λ ) 2 + o  (1 − r ) 3  . Proof. Let f : [0 , 1] → [0 , 1] and g : [0 , 1] → [0 , 1] be given by f ( r ) := 1 − r 2 1 + r 2 + 2 rλ , g ( r ) := 1 − p 1 − r 2 , and set h := g ◦ f = 1 − φ . W rite r = 1 − ε with ε ↓ 0 . Then f (1 − ε ) = 2 ε − ε 2 2(1 + λ ) − (2 + 2 λ ) ε + ε 2 = ε 1 + λ + ε 2 2(1 + λ ) + O ( ε 3 ) . Further , using the T aylor expansion of √ 1 − x at x = 0 and substituting x = y 2 , we obtain g ( y ) = 1 − p 1 − y 2 = y 2 2 + O ( y 4 ) , y ↓ 0 . Combining the last two displays yields 1 − φ (1 − ε ) = h (1 − ε ) = g  f (1 − ε )  = 1 2 f (1 − ε ) 2 + O  f (1 − ε ) 4  = 1 2  ε 1 + λ + ε 2 2(1 + λ ) + O ( ε 3 )  2 + O ( ε 4 ) = ε 2 2(1 + λ ) 2 + ε 3 2(1 + λ ) 2 + o ( ε 3 ) . Since ε = 1 − r , this proves the claim. 3.2 Criterion for visibility F or u ∈ S d − 1 and h ≥ 0 let S ◦ ( u, h ) = { x ∈ S d − 1 : ⟨ x, u ⟩ > h } be the open spherical cap centred at u with height h and denote by S ( u, h ) its closure. The following theorem is taken from [12, Theorem 4.1]. It is a reformulation of a result of Hoffmann-Jørgensen, see [14, F ormulas (3.10), (5.3) and (5.5)]. Proposition 3.2. Let u 1 , u 2 , . . . be independent and uniformly distributed on S d − 1 . Let the sequence h 1 , h 2 , . . . of real numbers in (0 , 1) be such that lim n →∞ nσ d − 1 ( S ◦ ( u n , h n )) = a ∈ [0 , ∞ ] . (a) If a > 1 , then P [lim sup n →∞ S ◦ ( u n , h n ) = S d − 1 ] = 1 , i.e., with probability one, each point of the sphere is covered infinitely often by the caps. (b) If a < 1 , P [lim sup n →∞ S ◦ ( u n , h n ) = S d − 1 ] = 0 and P  ∞ [ n =1 S ( u n , h n )  = S d − 1  > 0 . In par- ticular , the probability that there exist points on the sphere that are not covered by the closed caps is positive. Recall that we are interested in the behaviour of S ◦ ( u n , φ ( r n )) , where, by (2.2), u 1 , u 2 , . . . are uniformly distributed on S d − 1 and 0 < r 1 < r 2 < . . . form an inhomo- geneous P oisson point process on (0 , 1) with intensity function f ( r ) := 2 γ (1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d , 0 < r < 1 . (3.2) 8 W e note that R 1 0 f ( r ) d r = + ∞ . In what follows, we use the notation f ( x ) ∼ g ( x ) as x → x 0 for two functions f and g and some x 0 ∈ R if lim x → x 0 f ( x ) g ( x ) = 1 . Moreover , for a sequence of random variables ( X n ) n ≥ 1 and a positive deterministic sequence ( a n ) n ≥ 1 , we write X n = O ( a n ) almost surely if , with probability one, there exist a constant C > 0 and an integer N ≥ 1 such that | X n | ≤ C a n for all n ≥ N . Lemma 3.3. Almost surely , it holds that 1 − r n ∼  γ ( d − 1) n  1 / ( d − 1) (1 + λ ) as n → ∞ . Moreover , in the case d = 2 we have almost surely 1 − r n = γ (1 + λ ) n + O r log log( n ) n 3 ! , (3.3) as n → ∞ . Proof. The first part of the following proof is analogous to the proof of [12, Lemma 4.2] for λ = 0 . Let P 1 < P 2 < . . . be arrivals of a homogeneous Poisson point process on (0 , ∞ ) with intensity 1 . W e use the distributional representation τ ( P n ) = r n , where τ : (0 , ∞ ) → (0 , 1) is monotone increasing with Z τ ( y ) 0 2 γ (1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d d r = y for all y ∈ (0 , ∞ ) . By L ’Hospitals rule we have lim z ↑ 1 R z 0 2 γ (1+2 λr + r 2 ) d − 1 (1 − r 2 ) d d r γ (1+ λ ) d − 1 ( d − 1) (1 − z ) 1 − d = lim z ↑ 1 2 γ (1+2 λz + z 2 ) d − 1 (1 − z 2 ) d γ (1 + λ ) d − 1 (1 − z ) − d = lim z ↑ 1 2(1 + 2 λz + z 2 ) d − 1 (1 + z ) d (1 + λ ) d − 1 = 1 . Thus, Z z 0 2 γ (1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d d r ∼ γ (1 + λ ) d − 1 ( d − 1) (1 − z ) 1 − d as z ↑ 1 . Since τ ( y ) → 1 as y → ∞ we have γ (1 + λ ) d − 1 ( d − 1) (1 − τ ( y )) 1 − d ∼ y as y → ∞ . Since P n ∼ n almost surely as n → ∞ and τ ( P n ) = r n , it follows that almost surely 1 − r n = 1 − τ ( P n ) ∼  γ ( d − 1) P n  1 / ( d − 1) (1 + λ ) ∼  γ ( d − 1) n  1 / ( d − 1) (1 + λ ) , which proves the first claim. Assume now that d = 2 and set F ( z ) := Z z 0 2 γ (1 + 2 λr + r 2 ) (1 − r 2 ) 2 d r , z ∈ (0 , 1) . A direct computation shows that F ( z ) = 2 γ z (1 + λz ) 1 − z 2 , z ∈ (0 , 1) . (3.4) 9 Indeed, differentiating the right-hand side yields the integrand and F (0) = 0 . W e can now argue as before, but in view of the explicit form of F ( z ) in (3.4) the error terms can be controlled more precisely . Namely , by definition of τ , we have F ( τ ( y )) = y for all y > 0 . W rite ε := 1 − z and expand (3.4) at z = 1 to obtain, as ε ↓ 0 , F (1 − ε ) = γ (1 + λ ) ε − γ 2 (1 + 3 λ ) + O ( ε ) . (3.5) Since τ ( y ) ↑ 1 as y → ∞ , setting ε ( y ) := 1 − τ ( y ) and inserting z = τ ( y ) into (3.5) yields y = γ (1 + λ ) ε ( y ) − γ 2 (1 + 3 λ ) + O ( ε ( y )) , y → ∞ . Inverting this relation gives 1 − τ ( y ) = γ (1 + λ ) y − γ 2 2 (1 + λ )(1 + 3 λ ) y 2 + o  1 y 2  , y → ∞ . (3.6) Moreover , by the law of iterated logarithm it holds P n = n + O ( p n log log( n )) almost surely as n → ∞ . Thus, using T aylor expansion of the function g ( x ) = 1 x , we get almost surely , 1 P n = 1 n + O p n log log( n ) n 2 ! = 1 n + O r log log( n ) n 3 ! . Finally , since r n = τ ( P n ) , substituting y = P n into (3.6) yields almost surely 1 − r n = 1 − τ ( P n ) = γ (1 + λ ) P n − γ 2 2 (1 + λ )(1 + 3 λ ) P 2 n + o  1 P 2 n  = γ (1 + λ ) n + O r log log( n ) n 3 ! , which proves (3.3). W e are now prepared to derive the following result, which in particular covers part (a) of Theorem 1.1 stated in the introduction. Theorem 3.4. Let u 1 , u 2 , . . . be independent and uniformly distributed on S d − 1 , let 0 < r 1 < r 2 < . . . be such that they form an inhomogeneous Poisson point process on (0 , 1) with intensity function f , where f is defined as in (3.2) , and assume that the sequences ( u n ) n ≥ 1 and ( r n ) n ≥ 1 are independent. Let γ crit be the constant in (1.2) . a) If γ > γ crit , then P [lim sup n →∞ S ◦ ( u n , φ ( r n )) = S d − 1 ] = 1 . Thus, the visibility set Z γ ,λ,d in (1.1) is bounded almost surely . b) If γ < γ crit , then P [lim sup n →∞ S ◦ ( u n , φ ( r n )) = S d − 1 ] = 0 and P  ∞ [ n =1 S ( u n , φ ( r n ))  = S d − 1  > 0 . In particular , the probability that the visibility set Z γ ,λ,d is unbounded is strictly positive. Proof. Let u ∈ S d − 1 and 0 ≤ h ≤ 1 . Then, σ d − 1 ( S ◦ ( u, h )) = Γ( d 2 ) √ π Γ( d − 1 2 ) Z 1 h (1 − s 2 ) d − 3 2 d s 10 according to [19, Equation (2.7)]. It follows that, σ d − 1 ( S ◦ ( u, h )) ∼ c d (1 − h ) d − 1 2 , h ↑ 1 , for c d := Γ( d 2 ) √ π Γ( d − 1 2 ) · 2 d − 1 2 d − 1 = 2 d − 1 2 Γ( d 2 ) √ π ( d − 1)Γ( d − 1 2 ) . Since almost surely r n → 1 as n → ∞ , combining Lemma 3.1 and Lemma 3.3 leads to 1 − φ ( r n ) ∼ (1 − r n ) 2 2(1 + λ ) 2 ∼ 1 2  γ ( d − 1) n  2 d − 1 (3.7) as n → ∞ . It follows that almost surely σ d − 1 ( S ◦ ( u n , φ ( r n ))) ∼ c d γ 2 ( d − 1) / 2 ( d − 1) n = γ Γ( d 2 ) √ π ( d − 1) 2 Γ( d − 1 2 ) n as n → ∞ . Applying Proposition 3.2 completes the proof with γ crit as in (1.2). The translation to the visibility set follows from the construction described in Section 3.1. Remark 1. It is natural to ask whether the visibility event ’ there exists a geodesic ray from o avoiding all λ –geodesic hyperplanes of η γ ,λ ’ is monotone in λ in a suitable coupling. On an abstract level, if two locally finite intensity measures on the same state space satisfy ν 1 ≤ ν 2 , then one can construct Poisson point processes Π 1 and Π 2 with intensity measures ν 1 and ν 2 , respectively , on a common probability space such that Π 1 ⊆ Π 2 almost surely . In our setting, this idea can be applied to the induced Poisson process on S d − 1 × (0 , 1) that generates the random family of shadows {S ( u n , φ ( r n )) } n ≥ 1 . Indeed, using (3.1) and (3.2) one verifies that the intensity of the P oisson point process ( φ ( r n )) n ≥ 1 on (0 , 1) is given by g ( s ; λ ) = γ s (1 − s 2 ) d +1 2 p s 2 + λ 2 (1 − s 2 ) , 0 < s < 1 . F or fixed s , this function is decreasing in λ , which means that there exists a coupling of the shadow processes on S d − 1 such that, for all 0 ≤ λ 1 < λ 2 ≤ 1 , every shadow present in the process with parameter λ 2 is also present in the process with parameter λ 1 . However , this does not yield a coupling of the underlying λ -hyperplane processes { η γ ,λ } λ ∈ [0 , 1] on the state spaces Hyp o λ . Another natural attempt to construct a coupling of the λ -hyperplane processes is to use that, for λ ∈ (0 , 1) , a λ –geodesic hyperplane is an equidistant hypersurface to a totally geodesic hyperplane. However , passing to an equidistant may change on which side of the hypersurface the origin lies. Hence a naive attempt to couple across λ by mapping each H ∈ Hyp o 0 to one of its equidistants may leave the state space Hyp o λ , and any subsequent correction (for instance by discarding such hypersurfaces) would alter the intensity in a λ –dependent way . 3.3 The case γ = γ crit A natural question is what happens at the critical value γ = γ crit . In continuum per- colation models, the behaviour at criticality is typically the most delicate part of the phase transition and has been the focus of substantial recent work. In particular , for the isometry-invariant Poisson hyperplane tessellations in H d it was recently shown that at the critical intensity there are no unbounded cells, thus settling the critical case in all dimensions, see [7, Theorem 1.1]. This naturally raises the analogous question in 11 our setting: whether , for λ ∈ [0 , 1] , the visibility set Z γ ,λ,d is almost surely bounded at γ = γ crit . Our argument based on Proposition 3.2 does not apply at the boundary value a = 1 , and the strategy of [7, Section 3] relies on working with an isometry-invariant in- tensity measure. Since we restrict to Hyp o λ , the measure µ γ ,λ is not isometry-invariant, and the critical-case arguments from the isometry-invariant model do not transfer di- rectly . F or this reason, we do not address the critical case in full generality here. Nev- ertheless, using a criterion from [25], we can show that in dimension d = 2 the visibility region Z γ ,λ, 2 is almost surely bounded for every λ ∈ [0 , 1] even at γ = γ crit , proving the last claim in Theorem 1.1 (a). Proposition 3.5. W e assume the same set-up as in Theorem 3.4. F or d = 2 and γ = γ crit it holds P [lim sup n →∞ S ◦ ( u n , φ ( r n )) = S 1 ] = 1 . Thus, the visibility set Z γ crit ,λ, 2 in (1.1) is almost surely bounded. Proof. As before, let 0 < r 1 < r 2 < . . . be such that they form an inhomogeneous P oisson point process on (0 , 1) with intensity function f defined in (3.2). Then, in the P oincaré disc model a λ -geodesic hyperplane H ( r n , u ) of distance r n to o in direction u ∈ S 1 covers an arc of length 2 arccos( φ ( r n )) for n ∈ N (see Figure 2). Now , by [25, Equation (1)], Proposition 3.5 follows once we have shown that ∞ X n =1 1 n 2 e ℓ 1 + ... + ℓ n = ∞ , where 2 π ℓ i is the arc length covered by r i , i.e. ℓ i = 1 π arccos( φ ( r i )) for i ∈ N . Combining (3.7), γ crit = π and the refined d = 2 asymptotics from Lemma 3.3 we can sharpen the estimate for the arc lengths ℓ i . Indeed, for d = 2 and γ = γ crit = π , Lemma 3.3 yields almost surely 1 − r i = π (1 + λ ) i + O r log log( i ) i 3 ! , i → ∞ . T ogether with Lemma 3.1 this implies almost surely δ i := 1 − φ ( r i ) = (1 − r i ) 2 2(1 + λ ) 2 + O  (1 − r i ) 3  = π 2 2 i 2 + O r log log( i ) i 5 ! , i → ∞ . Using the T aylor expansions arccos(1 − x ) = √ 2 x + O ( x 3 / 2 ) as x ↓ 0 and √ 1 + x = 1 + x 2 + O ( x 2 ) as x ↓ 0 , we obtain almost surely arccos( φ ( r i )) = arccos(1 − δ i ) = p 2 δ i + O ( δ 3 / 2 i ) = π i + O r log log( i ) i 3 ! , i → ∞ . Consequently , ℓ i = 1 π arccos( φ ( r i )) = 1 i + O r log log( i ) i 3 ! , i → ∞ , almost surely . Hence, as P ∞ k =1 q log log( k ) k 3 < ∞ , this implies that there exists c ∈ R such that n X k =1 ℓ k ≥ n X k =1 1 k + c ≥ ln( n ) + c, where we used P n k =1 1 k ≥ ln( n ) . Thus, ∞ X n =1 1 n 2 e ℓ 1 + ... + ℓ n ≥ ∞ X n =1 1 n 2 e ln( n )+ c = ∞ X n =1 e c n = ∞ , which completes the proof . 12 4 Measure of λ -geodesic hyperplanes hitting a segment 4.1 Integral representation If we wish to compute the expected volume of the set of points in H d that are visible from o in the presence of a P oisson process of λ -geodesic hyperplanes in Section 5, we are naturally led to a purely geometric problem: What is the measure ν o λ of λ -geodesic hyperplanes intersecting a geodesic segment of a given length starting in o ? When λ = 0 , each totally geodesic hyperplane can intersect a geodesic segment at most once, and the corresponding measure is a linear function of the segment length, see [8]. However , for every λ > 0 , a λ -geodesic hyperplane may intersect a geodesic segment in 0 , 1 , or 2 points. In particular , a transition from an indicator function to the Euler characteristic as in (1.4) is no more possible. Showing that the measure remains a linear function of the segment length in this setting is a nontrivial problem, and resolving it is the aim of this section. T o tackle this, we first need to characterize exactly which λ -geodesic hyperplanes H ( r , u ) intersect a reference geodesic segment. It suffices to consider a segment of hyperbolic length h starting at the origin and aligned with the first coordinate vector e 1 := (1 , 0 , . . . , 0) ∈ R d . In the P oincaré ball model, this corresponds to the Euclidean segment connecting the origin to the point tanh( h/ 2) e 1 . The following lemma estab- lishes the precise analytic conditions on the parameters r and u that are necessary and sufficient for such an intersection to occur . Lemma 4.1. Let h > 0 , 0 ≤ r < 1 , u ∈ S d − 1 and u 1 := ⟨ u, e 1 ⟩ . Then, it holds that H ( r , u ) ∩  0 , tanh( h 2 ) e 1   = ∅ if and only if one of the following cases occurs: 1. r = 0 , 2. λ = 0 , r ∈ (0 , tanh  h 2  ] and u 1 ≥ tanh 2 ( h 2 ) r + r tanh ( h 2 ) ( r 2 +1) , 3. λ ∈ (0 , 1] , r ∈ (0 , r c ) and u 1 ≥ 2 √ ( r 2 λ + r )( λ + r ) r 2 +2 λr +1 , 4. λ ∈ (0 , 1] , r ∈  r c , tanh  h 2  and u 1 ≥ tanh 2 ( h 2 ) ( λ + r )+ r 2 λ + r tanh ( h 2 ) ( r 2 +2 λr +1) , where r c := tanh 2 ( h 2 ) − 1 2 λ + r ( 1 − tanh 2 ( h 2 )) 2 4 λ 2 + tanh 2  h 2  . Proof. Note at first that in the Poincaré ball model for H d , H ( r, u ) corresponds to the intersection of B d with a ( d − 1) -dimensional sphere of radius R with R = 1 − r 2 2( λ + r ) and centre ( r + R ) u , see Section 3.1 and Figure 2. Thus, H ( r , u ) ∩  0 , tanh( h 2 ) e 1   = ∅ if there exists x ∈  0 , tanh( h 2 )  such that R 2 = ∥ ( r + R ) u − xe 1 ∥ 2 = ( r + R ) 2 − 2( r + R ) xu 1 + x 2 , where u 1 = ⟨ u, e 1 ⟩ . Hence, H ( r , u ) ∩  0 , tanh( h 2 ) e 1   = ∅ if and only if the equation p ( x ) := x 2 − 2( r + R ) xu 1 + ( r + R ) 2 − R 2 = 0 has a solution in  0 , tanh( h 2 )  . It holds that p ( x ) = 0 if and only if x 1 , 2 = ( r + R ) u 1 ± q ( r + R ) 2 u 2 1 − ( r + R ) 2 + R 2 . If r = 0 , x = 0 is a solution of p ( x ) = 0 . Thus, H (0 , u ) ∩  0 , tanh( h 2 ) e 1   = ∅ . 13 F or r > 0 note at first that if real solutions exist, it holds x 1 , 2 > 0 if u 1 > 0 and x 1 , 2 < 0 if u 1 < 0 . Thus, p ( x ) = 0 has a solution in  0 , tanh( h 2 )  if and only if u 1 > 0 and ( r + R ) 2 u 2 1 − ( r + R ) 2 + R 2 ≥ 0 (4.1) ( r + R ) u 1 − q ( r + R ) 2 u 2 1 − ( r + R ) 2 + R 2 ≤ tanh  h 2  (4.2) are fulfilled. Condition (4.1) is equivalent to ( r + R ) 2 u 2 1 ≥ r 2 + 2 Rr , ⇐ ⇒ u 1 ≥ √ r 2 + 2 Rr r + R =: f 1 ( r ) Condition (4.2) is fulfilled if either ( r + R ) u 1 − tanh  h 2  ≤ 0 ⇐ ⇒ u 1 ≤ tanh  h 2  r + R =: f 2 ( r ) or u 1 ≥ tanh ( h 2 ) r + R and  ( r + R ) u 1 − tanh  h 2  2 ≤ ( r + R ) 2 u 2 1 − r 2 − 2 Rr ⇐ ⇒ − 2( r + R ) u 1 tanh  h 2  + tanh 2  h 2  ≤ − r 2 − 2 Rr ⇐ ⇒ u 1 ≥ tanh 2  h 2  + r 2 + 2 Rr 2( r + R ) tanh  h 2  =: f 3 ( r ) . Since 2 ab ≤ a 2 + b 2 for a, b ∈ R , we have f 1 ( r ) = 2 tanh  h 2  √ r 2 + 2 Rr 2 tanh  h 2  ( r + R ) ≤ tanh 2  h 2  + r 2 + 2 Rr 2( r + R ) tanh  h 2  = f 3 ( r ) . Altogether , in case that r 2 + 2 Rr < tanh 2  h 2  , it holds that f 1 ( r ) ≤ f 3 ( r ) ≤ 2 tanh 2  h 2  2( r + R ) tanh  h 2  = f 2 ( r ) . Thus, in this case, (4.2) is automatically fulfilled and H ( r, u ) ∩  0 , tanh( h 2 ) e 1   = ∅ if u 1 ≥ f 1 ( r ) . In the case that r 2 + 2 Rr ≥ tanh 2  h 2  , we find f 2 ( r ) ≤ √ r 2 + 2 Rr r + R = f 1 ( r ) ≤ f 3 ( r ) . Hence, in this case (4.1) and (4.2) are fulfilled, i.e., H ( r, u ) ∩  0 , tanh( h 2 ) e 1   = ∅ , if u 1 ≥ f 3 ( r ) . Inserting R = 1 − r 2 2( λ + r ) leads to r 2 + 2 Rr = r 2 + 2 r (1 − r 2 ) 2( λ + r ) = r 2 λ + r λ + r , r + R = r + 1 − r 2 2( λ + r ) = r 2 + 2 λr + 1 2( λ + r ) , f 1 ( r ) = r r 2 λ + r λ + r · 2( λ + r ) r 2 + 2 λr + 1 = 2 p ( r 2 λ + r )( λ + r ) r 2 + 2 λr + 1 , f 3 ( r ) = tanh 2  h 2  ( λ + r ) + r 2 λ + r ( λ + r ) · ( λ + r ) tanh  h 2  ( r 2 + 2 λr + 1) = tanh 2  h 2  ( λ + r ) + r 2 λ + r tanh  h 2  ( r 2 + 2 λr + 1) . 14 Then, for λ ∈ (0 , 1] , the condition r 2 + 2 Rr < tanh  h 2  can be rewritten as follows: r 2 λ + r λ + r < tanh 2  h 2  ⇐ ⇒ r 2 λ + r < tanh 2  h 2  ( λ + r ) ⇐ ⇒ r 2 λ + r  1 − tanh 2  h 2  − λ tanh 2  h 2  < 0 ⇐ ⇒ r 2 + 1 − tanh 2  h 2  λ r − tanh 2  h 2  < 0 . Note that the equation r 2 + 1 − tanh 2 ( h 2 ) λ r − tanh 2  h 2  = 0 has solutions r 1 , 2 = tanh 2  h 2  − 1 2 λ ± s  1 − tanh 2  h 2  2 4 λ 2 + tanh 2  h 2  . Then, r 1 < 0 and, since √ a 2 + b 2 ≤ a + b for a, b ≥ 0 , it holds that r 2 = tanh 2  h 2  − 1 2 λ + s  1 − tanh 2  h 2  2 4 λ 2 + tanh 2  h 2  ≤ tanh 2  h 2  − 1 2 λ + 1 − tanh 2  h 2  2 λ + tanh  h 2  = tanh  h 2  . Finally , with the fact that H ( r, u ) ∩  0 , tanh( h 2 ) e 1   = ∅ is only possible for r ∈ [0 , tanh  h 2  ] , the lemma follows by using the equations above for λ ∈ (0 , 1] to rewrite r ∈ [0 , tanh  h 2  ] with r 2 λ + r λ + r < tanh 2  h 2  as r ∈ [0 , r 2 ] and, similarly , r ∈ [0 , tanh  h 2  ] with r 2 λ + r λ + r ≥ tanh 2  h 2  as r ∈ [ r 2 , tanh  h 2  ] . F or λ = 0 we always have r 2 + 2 Rr = 1 ≥ tanh  h 2  . Thus, in this case H ( r , u ) ∩  0 , tanh( h 2 ) e 1   = ∅ if r ∈ [0 , tanh  h 2  ] and u 1 ≥ f 3 ( r ) . F or a closed set A ⊂ H d let [ A ] λ denote the collection of λ -geodesic hyperplanes having non-empty intersection with A and o on its non-convex side, i.e., [ A ] λ := { H ∈ Hyp o λ : H ∩ A  = ∅} . (4.3) T o determine the measure of [ A ] λ , where A is a geodesic segment of given length, we need a formula for the spherical Lebesgue measure of spherical caps. As before, we denote by S ( u, h ) = { x ∈ S d − 1 : ⟨ u, x ⟩ ≥ h } the closed spherical cap centred at u ∈ S d − 1 with height 0 < h < 1 . Then, σ d − 1 ( S ( u, h )) = Γ( d 2 ) √ π Γ( d − 1 2 ) Z 1 h (1 − s 2 ) d − 3 2 d s, (4.4) see [19, Equation (2.7)]. Further , we recall the notation r c = tanh 2  h 2  − 1 2 λ + s  1 − tanh 2  h 2  2 4 λ 2 + tanh 2  h 2  . The first main result of this subsection is the following integral representation for the intersection measure. Corollary 4.2. Let h > 0 and ℓ ( h ) be an arbitrary geodesic segment of length h starting at o . 15 (a) If λ = 0 then µ γ , 0 ([ ℓ ( h )] 0 ) = γ Z tanh ( h 2 ) 0 σ d − 1 S e 1 , tanh 2  h 2  r + r tanh  h 2  ( r 2 + 1) !! 2(1 + r 2 ) d − 1 (1 − r 2 ) d d r . (b) If 0 < λ ≤ 1 then µ γ ,λ ([ ℓ ( h )] λ ) = γ Z r c 0 σ d − 1 S e 1 , 2 p ( r 2 λ + r )( λ + r ) r 2 + 2 λr + 1 !! 2(1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d d r + γ Z tanh ( h 2 ) r c σ d − 1 S e 1 , tanh 2  h 2  ( λ + r ) + r 2 λ + r tanh  h 2  ( r 2 + 2 λr + 1) !! 2(1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d d r . Proof. F or 0 ≤ λ ≤ 1 and by (2.2) we have that µ γ ,λ ([ ℓ ( h )] λ ) is the same as γ Z 1 0 Z S d − 1 1 n H ( r , u ) ∩ h 0 , tanh  h 2  e 1 i  = ∅ o 2(1 + 2 λr + r 2 ) d − 1 (1 − r 2 ) d σ d − 1 (d u )d r . The result follows from this and Lemma 4.1. 4.2 Evaluation of the integrals for λ = 0 and λ = 1 W e were not able to analytically evaluate the integrals describing µ γ ,λ ([ ℓ ( h )] λ ) in Corol- lary 4.2 for general 0 ≤ λ ≤ 1 . However , for the two special cases λ = 0 and λ = 1 such an evaluation is possible as we shall demonstrate in this subsection. W e start with the case λ = 0 . Lemma 4.3. Suppose λ = 0 . Then µ γ , 0 ([ ℓ ( h )] 0 ) = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h, h ≥ 0 . Proof. Recall from Corollary 4.2 (a) that for λ = 0 we have µ γ , 0 ([ ℓ ( h )] 0 ) = γ Z tanh ( h 2 ) 0 σ d − 1 S e 1 , tanh 2  h 2  r + r tanh  h 2  ( r 2 + 1) !! 2(1 + r 2 ) d − 1 (1 − r 2 ) d d r . W e apply the substitution r = tanh x 2 and simplify . This yields µ γ , 0 ([ ℓ ( h )] 0 ) = γ Z h 0 σ d − 1  S  e 1 , tanh x tanh h  cosh d − 1 x d x. Now , take the derivative with respect to h and use the integral representation (4.4) for σ d − 1 ( S ( u, h )) . This gives d d h µ γ , 0 ([ ℓ ( h )] 0 ) = γ Γ( d 2 ) √ π Γ( d − 1 2 ) Z h 0  1 − tanh 2 x tanh 2 h  d − 3 2 tanh x sinh 2 h cosh d − 1 x d x. Next, we applying the substitution u = tanh 2 x tanh 2 h and then use the Euler integral [22, Equation (15.6.1)]: d d h µ γ , 0 ( F ℓ ( h ) ) = γ Γ( d 2 ) √ π Γ( d − 1 2 ) tanh 2 h 2 sinh 2 h Z 1 0 (1 − u ) d − 3 2 (1 − u tanh 2 h ) − d +1 2 d u = γ Γ( d 2 ) √ π Γ( d − 1 2 ) tanh 2 h 2 sinh 2 h B  1 , d − 1 2  2 F 1  d + 1 2 , 1; d + 1 2 ; tanh 2 h  16 with the beta and the Gauß hypergeometric function. W e have B (1 , d − 1 2 ) = 2 d − 1 and since the first and the third argument of the hypergeometric function are the same, 2 F 1 ( a, b ; a ; z ) = (1 − z ) − b by [22, Equation (15.4.6)], and this term simplifies to 1 / (1 − tanh 2 h ) . As a result, d d h µ γ , 0 ([ ℓ ( h )] 0 ) = γ Γ( d 2 ) √ π Γ( d − 1 2 ) tanh 2 h 2 sinh 2 h · 2 d − 1 · 1 1 − tanh 2 h = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) , independently of h , implying that µ γ , 0 ([ ℓ ( h )] 0 ) must be an affine-linear function in h . Since µ γ , 0 ([ ℓ (0)] 0 ) = 0 , it follows that µ γ , 0 ([ ℓ ( h )] 0 ) = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h for all h ≥ 0 . Next, we deal with the case λ = 1 corresponding to horospheres. Lemma 4.4. Suppose λ = 1 . Then µ γ , 1 ([ ℓ ( h )] 1 ) = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h, h ≥ 0 . Proof. W e have µ γ , 1 ([ ℓ ( h )] 1 ) = I 1 ( h ) + I 2 ( h ) with I 1 ( h ) := γ Z t 0 σ d − 1  S  e 1 , 2 √ r r + 1  W ( r ) d r , I 2 ( h ) := γ Z t t 2 σ d − 1  S  e 1 , t 2 + r t (1 + r )  W ( r ) d r , where we put t := tanh  h 2  and W ( r ) := 2 (1+ r ) 2( d − 1) (1 − r 2 ) d . Next, we set z ( r , t ) := t 2 + r t (1+ r ) , B ( r, t ) := σ d − 1 ( S ( e 1 , z ( r, t ))) W ( r ) and A ( r ) := σ d − 1  S  e 1 , 2 √ r r +1  W ( r ) . Then I 1 ( h ) and I 2 ( h ) can be written as a function of t as I 1 ( t ) = γ Z t 2 0 A ( r ) d r and I 2 ( t ) = γ Z t t 2 B ( r, t ) d r. Note that A ( t 2 ) = B ( t 2 , t ) , z ( t, t ) = 1 and so B ( t, t ) = 0 . W e can now differentiate with respect to t according to the Leibniz rule: d d t ( I 1 ( t ) + I 2 ( t )) = γ Z t t 2 ∂ t B ( r, t ) d r. By definition of B ( r, t ) , the chain rule and (4.4) we get ∂ t B ( r, t ) = − Γ( d 2 ) √ π Γ( d − 1 2 ) (1 − z ( r, t ) 2 ) d − 3 2 ∂ t z ( r , t ) W ( r ) . From ∂ t z ( r , t ) = t 2 − r t 2 (1+ r ) and the definitions of z ( r, t ) and W ( r ) it follows after simplifica- tion that ∂ t B ( r, t ) = − Γ( d 2 ) √ π Γ( d − 1 2 ) 2(1 − t 2 ) d − 3 2 t d − 1 ( t 2 − r 2 ) d − 3 2 ( t 2 − r ) (1 − r ) d . Thus, d d t ( I 1 ( t ) + I 2 ( t )) = − γ Γ( d 2 ) √ π Γ( d − 1 2 ) 2(1 − t 2 ) d − 3 2 ( d − 1) t d − 1 Z t t 2 H ′ ( r ) d r 17 if we put H ( r ) := ( t 2 − r 2 ) d − 1 2 (1 − r ) d − 1 . But Z t t 2 H ′ ( r ) d r = H ( t ) − H ( t 2 ) = 0 − t d − 1 (1 − t 2 ) d − 1 2 and hence d d t ( I 1 ( t ) + I 2 ( t )) = γ Γ( d 2 ) √ π Γ( d − 1 2 ) 2(1 − t 2 ) d − 3 2 ( d − 1) t d − 1 t d − 1 (1 − t 2 ) d − 1 2 = γ Γ( d 2 ) √ π Γ( d − 1 2 ) 2 d − 1 1 1 − t 2 . Now , we recall that t = tanh  h 2  . Then the chain rule implies that d d h ( I 1 ( h ) + I 2 ( h )) = d d t ( I 1 ( t ) + I 2 ( t )) d t d h  tanh  h 2  = γ Γ( d 2 ) √ π Γ( d − 1 2 ) 2 d − 1 1 1 − t 2 1 − t 2 2 = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) , implying that µ γ , 1 ([ ℓ ( h )] 1 ) must be an affine-linear function of h . But again, since µ γ , 1 ([ ℓ (0)] 1 ) = 0 , we conclude that µ γ , 1 ([ ℓ ( h )] 1 ) = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h for all h ≥ 0 . 4.3 Linearity for all 0 ≤ λ ≤ 1 As mentioned in the previous subsection, for 0 < λ < 1 we were not able to analytically evaluate the integrals in the representation for µ γ ,λ ([ ℓ ( h )] λ ) in Corollary 4.2. T o show that also in this case µ γ ,λ ([ ℓ ( h )] λ ) is a linear function and the same as for λ = 0 in Lemma 4.3 and λ = 1 in Lemma 4.4 we take a different route. W e start with the following result. Lemma 4.5. Suppose that 0 ≤ λ ≤ 1 . Then lim h ↓ 0 µ γ ,λ ([ ℓ ( h )] λ ) h = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) . Proof. Using tanh( x ) = x − x 3 / 3 + O ( x 5 ) as x → 0 one first obtains r c = λ 4 h 2 + λ 48 (1 − 3 λ 2 ) h 4 + O ( h 6 ) , h ↓ 0 . Call F ( r ) the integrand in the first integral I 1 ( h ) from 0 to r c in Corollary 4.2 (b). Then F (0) = 1 2 · 2 = 1 by (4.4). Moreover , 2 p ( r 2 λ + r )( λ + r ) r 2 + 2 λr + 1 = 2 √ λr + (1 − λ 2 ) √ λ r 3 / 2 + O ( r 5 / 2 ) , implying that F ( r ) = F (0) + O ( √ r ) as r ↓ 0 . It follows that 1 γ I 1 ( h ) = Z r c ( h ) 0 F ( r ) d r = F (0) r c ( h ) + Z r c ( h ) 0 F ( r ) − F (0) d r = r c ( h ) + O ( r c ( h ) 3 / 2 ) . 18 Thus, I 1 ( h ) = γ λ 4 h 2 + o ( h 2 ) , h ↓ 0 . (4.5) Next, we consider the integral I 2 ( h ) from r c to tanh( h 2 ) in Corollary 4.2 (b). Apply the substitution r = sh with r c /h ≤ s ≤ tanh( h 2 ) /h . Then, 1 γ I 2 ( h ) = h Z 1 h tanh( h 2 ) r c /h σ d − 1 ( S ( e 1 , A ( h, s ))) B ( h, s ) d s with the abbreviations A ( h, s ) := tanh 2  h 2  ( λ + sh ) + s 2 h 2 λ + sh tanh  h 2  (1 + 2 λsh + s 2 h 2 ) , B ( h, s ) := 2(1 + 2 λsh + s 2 h 2 ) d − 1 (1 − s 2 h 2 ) d . Using the expansion of the hyperbolic tangent function from above, A ( h, s ) = 2 s (1 + O ( h )) and B ( h, s ) = 2 + O ( h ) as h ↓ 0 uniformly for 0 ≤ s < 1 / 2 (note that 1 h tanh  h 2  → 1 / 2 , whereas r c /h → 0 ). It follows that I 2 ( h ) = 2 γ h Z 1 / 2 0 σ d − 1 ( S ( e 1 , 2 s )) d s + o ( h ) . Now , use the formula (4.4) for σ d − 1 ( S ( e 1 , 2 s )) and Fubini’s theorem: 2 Z 1 / 2 0 σ d − 1 ( S ( e 1 , 2 s )) d s = Z 1 0 σ d − 1 ( S ( e 1 , u )) d u = Γ( d 2 ) √ π Γ( d − 1 2 ) Z 1 0 (1 − t 2 ) d − 3 2  Z t 0 d u  d t = Γ( d 2 ) √ π Γ( d − 1 2 ) Z 1 0 (1 − t 2 ) d − 3 2 t d t = Γ( d 2 ) 2 √ π Γ( d − 1 2 ) Z 1 0 (1 − x ) d − 3 2 d x = Γ( d 2 ) 2 √ π Γ( d +1 2 ) , where we applied the substitutions u = 2 s and x = t 2 . As a result, I 2 ( h ) = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h + o ( h ) . If we combine this with (4.5), divide by h and take the limit as h ↓ 0 we get lim h ↓ 0 µ γ ,λ ([ ℓ ( h )] λ ) h = lim h ↓ 0 I 1 ( h ) + I 2 ( h ) h = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) and the proof is complete. The next step consists in establishing the following fact. Lemma 4.6. Suppose that 0 ≤ λ ≤ 1 . Then µ γ ,λ ([ ℓ ( h )] λ ) is a linear function in h (with zero intercept). 19 Proof. Since the case λ = 0 was handled in Lemma 4.3 and also in [8], we restrict to the case 0 < λ ≤ 1 in what follows. Let e 1 be a basis vector in R d and consider the geodesic line L = { ce 1 : c ∈ ( − 1 , 1) } in the P oincaré ball model B d . W e identify ℓ ( h ) with the geodesic segment [0 , (tanh h 2 ) e 1 ] ⊂ L . W e say that the point ce 1 ∈ L is below de 1 ∈ L if − 1 < c < d < 1 . Every λ -geodesic hyperplane H ∈ Hyp λ (having o on its convex side, or not) either intersects L in two points, denoted by C ( H ) ∈ L and D ( H ) ∈ L with the convention that C ( H ) is below D ( H ) , or does not intersect L at all, or intersects L in one point. Since the latter possibility has measure zero, it will be ignored. Let Q λ ⊂ Hyp λ be the set of all λ -geodesic hyperplanes H ∈ Hyp λ that intersect L in two points. Note that Q λ is invariant under hyperbolic isometries that map L to L . The map Ψ : Q λ → L , H 7→ C ( H ) e 1 , that maps every λ -geodesic hyperplane H ∈ Q λ to the “lower” intersection point with L commutes with all hyperbolic isometries that map L to L and are orientation-preserving. Let m λ be the push-forward of the invariant measure ν λ (restricted to Q λ ) under the map Ψ . Then m λ is a measure on L invariant under hyperbolic orientation-preserving isometries of L . Also, m λ is locally finite, since for any interval I a of length a centred at o we have m λ ( I a ) ≤ ν λ ( { H ∈ Hyp λ : H ∩ I a  = ∅ } ) ≤ Z a/ 2 − a/ 2 (cosh( s ) + λ sinh( s )) d − 1 d s < ∞ . It follows that m λ is a multiple of the hyperbolic length measure on L . In particular , m λ ( ℓ ( h )) is a linear function of h > 0 . Next we observe that for a hyperplane H ∈ Q λ , Ψ( H ) ∈ ℓ ( h ) is equivalent to H ∈ [ ℓ ( h )] λ . Indeed, if Ψ( H ) ∈ ℓ ( h ) then both intersection points of H with L are “above” the origin and hence the origin is not on the convex side of H . So, H ∈ Hyp o λ and since H intersects the geodesic segment ℓ ( h ) at Ψ( H ) , we conclude that H ∈ [ ℓ ( h )] λ . Conversely , if H ∈ [ ℓ ( h )] λ , then by definition of [ ℓ ( h )] λ , 0 is not on the convex side of H and at least one of the intersection points of H with L belongs to ℓ ( h ) . This implies that, in fact, the lower intersection point belongs to ℓ ( h ) , that is Ψ( H ) ∈ ℓ ( h ) , proving the equivalence. The equivalence we just proved shows that µ γ ,λ ([ ℓ ( h )] λ ) = γ m λ ( ℓ ( h )) . Since m λ ( ℓ ( h )) is linear in h > 0 , the proof is complete. If we combine Lemma 4.5 with Lemma 4.6 we obtain the following generalization of Lemma 4.3 and Lemma 4.4. Corollary 4.7. Suppose 0 ≤ λ ≤ 1 . Then µ γ ,λ ([ ℓ ( h )] λ ) = γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h, h ≥ 0 . 5 Expected volume of the visibility region In this section we derive the formula for the expected volume of the visibility region Z γ ,λ,d . The calculations are analogous to the ones in [8] and will eventually show that E v ol d ( Z γ ,λ,d ) is independent of λ . F or u ∈ S d − 1 let s u ( η γ ,λ ) = sup  h ≥ 0 :  r u : 0 ≤ r ≤ tanh  h 2  ∩ H = ∅ for all H ∈ η γ ,λ  . Here, the set { r u : 0 ≤ r < 1 } represents the geodesic ray in the P oincaré ball model starting at o with direction u , and the upper bound tanh( h/ 2) corresponds to the Eu- clidean distance of a point with hyperbolic distance h from the origin. 20 Proof of Theorem 1.1 (b). Fix u ∈ S d − 1 and let ℓ ( h ) be a geodesic segment of length h > 0 starting at o in an arbitrary direction. Then by Corollary 4.7 we have P ( s u ( η γ ,λ ) > h ) = P ( η γ ,λ ([ ℓ ( h )] λ ) = 0) = exp  − µ γ ,λ ([ ℓ ( h )] λ )  = exp  − γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) h  , independently of λ . Hence, the random variable s u ( η γ ,λ ) is exponentially distributed with parameter γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) and by the polar decomposition of hyperbolic space we have E v ol d ( Z γ ,λ,d ) = 2 π d/ 2 Γ( d 2 ) E " Z S d − 1 Z s u ( η γ ,λ ) 0 sinh d − 1 ( s ) d sσ d (d u ) # = 2 π d/ 2 Γ( d 2 ) Z S d − 1 Z ∞ 0 P ( s u ( η γ ,λ ) ≥ s ) sinh d − 1 ( s ) d sσ d (d u ) # = 2 π d/ 2 Γ( d 2 ) Z ∞ 0 exp  − γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) s  sinh d − 1 ( s ) d s, where σ d denotes the normalized spherical Lebesgue measure on S d − 1 . According to Identity 3.541.1 in [13] (see also [8, Equation (6.9)]) it holds that Z ∞ 0 sinh d − 1 ( s ) e − as d s = ( d − 1)! 2 d Γ( a − d +1 2 ) Γ( a + d +1 2 ) , (5.1) whenever a > d − 1 . Thus, the expected volume is finite if and only if γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) > d − 1 or γ > 2( d − 1) √ π Γ( d +1 2 ) Γ( d 2 ) = ( d − 1) 2 √ π Γ( d − 1 2 ) Γ( d 2 ) = γ crit and, in this case, E v ol d ( Z γ ,λ,d ) = E v ol d ( Z γ , 0 ,d ) = 2 1 − d π d/ 2 ( d − 1)! Γ( d 2 ) Γ  γ ∗ d − d +1 2  Γ  γ ∗ d + d +1 2  with γ ∗ d := γ Γ( d 2 ) 2 √ π Γ( d +1 2 ) . T o arrive at the desired expression, we simplify the factor 2 1 − d π d/ 2 ( d − 1)! Γ( d 2 ) using the Leg- endre duplication formula. It follows that 2 1 − d π d/ 2 ( d − 1)! Γ( d 2 ) = 2 1 − d π d/ 2 ( d − 1)! Γ( d +1 2 ) 2 1 − d √ π Γ( d ) = π d − 1 2 Γ  d + 1 2  . Plugging this into the expression for E v ol d ( Z γ ,λ,d ) completes the proof. Remark 2. Note that, compared to the constant γ ∗ d obtained in [8, Theorem 7.1], our expression for γ ∗ d differs by an additional factor of 1 2 . This is due to our choice of considering the space Hyp o λ rather than Hyp λ , i.e., the choice of omitting all those λ - geodesic hyperplanes whose compact side contains o . F or the case λ = 0 , this restriction leads to a factor of 1 2 in the measure considered on Hyp o 0 compared to the measure on Hyp 0 used in [8]. Remark 3. The notion of λ -geodesic hyperplanes as totally umbilical hypersurfaces of H d can be extended to the case λ > 1 . In that situation, a λ -geodesic ’hyperplane’ is a hyperbolic sphere of radius R λ := artanh  1 λ  . Hence, in this regime the visibility question in presence of a Poisson process of such spheres reduces to that in a Boolean model of balls with fixed radius R λ , which in turn 21 has been studied in [21]. As for λ ∈ [0 , 1] , there exists a critical intensity γ B , crit > 0 such that the visibility region is unbounded with positive probability if γ < γ B , crit and almost surely bounded for γ > γ B , crit . 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