A Furstenberg type formula for the speed of distance stationary sequences
AA Furstenberg type formula for the speed ofdistance stationary sequences
Mat´ıas Carrasco Pablo Lessa Elliot PaquetteOctober 4, 2017
Abstract
We prove a formula for the speed of distance stationary random sequences.A particular case is the classical formula for the largest Lyapunov exponent ofan i.i.d. product of two by two matrices in terms of a stationary measure onprojective space. We apply this result to Poisson-Delaunay random walks onRiemannian symmetric spaces. In particular, we obtain sharp estimates forthe asymptotic behavior of the speed of hyperbolic Poisson-Delaunay randomwalks when the intensity of the Poisson point process goes to zero. Thisallows us to prove that a dimension drop phenomena occurs for the harmonicmeasure associated to these random walks. With the same technique we giveexamples of co-compact Fuchsian groups for which the harmonic measure ofthe simple random walk has dimension less than one.
Keywords:
Lyapunov exponents, Random walk speed, Riemannian symmet-ric space, Poisson-Delaunay graph.
AMS2010:
Primary 60G55, 60Dxx, 51M10, 05C81, 34D08.
Contents
I A Furstenberg type formula for speed 4 a r X i v : . [ m a t h . P R ] O c t Applications 11
II Distance stationarity of Poisson-Delaunay random walks14
III Zero-one laws for Poisson–Delaunay random walks 21IV Graph and ambient speed of Poisson-Delaunay ran-dom walks 26
V Hyperbolic Poisson-Delaunay random walks 28
VI Dimension drop phenomena 42
10 Tools for proving dimension drop 4311 Dimension drop for some co-compact Fuchsian groups 4612 Dimension drop for low intensity hyperbolic Poisson-Delaunay ran-dom walks 49 Introduction
A random sequence is said to be distance stationary if the distribution of distancesbetween its points is shift invariant.The speed, or linear drift, of a distance stationary sequence is the limit (cid:96) = lim n → + ∞ n d ( x , x n ) , where d ( x , x n ) is the distance between the initial point and the n -th point of therandom sequence.From Kingman’s subadditive ergodic theorem the speed exists almost surely andin mean under the mild assumption that the expected distance between the firsttwo points of the sequence is finite.In Part I of this article we prove the following integral formula for the speed of adistance stationary sequence (which we call a ‘Furstenberg type formula’) E ( (cid:96) ) = − E ( ξ ( x ) − ξ ( x ))where ξ is a random horofunction depending on the past tail of the sequence (The-orem 3).The formula is most useful when one knows that the random horofunction ξ ison the horofunction boundary of the space under consideration (see Proposition 1).When this is the case one can sometimes decide whether the speed is zero or positive,and obtain explicit estimates.In the case where ( A n ) n ∈ Z is a sequence of i.i.d. matrices in SL(2 , R ) not supportedon a compact subgroup, and one considers the sequence of positive parts in the polardecomposition of the products A n · · · A , our result implies the classical formula forthe largest Lyapunov exponent in terms of a stationary probability measure onprojective space originated by Furstenberg (see [BL85, Chapter 2, Theorem 3.6]and [Fur63]). We discuss this briefly in Section 4, together with a more elementaryexample.In the rest of the article we show that Poisson-Delaunay random walks on Rie-mannian symmetric spaces (previously studied in [BPP14], [Paq17], and [CPL16])may be fruitfully seen as distance stationary sequences. In particular, we show thatone may obtain interesting results about their speed by using the Furstenberg typeformula.For this purpose we establish, in Part II, that these walks are distance stationaryunder an appropriate bias (Theorem 4). We also prove, in Part III, an ergodicitytheorem (Theorem 5) for Poisson-Delaunay random walks which implies in particularthat their graph speed and ambient speed are almost surely constant. In Part IV3e show show that the graph speed and ambient speed are simultaneously eitherpositive or zero (Proposition 4).In Part V of the article we prove a sharp estimate for the ambient speed of thePoisson-Delaunay random walk in hyperbolic space when the intensity of the Poissonpoint process is small (Theorem 6).From this approach we obtain an alternative proof that the graph speed of hy-perbolic Poisson-Delaunay random walks is positive for small intensities (Corollary4). Positivity of the graph speed was proved in the two dimensional case for allintensities in [BPP14], by showing that the graph satisfies anchored expansion.For all non-compact type Riemannian symmetric spaces, positivity of the speedof Poisson-Delaunay random walks was established in [Paq17] by using the theoryof unimodular random graphs and invariant non-amenability.We also prove that the graph speed of hyperbolic Poisson-Delaunay random walksgoes to its maximal possible value 1, when the intensity of the Poisson process goesto zero (Corollary 5). This was previously unknown, and answers a question posedin [BPP14].In Part VI of the article we discuss dimension drop phenomena. In particular weuse the Furstenberg type formula to estimate the speed of certain simple randomwalks on co-compact Fuchsian groups well enough to show that the dimension oftheir harmonic measure is less than one. We also prove, using the previously ob-tained estimate for speed, that for hyperbolic Poisson-Delaunay random walks withlow enough intensity the dimension drop phenomena also occurs. Both of theseresults are new as far as the authors are aware.The Furstenberg type formula for speed we prove in this article (Theorem 3)is related to previous work of Gou¨ezel, Karlsson, and Ledrappier (see [KL06] and[GK15]). We will discuss this in detail in the introduction to Part I. Part I
A Furstenberg type formula forspeed
The purpose of this part of the article is to construct, given a distance stationaryrandom sequence ( x n ) n ∈ Z , a random horofunction ξ which captures its linear drift,and whose increments along the sequence are stationary (see below for definitions).We do so under the hypothesis that there exists a random variable u which isuniform on [0 ,
1] and independent from the given random sequence ( x n ) n ∈ Z .4f course, such a random variable can be constructed if one extends the proba-bility space on which the random sequence ( x n ) n ∈ Z is defined. And, any conclusionabout the sequence stated only in terms of its distribution (e.g. any almost sureproperty) which is proved in such an extension (possibly using the random horo-function) will be valid in the original probability space as well.In previous results of Gou¨ezel, Karlsson, and Ledrappier (see [KL06] and [GK15])a random horofunction capturing the rate of escape of a distance stationary sequenceis constructed without imposing any condition on the underlying probability space.However, in their results the sequence of increments of such a horofunction is notguaranteed to be stationary.Our result is very much related to the decomposition of stationary subadditiveprocesses into the sum of a stationary additive process and a stationary purelysubadditive one. This result was first proved by Kingman to obtain the subadditiveergodic theorem (see [Kin68], [Kin73], and [dJ77]). In essence we modify the proofof the decomposition theorem via Komlos’ theorem due to Burkholder (see thediscussion by Burkholder in [Kin73]). In this context the extension of the baseprobability space seems to be required to interpret the stationary additive processin the decomposition as the sequence of increments of a random horofunction.The existence of a horofunction capturing the rate of escape of a sequence hasseveral implications. Notably, in spaces of negative curvature it implies the existenceof a geodesic tracking the sequence. As shown by Kaimanovich [Ka˘ı87] this issufficient to obtain Oseledet’s theorem. A wealth of other applications are exhibitedin previously cited works (also see [KM99]).The additional fact that the increments of the horofunction are stationary, allowsone to obtain explicit estimates on the speed in certain situations, and also torecapture the original formula due to Furstenberg of the Lyapunov exponent of aproduct of 2 × In what follows (
M, d ) denotes a complete separable metric space and o ∈ M a basepoint which is fixed from now on.A random sequence ( x n ) n ∈ Z of points in M is said to be distance stationary if thedistribution of ( d ( x m , x n )) m,n ∈ Z coincides with that of ( d ( x m +1 , x n +1 )) m,n ∈ Z .5 .2 Horofunctions To each point x ∈ M we associate a horofunction ξ x : M → R defined by ξ x ( y ) = d ( x, o ) − d ( x, y ) . The horofunction compactification of M is the space (cid:99) M obtained as the closure ofthe functions of the form ξ x in the topology of uniform convergence on compact sets.Compactness of (cid:99) M follows from the Arsel`a-Ascoli theorem and the fact that allfunctions ξ x are 1-Lipschitz. A horofunction on M is an element of (cid:99) M .Horofunctions which are not of the form ξ x will be called boundary horofunctions and the set of boundary horofunctions is the horofunction boundary of M , whichmight sometimes be written (cid:99) M \ M abusing notation slightly. If ( x n ) n ∈ Z is a distance stationary sequence in M satisfying E ( d ( x , x )) < + ∞ ,then by Kingman’s subadditive ergodic theorem the random limit (cid:96) = lim n → + ∞ n d ( x , x n )exists almost surely and in L .We call this limit the speed or linear drift of ( x n ) n ∈ Z . Recall that a random sequence ( s n ) n ∈ Z is stationary if its distribution coincides withthat of ( s n +1 ) n ∈ Z .If ( s n ) n ∈ Z is stationary and E ( | s | ) < + ∞ , then by Birkhoff’s ergodic theoremthe limit lim n → + ∞ n n − (cid:88) i =0 s i exists almost surely and in L . We call this limit the Birkhoff limit of the sequence. Given a Polish space X we use P ( X ) to denote the space of Borel probabilitymeasures on X endowed with the topology of weak convergence (i.e. a sequence6onverges if the integral of each continuous bounded function from X to R does).This space is also Polish and is compact if X is compact.We will use the following result due to Blackwell and Dubins (see [BD83]): Theorem 1 (Continuous representation of probability measures) . For any Polishspace X there exists a function F : P ( X ) × [0 , → X such that if u is a uniformrandom variable on [0 , the following holds:1. For each µ ∈ P ( X ) The random variable F ( µ, u ) has distribution µ .2. If µ n → µ then F ( µ n , u ) → F ( µ, u ) almost surely. We call a function F satisfying the properties in the above theorem a continuousrepresentation of P ( X ). Recall that a sequence ( a n ) n ≥ is Cesaro convergent if the limit lim n → + ∞ n n (cid:80) k =1 a k exists.We restate the main result [Kom67]. Theorem 2 (Komlos’ theorem) . Let ( X n ) n ≥ be a sequence of random variableswith sup n E ( | X n | ) < + ∞ . Then there exists a subsequence ( Y n ) n ≥ of ( X n ) n ≥ whichCesaro converges almost surely to a random variable Y with finite expectation, andfurthermore any subsequence of ( Y n ) n ≥ has the same property. We will need the following corollary of Komlos’ theorem.
Corollary 1.
Let ( µ n ) n ≥ be a sequence of random probabilities on a compact metricspace ( X, d ) . There exists a subsequence ( µ n k ) k ≥ which Cesaro converges almostsurely to a random probability µ on X .Proof. In this proof we use the notation ν ( f ) = (cid:82) X f ( x ) dν ( x ).Let ( f n ) n ≥ be a dense sequence in the space of continuous functions from X to R (with respect to the topology of uniform convergence).Applying Komlos’ theorem to ( µ n ( f )) n ≥ one obtains a subsequence n ,k → + ∞ such that µ n ,k ( f ) Cesaro converges almost surely and any further subsequence hasthe same property.For i = 1 , , , , . . . , inductively applying Komlos’ theorem to ( µ n i,k ( f i +1 )) k ≥ weobtain a subsequence ( n i +1 ,k ) k ≥ of ( n i,k ) k ≥ such that µ n i +1 ,k ( f i +1 ) Cesaro convergesalmost surely and any further subsequence has the same property.7etting n k = n k,k one obtains that ( µ n k ) n ≥ Cesaro converges to a random prob-ability µ almost surely. Theorem 3 (Furstenberg type formula for distance stationary sequences) . Let ( x n ) n ∈ Z be a distance stationary sequence in a complete separable metric space ( M, d ) satisfying E ( d ( x , x n )) < + ∞ and (cid:96) be its linear drift.Suppose there exists a random variable u which is uniformly distributed on [0 , and independent from ( x n ) n ∈ Z . Then the following holds:1. The sequence of random probability measures on (cid:99) M defined by µ n = n n (cid:80) i =1 δ ξ x − i has a subsequence which is almost surely Cesaro convergent to a random prob-ability µ .2. There exists a random horofunction ξ which is measurable with respect to σ ( u, µ ) and whose conditional distribution given ( x n ) n ∈ Z is µ .3. The sequence of increments ( ξ ( x n ) − ξ ( x n +1 )) n ∈ Z is stationary and its Birkhofflimit equals (cid:96) almost surely. In particular, E ( (cid:96) ) = E ( ξ ( x ) − ξ ( x )) .Proof. The fact that ( µ n ) n ≥ has an almost surely Cesaro convergent subsequencefollows directly from the version of Komlos’ theorem for random probabilities givenabove (see Corollary 1). Let ( µ n j ) j ≥ be such a subsequence and µ be its almostsure Cesaro limit.Let F : P ( (cid:99) M ) × [0 , → (cid:99) M be continuous representation of P ( (cid:99) M ), as givenby Theorem 1 and define ξ = F ( u, µ ). Clearly ξ is σ ( u, µ )-measurable and itsconditional distribution given ( x n ) n ∈ Z is µ .We will now show that E ( ξ ( x ) − ξ ( x )) = (cid:96) .For this purpose let ξ k = F ( u, k k (cid:80) j =1 µ n j ) and notice that ξ k → ξ almost surely when k → + ∞ . Because horofunction are 1-Lipschitz one has | ξ k ( x ) − ξ k ( x ) | ≤ d ( x , x ).Since E ( d ( x , x )) < + ∞ this implies that the sequence is uniformly integrable andone obtains E ( ξ ( x ) − ξ ( x )) = lim k → + ∞ E ( ξ k ( x ) − ξ k ( x )).For the sequence on the right hand side using distance stationarity one obtains8 ( ξ k ( x ) − ξ k ( x )) = 1 k k (cid:88) j =1 E (cid:32) n j n j (cid:88) i =1 ξ x − i ( x ) − ξ x − i ( x ) (cid:33) = 1 k k (cid:88) j =1 E (cid:32) n j n j (cid:88) i =1 − d ( x − i , x ) + d ( x − i , x ) (cid:33) = 1 k k (cid:88) j =1 E (cid:32) n j n j (cid:88) i =1 − d ( x , x i ) + d ( x , x i +1 ) (cid:33) = 1 k k (cid:88) j =1 E (cid:18) − d ( x , x ) + d ( x , x n j +1 ) n j (cid:19) . Taking the limit when k → + ∞ above it follows that E ( ξ ( x ) − ξ ( x )) = E ( (cid:96) ) asclaimed.We will now prove that ( ξ ( x n ) − ξ ( x n +1 )) n ∈ Z is stationary.Suppose F : R Z → R is continuous and bounded and notice that by distancestationarity one has E ( F (( ξ k ( x n +1 ) − ξ k ( x n +2 )) n ∈ Z ))= 1 k k (cid:88) j =1 n j n j (cid:88) i =1 E (cid:16) F (cid:16)(cid:0) ξ x − i ( x n +1 ) − ξ x − i ( x n +2 ) (cid:1) n ∈ Z (cid:17)(cid:17) = 1 k k (cid:88) j =1 n j n j (cid:88) i =1 E (cid:16) F (cid:16)(cid:16) ξ x − ( i − ( x n ) − ξ x − ( i − ( x n +1 ) (cid:17) n ∈ Z (cid:17)(cid:17) = 1 k k (cid:88) j =1 n j n j − (cid:88) i =0 E (cid:16) F (cid:16)(cid:0) ξ x − i ( x n ) − ξ x − i ( x n +1 ) (cid:1) n ∈ Z (cid:17)(cid:17) = C k max | F | + E (cid:0) F (cid:0) ( ξ k ( x n ) − ξ k ( x n +1 )) n ∈ Z (cid:1)(cid:1) where | C k | ≤ k k (cid:80) j =1 2 n j → k → + ∞ .From this the stationarity of the increments of ξ along the sequence ( x n ) n ∈ Z ,follows directly taking limit when k → + ∞ .By Birkhoff’s theorem the Birkhoff averages of the increments of ξ along thesequence exist almost surely and in L . Additionally, because horofunctions are1-Lipschitz, one has lim n → + ∞ n n − (cid:88) k =0 ξ ( x k ) − ξ ( x k +1 ) ≤ (cid:96) E ( ξ ( x ) − ξ ( x )) = E ( (cid:96) ). Hence both sides coincide almost surely. This concludesthe proof. The question of whether the random horofunction ξ given by Theorem 3 is almostsurely on the horofunction boundary of M sometimes arises.A trivial example where this is not the case is obtained by letting ( x n ) n ∈ Z bean i.i.d. sequence of uniformly distributed random variables on [0 , ξ given by Theorem 3 will be uniformly distributed on [0 ,
1] andindependent from the sequence.In the previous example the linear drift (cid:96) was 0 almost surely. It is not difficultto show that if (cid:96) > ξ must be a boundary horofunction almostsurely.However, in many examples Theorem 3 can be used to decide whether or not (cid:96) ispositive. Hence it is useful to have a criteria for establishing that ξ is almost surelyon the horofunction boundary without knowledge of (cid:96) . The following proposition issuch a result. Proposition 1.
Assume ( x n ) n ∈ Z is a distance stationary sequence with E ( d ( x , x )) < + ∞ and u is a random variable which is uniformly distributed on [0 , and indepen-dent from ( x n ) n ∈ Z . If P ( x n ∈ K ) → when n → −∞ for all bounded sets K , thenthe random horofunction ξ given by Theorem 3 is almost surely on the horofunctionboundary.Proof. We will use the notation from Theorem 3. Let ν n denote the sequence ofaverages of the subsequence of the probabilities ( µ n ) which Cesaro converges to µ almost surely.Given a bounded set K pick a bounded open set U containing the closure of K . From the hypothesis it follows that E ( µ n ( U )) → n → + ∞ . Therefore E ( ν n ( U )) → ν n → µ one has µ ( U ) ≤ lim inf n ν n ( U ) almost surely. Combining this withFatou’s lemma one obtains P ( ξ ∈ K ) = E ( µ ( K )) ≤ E ( µ ( U )) ≤ E (cid:16) lim inf n ν n ( U ) (cid:17) ≤ lim inf n E ( ν n ( U )) = 0 , and hence ξ / ∈ K almost surely. 10igure 1: A realization of a right angled random walk in the Poincar´e disk for stepsize r = 0 .
1. Hyperbolic segments have been added between consecutive points ofthe walk, notice that there are no squares in the hyperbolic plane, so any apparentsquare in the figure does not in fact close up.
To illustrate Theorem 3 consider a right angled random walk on the hyperbolic plane(see also [Gru08]). That is, starting with a unit tangent vector ( o, v ) consider theMarkov process where at each step one rotates the vector a random multiple of90 o (each value − , , ,
180 having the same probability) and then advances indirection of the geodesic a distance r >
0. Suppose the sequence of base points thusobtained is x = o, x , x , . . . . One can extend this to a bi-infinite distance stationarysequence by letting x , x − , x − , . . . be an independent random walk constructed inthe same way.For n = 5 , , , . . . let r n be the side of the regular n -gon with interior right anglesin the hyperbolic plane. Notice that if r = r n , then the walk ( x n ) n ∈ Z remains onthe vertices of a tessellation by regular n -gons with 4 meeting at each vertex.Setting r ∞ = lim n → + ∞ r n one may show (via a ping-pong argument on the boundary)that if r ≥ r ∞ the random walk remains on the vertices of an embedded regular treeof degree 4.For all other values of r (smaller than r ∞ but not one of the r n ) it seems clearthat the set of points attainable by the random walk is dense in the hyperbolic plane11though a short argument is not known to the authors). For example, for r smallenough this follows by Margulis’ lemma, while for a dense set of values of r theelliptic element relating the initial unit vector to the one obtained by advancing r and then rotating 90 o is an irrational rotation.The speed (cid:96) r = lim n → + ∞ d ( x ,x n ) n exists for all r since d ( x , x ) = r almost surely.We will now sketch how Theorem 3 may be used to show that (cid:96) r > r > P ( d ( x , x n ) < C ) → n → + ∞ for all C >
0. This can be shown by first observing that there exists a sequence ( g n ) n ∈ Z ofi.i.d. isometries of the hyperbolic plane such that x n = g ◦ · · · ◦ g n ( x ) for all n ≥ g is not supported on a compact subgroup of the isometrygroup the distribution of g ◦ · · · ◦ g n goes to zero on any compact set as follows forexample from [Der76, Theorem 8].Furthermore, since (cid:96) is tail measurable with respect to the sequence ( g n ) n ∈ Z , itfollows from Kolmogorov’s zero-one law that (cid:96) is almost surely constant.By Theorem 3 and Proposition 1, one obtains that there exists a random boundaryhorofunction ξ which is independent from x and such that (cid:96) = E ( (cid:96) ) = − E ( ξ ( x )) . Even though the distribution of ξ is unknown (e.g. a priori it need not be uniformon the boundary circle, though it must be invariant under 90 degree rotation), onemay use the existence of ξ to show that (cid:96) > x takes four values, say a, b, c, d with equal proba-bility 1 /
4. Conditioning on ξ (using the independence of ξ and x ) one obtains: (cid:96) = − E (cid:18) ξ ( a ) + ξ ( b ) + ξ ( c ) + ξ ( d )4 (cid:19) . Finally the result follows because ξ ( a ) + ξ ( b ) + ξ ( c ) + ξ ( d ) < ξ . To see this one may calculate in a concrete model. For example inthe Poincar´e disk if the starting point is 0 and the initial unit tangent vector pointstowards the positive real axis, one may take a = x, b = ix, c = − x, d = − ix where x = tanh( r/ ξ ( z ) = log (cid:16) −| z | | z − e iθ | (cid:17) for θ ∈ [0 , π ] (see for example [BH99, Section 8.24]).Hence one obtains ξ ( a ) + ξ ( b ) + ξ ( c ) + ξ ( d ) = log (cid:18) (1 − x ) | e iθ − x | (cid:19) ≤ log (cid:18) (1 − x ) (1 + x ) (cid:19) = − (cid:90) r tanh( s )2 ds < . r >
0. The lower boundis equivalent to r / r → r/ r → + ∞ . For large r the boundis close to optimal because the random walk on a regular tree of degree 4 has lineardrift 1 / ξ is independentfrom x . × i.i.d. matrix products Suppose that ( A n ) n ∈ Z is an i.i.d. sequence of matrices in SL(2 , R ) with the additionalproperty that E (log( | A | )) < + ∞ where | A | denotes the operator norm of the matrix A .The largest Lyapunov exponent of the sequence ( A n ) is defined by χ = lim n → + ∞ n log ( | A n · · · A | ) , and is almost surely constant since it is a tail function of the sequence.Notice that if one writes A n · · · A = O n P n where O n is orthogonal and P n issymmetric with positive eigenvalues, one obtains χ = lim n → + ∞ n log ( | P n | ) . This implies that ξ depends only on the sequence of projections of A n · · · A tothe left quotient M = SO(2) \ SL(2 , R ). Let [ A ] denote the equivalence class of amatrix A ∈ SL(2 , R ) in the quotient above.The quotient space admits a (unique up to homotethy) Riemannian metric forwhich the transformations [ A ] (cid:55)→ [ AB ] are isometries for all B ∈ SL(2 , R ). One maychoose such a metric so that the distance d ([Id] , [ A ]) = (cid:112) log( σ ) + log( σ ) where σ , σ are the singular values of A and Id denotes the identity matrix. In particular,since σ = | A | and A has determinant 1, one obtains d (Id , [ A ]) = √ | A | ).With the Riemannian metric under consideration the sequence . . . , x − = [ A − − A − ] , x − = [ A − ] , x = [Id] , x = [ A ] , x = [ A A ] , . . . is distance stationary and satisfies E ( d ( x , x )) < + ∞ . Furthermore, its rate ofescape is (cid:96) = √ χ .The boundary horofunctions on M are of the form ξ ([ A ]) = −√ | Av | ) forsome | v | = 1 (see for example [Hat00]). 13f A is not contained almost surely in a compact subgroup of SL(2 , R ) then onemay use [Der76, Theorem 8] to show that P ( x n ∈ K ) → n → + ∞ for allcompact sets K ⊂ M .Hence, Theorem 3 and Proposition 1 imply the existence of a random unit vector v ∈ R which is independent from A and such that χ = E (log ( | A v | )) . In particular, letting µ be the distribution of A , there is a probability ν on theunit circle S ⊂ R such that χ = (cid:90) SL(2 , R ) (cid:90) S log ( | Av | ) d ν ( v )d µ ( A ) . This is typically called Furstenberg’s formula for the largest Lyapunov exponent(see [BL85, Theorem 3.6]). It follows from Theorem 3 that ν is µ -stationary (wherethe action of SL(2 , R ) on S is by transformations of the form v (cid:55)→ Av/ | Av | ). Thismay be used as a starting point to establish a criteria for an i.i.d. random matrixproduct to have a positive Lyapunov exponent.Also, in some cases, formulas of this type can be used to give explicit estimates forthe largest Lyapunov exponent in a family of random matrix products dependingon some parameter (see for example [GGG17], and [DH83]).The reasoning above may be carried out in SL( n, R ) for larger n . What resultsis a formula for the sum of squares of the Lyapunov exponents of the random i.i.d.product of matrices. As above, the distribution of the random boundary horofunc-tion is unknown (in larger dimension horofunctions are determined by a choice of aflag and a sequence of weights adding up to zero). Part II
Distance stationarity ofPoisson-Delaunay random walks
Throughout this part of the article M will be a Riemannian symmetric space, o ∈ M a fixed base point, and P a homogeneous Poisson point process in M with constantintensity λ (i.e. λ points per unit volume).We say two distinct points x, y in a discrete subset X of M are Delaunay neighborsif there exits an open ball in M which is disjoint from X and contains x and y on14ts boundary. This gives the set X a graph structure by adding an undirected edgebetween each pair of Delaunay neighbors. We call this graph the Delaunay graphassociated to X .The Voronoi cell of a point x in a discrete set X is the set V x = { y ∈ M : d ( x, y ) = d ( X, y ) } . An alternative definition of the Delaunay graph is obtained by noticing that twodistinct point x, y ∈ X are Delaunay neighbors if and only if V x ∩ V y (cid:54) = ∅ .In what follows we will consider the Delaunay graph of the set P o = P ∪ { o } rooted at o . This is a Poisson-Delaunay random graph (see [BPP14]). See Figure 2for some examples in the hyperbolic plane.Such graphs are known to be unimodular, and stationary under a suitable bias.We will prove a slight generalization of these facts where we take into account theembedding of the graph in the ambient space M .Using this we will construct a distance stationary sequence related to the simplerandom walk on the Poisson-Delaunay random graph. For each x ∈ M we denote by g x a central symmetry exchanging o and x chosenmeasurably as a function of x (if M is of non-compact type g x is unique for all x ).The space of discrete subsets of M will be denoted by Discrete ( M ). We considerthe natural topology on Discrete ( M ) where each discrete set is identified with acounting measure and convergence is equivalent to convergence of the integrals ofall continuous functions with compact support. With this topology Discrete ( M ) isseparable and completely metrizable (i.e. a Polish space).We assume all random objects in this section are defined on the same fixed prob-ability space which we denote by (Ω , F , P ).In what follows we will use Slivyak’s formula (sometimes called Mecke’s formula)which allows one to calculate the expected values of the sum of a function over allpoints in P (where the function may depend on P ) as integrals over M . We refer to[CSKM13, Section 4.4] for a proof of this result (the context there is point processesin R n but the same arguments go through on a Riemannian homogeneous space).15 a) λ = 10 (b) λ = 1(c) λ = 0 . Figure 2: Illustration of three realizations for different values of λ of Poisson-Delaunay graphs in the hyperbolic plane. The neighbors of o are indicated in red.16 roposition 2 (Unimodularity) . For every Borel function F : Discrete ( M ) × M → [0 , + ∞ ) one has E (cid:32)(cid:88) x ∈ P F ( P o , x ) (cid:33) = E (cid:32)(cid:88) x ∈ P F ( g x P o , x ) (cid:33) . Proof.
By Slivnyak’s formula one has E (cid:32)(cid:88) x ∈ P F ( P o , x ) (cid:33) = (cid:90) M E ( F ( P ∪ { o, y } , y )) λdy, where integration is with respect to the volume measure on M .For each fixed y , one has that g y ( P ∪ { o, y } ) = g y P ∪ { o, y } which has the samedistribution as P ∪ { o, y } . Hence, the right-hand side of the equation above equals (cid:90) M E ( F ( g y P ∪ { o, y } , y )) λdy = E (cid:32)(cid:88) x ∈ P F ( g x P o , x ) (cid:33) . In the last inequality we used again Slivnyak’s formula.
Let deg( o ) denote the number of Delaunay neighbors of o in the Delaunay graph of P o . It follows from [Paq17, Theorem 3.3] that E (deg( o )) < + ∞ .We define the degree biased probability P dg on (Ω , F , P ) by d P dg d P = deg( o ) E (deg( o )) . Expectation with respect to the degree biased probability is denoted by E dg ( · ).Let x be a uniform random Delaunay neighbor of o in P o ; i.e. given P o , x hasuniform distribution among the neighbors of o . Proposition 3 (Reversibility) . Under the degree biased probability the distributionof ( P o , x ) is the same as that of ( g x P o , x ) .Proof. Given any Borel function F : Discrete ( M ) × M → [0 , + ∞ ) one has E dg ( F ( P o , x )) E (deg( o )) = E (cid:32)(cid:88) x ∈ P F ( P o , x ) [ x ∼ o ] (cid:33) = E (cid:32)(cid:88) x ∈ P F ( g x P o , x ) [ x ∼ o ] (cid:33) = E dg ( F ( g x P o , x )) E (deg( o )) , x ∼ o means that x isa Delaunay neighbor of o in the discrete set under consideration. Notice that x ∼ o in P o if and only if x ∼ o in g x P o .Since this is valid for all choices of F , the distributions must coincide as claimed. We say a discrete set X ⊂ M intersects all horoballs in M if for every sequenceof balls D , D , . . . such that the radius of D n goes to infinity with n , and all D n intersect some fixed compact set K ⊂ M , one has (cid:83) n D n ∩ X (cid:54) = ∅ . Lemma 1 (Poisson processes intersect all horoballs) . Almost surely, P intersectsall horoballs.Proof. If M compact the statement is trivial. We assume from now on that M isnon-compact.Consider for each n = 1 , , . . . a maximal n/ S n of the theboundary ∂B n of the ball of radius n centered at o , and let A n be the set of balls ofradius n/ S n .Notice that A n is an open covering of ∂B n . Furthermore if V ( r ) denotes thevolume of any ball of radius r in M one has that the number of elements N n in A n is at most V (4 n/ /V ( n/ n large enough every ball in A n intersects P .To see this we calculate P (cid:0) P ∩ B n/ ( x ) = ∅ for some x ∈ S n (cid:1) ≤ N n e − λV ( n/ ≤ V (4 n/ V ( n/ e − λV ( n/ . Since M is a non-compact symmetric space one has that either V ( r ) is boundedbetween two polynomials of the same degree which is at least 1 (if the only non-compact factor in the de Rham splitting of M is Euclidean) or there exist positiveconstants a < b such that e ar ≤ V ( r ) ≤ e br for all r large enough (if there is asymmetric space of non-compact type in the de Rham splitting of M ). In both casesthe right hand term above is summable in n . Hence, applying the Borel-CantelliLemma establishes the claim.Suppose now that c n is a sequence of points in M and r n an unbounded sequenceof radii such that the open balls D n = B r n ( c n ) of radius r n centered at c n satisfy for d ( o, D n ) ≤ C for some fixed positive constant C .18et R n be the integer part of d ( o, c n ) and x n a point in ∂B R n which minimizesthe distance to c n . Choose y n ∈ S n such that d ( x n , y n ) < R n / d ( y n , c n ) ≤ d ( y n , x n ) + d ( x n , c n ) ≤ R n /
3. On the other hand,picking a minimizing geodesic from o to c n , one has r n = d ( o, c n ) − d ( o, D n ) ≥ d ( o, c n ) − C ≥ R n − C . Hence for all n large enough B R n / ( y n ) ⊂ D n and thereforealmost surely there exists n such that D n ∩ P (cid:54) = ∅ .An important consequence of the above lemma is that almost surely every pointin P has a finite number of Delaunay neighbors. Recall that the Voronoi cell of apoint x in a discrete set X is the set of points y satisfying d ( x, y ) = d ( X, y ) (i.e. y at least as close to x as it is to any other point in X ). Corollary 2 (Poisson-Delaunay graphs are locally finite) . Almost surely, the Poisson-Delaunay graph in a symmetric space is locally finite and all Voronoi cells arebounded.Proof.
The corollary follows from the claim that if a discrete set X intersects allhoroballs then all its Voronoi cells are bounded and every point in X has a finitenumber of Delaunay neighbors.To establish the claim first suppose that some point x ∈ X has an infinite numberof Delaunay neighbors. Notice that for each neighbor y of X there exists an openball with x and y on its boundary which is disjoint from X . Since X is discretethis gives a sequence of balls with unbounded radii with x on their boundary anddisjoint from X . This would contradict the fact that X intersects all horoballs.On the other hand if the Voronoi cell of some point x were unbounded one maytake an unbounded sequence of points y n which are closer to x than to any otherpoint in X . In this case the sequence of balls centered at the y n and with x on theirboundary would contradict the fact that X intersects all horoballs. A Delaunay random walk on a discrete set X ⊂ M is a simple random walk on itsDelaunay graph. Such a walk is well defined only if the Delaunay graph of X islocally finite.Let ( x n ) n ∈ Z be defined so that ( x n ) n ≥ and ( x − n ) n ≥ , conditioned on P o , are twoindependent Delaunay random walks on P o starting at o . Let y n = x − n , then bydefinition ( P o , ( x n ) n ∈ Z ) and ( P o , ( y n ) n ∈ Z ) have the same distribution.We call a process ( x n ) n ∈ Z as defined in the previous paragraph a Delaunay randomwalk on P o starting at o , or a Poisson-Delaunay random walk.19 x x x x x -1 x -2 x -3 x -4 oz z z z -1 z -2 z -3 z -4 z -5 P o Q o g x Figure 3: This figure illustrates the proof of Theorem 4. Conditioned on P o and x , the sequence ( x n ) n ≥ is a Poisson-Delaunay random walk starting at x . Inparticular, (conditioned on Q o ) z is a uniformly distributed random neighbor of o in Q o which is independent from z − . Theorem 4 (Distance stationarity) . Let ( x n ) n ∈ Z be a Delaunay random walk start-ing at o on P o where P is a constant intensity Poisson point process on a Riemanniansymmetric space M with base point o .Then, under the degree biased probability, the distribution of ( P o , ( x n ) n ∈ Z ) is thesame as that of ( g x P o , ( g x x n +1 ) n ∈ Z ) .In particular, the sequence ( x n ) n ∈ Z is distance stationary under the degree biasedprobability.Proof. In all of what follows we will work under the probability P dg . Let z n = g x x n +1 and Q o = g x P o . By Proposition 3 the distribution of ( P o , x ) is the sameas that of ( g x P o , x ) = ( Q o , z − ). Hence z − is a uniformly distributed randomneighbor of o in Q o .By the Markov property, the conditional distribution of ( x n ) n ≥ given P o and x ,is that of a Delaunay random walk starting at x on P o . Applying g x one obtainsthat the conditional distribution of ( z n ) n ≥ given Q o and z − , is that of a Delaunayrandom walk starting at o on Q o . In particular, z is a uniformly distributed randomneighbor of o in Q o which is independent from z − conditioned on Q o .By definition, the conditional distribution of ( x − n ) n ≥ given P o and x is thatof a Delaunay random walk starting at o on P o . Applying g x one obtains thatthe conditional distribution of ( z − n ) n ≥ given Q o and z − is a Delaunay randomwalk starting at z − on Q o . This implies that ( z − n ) n ≥ is a Delaunay random walk20tarting at o in Q o which is independent from ( z n ) n ≥ .Notice that ( d ( x m , x n )) m,n ∈ Z has the same distribution as ( d ( z m , z n )) m,n ∈ Z , andby definition one has d ( z m , z n ) = d ( g x x m +1 , g x x n +1 ) = d ( x m +1 , x n +1 )where the last equality follows because g x is an isometry. This shows that ( x n ) n ∈ Z is distance stationary as claimed. Part III
Zero-one laws forPoisson–Delaunay random walks
We maintain the notation from Part II As before, M denotes a Riemannian sym-metric space, o ∈ M a base point, and P a homogeneous Poisson point process in M .We will now investigate various 0–1 laws. Our primary motivations are to showthat certain asymptotics of Poisson–Delaunay random walks are deterministic: through-out this section, we will let ( x n ) n ∈ Z be a Delaunay random walk on P o starting at o. Before delving deeper, we introduce some of the statistics we would like to say aredeterministic.
Graph speed
Let d G ( x, y ) denote the graph distance between x, y ∈ P o in the Delaunay graph of P o . By stationarity of the Poisson-Delaunay random graph under the degree biasedprobability (see [Paq17, Theorem 1.4] and [BC12, Proposition 2.5]) the graph speed (cid:96) G = lim n → + ∞ n d G ( x , x n )exists almost surely and in L . Furthermore (cid:96) G ≤ d G ( x , x ) = 1 almostsurely. Ambient speed
By [Paq17, Lemma 3.2, Theorem 3.3] one has that deg( o ) has moments of all order,and d ( x , x ) has all exponential moments, therefore one obtains that E dg ( d ( x , x )) < ∞ . Hence, by distance stationarity (Theorem 4) the ambient speed (cid:96) = lim n → + ∞ n d ( x , x n )also exists almost surely and in L under the degree biased probability. Asymptotic random walk entropy
Let p n ( x, y ) denote the probability that a Delaunay random walk on P o starting at x will arrive at y after n steps, conditional on P o . We define the asymptotic entropy h = lim n → + ∞ − n log( p n ( o, x n ))where ( x n ) n ≥ is a Delaunay random walk on P o starting at o . The limit existsalmost surely and in L by Kingman’s subadditive ergodic theorem. Recall that Discrete ( M ) denotes the space of discrete subsets of M . The topologyon Discrete ( M ) is that of weak convergence of point measures, by this we meanthat a neighborhood of a discrete set X ∈ Discrete ( M ) may be defined by pickingan open set U ⊂ M whose boundary is disjoint from X , a positive number (cid:15) , andconsidering all discrete subsets X (cid:48) with the same number of points in U as X andsuch that the Hausdorff distance between X (cid:48) ∩ U and X ∩ U is less than (cid:15) .Let π r : Discrete ( M ) → Discrete ( M ) be the mapping π r ( X ) = X \ B r where B r is the open ball centered at o with radius r . Let F r be the σ -algebra of Borelsubsets of Discrete ( M ) generated by π r . We define the tail σ -algebra by the equation F ∞ = (cid:84) r> F r .Informally the σ -algebra F r only allows one to distinguish events that happenoutside of the ball B r while the tail sets are characterized by properties of a discreteset X ∈ Discrete ( M ) which do not depend on any bounded subset of X . Forexample, the family of discrete subsets such that lim r → + ∞ | X ∩ B r | /V ( r ) exists is atail subset. Lemma 2 (Spatial zero-one law) . All tail Borel subsets of Discrete ( M ) are trivialfor the Poisson point process (that is they have probability equal to either or ).Proof. This is a corollary of Kolmogorov’s zero-one law.To see this let A n = B n +1 \ B n , and notice that if P is a Poisson process then thepoint processes P n = P ∩ A n are independent. Any event of the form { P ∈ T } with22igure 4: A Voronoi flower in the hyperbolic plane. T a tail subset of Discrete ( M ) belongs to the tail σ -algebra of the sequence P n andis therefore trivial by Kolmogorov’s zero-one law.Two graphs X and X (cid:48) are said to be finite perturbations of one another if thereexist finite subsets K ⊂ X and K (cid:48) ⊂ X (cid:48) such that X \ K and X (cid:48) \ K (cid:48) are isomorphic(here one removes the sets K and K (cid:48) and all edges having an endpoint in them).By Lemma 1 the following result implies that any property of the Poisson-Delaunaygraph which is stable under finite perturbations, is a tail event for the underlyingPoisson process. Lemma 3.
If two discrete sets which intersect all horoballs, coincide outside ofa bounded subset of M then their Delaunay graphs are finite perturbations of oneanother.Proof.
Let X and X (cid:48) be two discrete subsets of M with bounded Voronoi cells and r > X \ B r = X (cid:48) \ B r .By a Voronoi flower of a point x in X we mean an open set containing an open diskwhich is disjoint from X and contains x and y on its boundary for each Delaunayneighbor y of x in X . Similarly we define a Voronoi flower for a point in X (cid:48) . SeeFigure 4.Let us say a point x ∈ X is good if there is a Voronoi flower for x which is disjointfrom B r . Similarly we say a point x ∈ X (cid:48) is good if it admits a Voronoi flower withrespect to X (cid:48) which is disjoint from B r .We claim that that the set of good points with respect to X and X (cid:48) coincide. Tosee this notice that if x ∈ X is good, it must lie outside of B r and furthermore allits neighbors do as well. Hence x and all its neighbors are in X (cid:48) . Furthermore theVoronoi flower for x with respect to X which is disjoint from B r is also a Voronoiflower for x with respect to X (cid:48) . This establishes the claim (by symmetry).23urthermore notice that if two good points are Delaunay neighbors in X they arealso Delaunay neighbors in X (cid:48) .Hence to establish the lemma it remains only to show that the set of good pointshas a finite complement in X and X (cid:48) . We give the argument for X only since theclaim is symmetric with respect to exchanging X and X (cid:48) .Suppose for the sake of contradiction that there exists a sequence of distinct points x n ∈ X which are not good. Then for each n , every Voronoi flower of x n intersects B r . In particular, for each n , some disk with x n (and a certain Delaunay neighbor)on its boundary must be disjoint from X and intersect B r . The existence of such asequence of disks contracts the fact that X intersects all horoballs. Hence the setof good points is cofinite in X (and by the same argument also in X (cid:48) ). Let F sym ⊂ F be the σ -algebra of angularly invariant events, that is A ∈ F sym implies that for all isometries ψ of M in the stabilizer of o, P dg ( { ( P o , ( x n ) n ∈ Z ) ∈ A }(cid:52){ ( ψ ( P o ) , ( ψ ( x n )) n ∈ Z ) ∈ A } ) = 0 . For each ( X, ( y n ) n ∈ Z ) ∈ Discrete ( M ) × M Z such that y n ∈ X for all n , and y = o ,define T ( X, ( y n ) n ∈ Z ) = ( g y X, ( z n ) n ∈ Z )where z n = g y ( x n +1 ) for all n and g x denotes the central symmetry exchanging o and x .Notice that by Theorem 4, the transformation T preserves the distribution of( P o , ( x n ) n ∈ Z ). We will show that this measure is ergodic for the restriction of T to F sym . Theorem 5 (Ergodicity) . Let ( x n ) n ∈ Z be a Delaunay random walk starting at o on P o where P is a stationary Poisson point process on a Riemannian symmetric space M with base point o . With T as above there are P dg –nontrivial T –invariant eventsin F sym if and only if M is compact. Remark 1.
We do not use any special features of P being Poisson. This theoremalso holds for tail–trivial distributional lattices, as defined in [Paq17]. Before proceeding to the proof, we give a corollary.
Corollary 3.
For a Poisson–Delaunay random walk on a Riemannian symmetricspace, (cid:96), (cid:96) G , and h are all deterministic. roof. We discuss some details of the statement for (cid:96).
Similar arguments show theclaim for the other quantities. Recall that (cid:96) is the limit (cid:96) = lim n → + ∞ n d ( x , x n ) , whose existence was guaranteed to exist by the subadditive ergodic theorem. Ob-serve that (cid:96) = lim n → + ∞ n d ( x , x n ) = lim n → + ∞ n − d ( x , x n ) = lim n → + ∞ n − d ( g x ( x ) , g x ( x n )) . So (cid:96) is T –invariant. As (cid:96) carries no angular information, we have that (cid:96) is deter-ministic when M is noncompact. If M is compact, then (cid:96) = 0 as the diameter ofthe manifold is finite. Proof of Theorem 5. If M is compact, then the number of points in P o has thedistribution of 1 + X where X is Poisson with mean λ · Vol( M ) . Hence | P o | = k forany k > T –invariant and angularly invariant event.If M is noncompact, then the number of points in P o is almost surely infinite.Hence, its Poisson–Delaunay graph is infinite. Let E ∈ F sym be an arbitrary T –invariant event, so that P dg ( E (cid:52) T ( E )) = 0 . For k ∈ Z , let F k = σ (( x n ) n ≤ k , P o ) . The martingale Z k = E dg ( E |F k ) is uniformlyintegrable, and hence Z k → E as k → ∞ almost surely. On the other hand, usingthat T is measure preserving, we have an equality in distribution E dg ( E |F k ) (( P o , ( x n ) n ≤ k )) ( d ) = E dg ( E |F k ) (( g x P o , g x ( x n ) n ≤ k +1 )) . Changing variables, we can express E dg ( E |F k ) (( g x P o , g x ( x n ) n ≤ k +1 )) = E dg (cid:0) T − ( E ) |F k +1 (cid:1) (( P o , ( x n ) n ≤ k +1 )) . Using invariance of E, we conclude that E dg ( E |F k ) (( P o , ( x n ) n ≤ k )) ( d ) = E dg ( E |F k +1 ) (( P o , ( x n ) n ≤ k +1 )) . Therefore, on taking k → ∞ , we conclude that E dg ( E |F k ) ∈ { , } almost surely,for each k ∈ Z , which implies that E is measurable with respect to F k up tomodification by a P dg -null set.The same argument shows that E dg ( E | σ ( P o , ( x n ) n ≥ k )) ∈ { , } almost surely,and so E is measurable with respect to σ ( P o , ( x n ) n ≥ k ). As the left and right tails of( x n ) n ∈ Z are independent given P o , it must be that E is in σ ( P o ) up to modificationby a P dg –null set.In particular, there is some Borel set A in Discrete ( M ) so that E = { P o ∈ A } upto P dg –null events. Invariance of E implies that for each n ∈ Z , T n ( P o , ( x k ) x ∈ Z ) ∈ E = P o ∈ A dg –a.s.For any path p = u u u . . . u k − u k with u = o in the Delaunay graph, let g ∗ p bethe isometry of M defined inductively by g ∗ p = g g ∗ q ( u k ) ◦ g ∗ q where q is the path u u u . . . u k − and g ∗ u = Id . Observe that g ∗ p is an isometrythat takes u k to o, and therefore that g ∗ p ◦ g − u k is in the stabilizer of o. Taking conditional expectations with respect to σ ( P o ) , we can write P o ∈ A = E dg ( T n ( P o , ( x k ) x ∈ Z ) ∈ E | σ ( P o ))= (cid:88) p g ∗ p ( P o ) ∈ A P dg (( x k ) nk =0 = p | σ ( P o )) P dg –a.s., where the sum is over all paths p started from o in the Delaunay graph on P o . As these are indicators and P dg (( x k ) nk =0 = ( · ) | σ ( P o )) is a probability measure onpaths, it follows that P o ∈ A = g ∗ p ( P o ) ∈ A P dg –a.s. for all paths p of length n started at o. As n is arbitrary, and each pointin P o can be reached with positive probability by ( x n ) n ∈ Z , we conclude by angularinvariance of A that P o ∈ A = g u ( P o ) ∈ A P dg –a.s. for all u ∈ P o . As this holds P dg –a.s. it also holds P –a.s.Fix K > . For any (cid:15) > , we may approximate A by A (cid:48) in the Borel algebraof Discrete ( B R ( o )) with P ( { P o ∈ A (cid:48) }(cid:52){ P o ∈ A } ) < (cid:15) for R sufficiently large. Let y ∈ P o be the point minimizing d ( o, y ) among P o \ B R + K ( o ) . Then by stationarityof P P ( { g y ( P o ) ∈ A (cid:48) }(cid:52){ g y ( P o ) ∈ A } ) < (cid:15) Hence by invariance of A P ( { g y ( P o ) ∈ A (cid:48) }(cid:52){ P o ∈ A } ) < (cid:15) Therefore, we have approximated A by an event measurable with respect to F K . As (cid:15) and K were arbitrary, it follows that { P o ∈ A } is in the tail F ∞ up to modificationby a null–set. 26 art IV Graph and ambient speed ofPoisson-Delaunay random walks
We maintain the notation of Part II and Part III but restrict ourselves from now onto the case where M is a Riemannian symmetric space of non-compact type.The point of this section is to show that the ambient speed (cid:96) and graph speed (cid:96) G of the Poisson-Delaunay random walk are zero or positive simultaneously. Proposition 4 (Graph and ambient speed comparison) . For any Poisson-Delaunayrandom walk on a Riemannian symmetric space of non-compact type one has (cid:96) G = 0 almost surely if and only if (cid:96) = 0 almost surely.Proof. We will begin by showing that if (cid:96) G = 0 almost surely then (cid:96) = 0 almostsurely.By [Paq17, Proposition 4.1] there exist positive constants t and δ depending onlyon M and λ such that P (cid:0) B Gn (cid:54)⊆ B tn (cid:1) ≤ e − ne δt for all n = 1 , , . . . and t > t , where B Gn denotes the graph ball of radius n centeredat o in P o with respect to d G , and B r the ball of radius r centered at o in M withrespect to d .By the Borel-Cantelli Lemma one has for any fixed t > t that B Gn ⊆ B tn for all n large enough almost surely. This implies that t(cid:96) G ≥ (cid:96) almost surely. Hence, wehave shown that if (cid:96) G = 0 almost surely then (cid:96) = 0 almost surely as claimed.We will now show that if (cid:96) = 0 almost surely then (cid:96) G = 0 almost surely.Recall that the Poisson-Delaunay graph is stationary under degree biased proba-bility. Furthermore since B Gn ⊆ B tn for all n large enough one has thatlim sup n → + ∞ n log (cid:0) | B Gn | (cid:1) ≤ lim n → + ∞ n log ( | P o ∩ B tn | ) < + ∞ so the Poisson-Delaunay graph has finite exponential growth almost surely.By [CPL16, Lemma 5.1] one has (cid:96) G ≤ h almost surely. Hence, it suffices toestablish that h = 0 almost surely to obtain that (cid:96) G = 0 almost surely. We will in27act show that the conditional expectation of h given P o is 0, which suffices because h is non-negative.From L convergence one obtains E ( h | P o ) = lim n → + ∞ n (cid:88) x ∈ P o − p n ( o, x ) log( p n ( o, x ))almost surely.Since (cid:96) = 0 one may choose a deterministic sequence r n → + ∞ such that r n = o ( n ) such that p n = P ( x n ∈ B r n ) → n → + ∞ .Conditioning on the event that x n ∈ B r n , and using the fact that the entropy of arandom variable is at most the logarithm of the number of distinct possible values,one obtains (cid:88) x ∈ P o − p n ( o, x ) log( p n ( o, x )) ≤ p n log ( | P o ∩ B r n | ) + (1 − p n ) log (cid:0) | B Gn | (cid:1) . To bound the first term on the right hand side notice that | P o ∩ B r | /V ( r ) → λ almost surely when r → + ∞ , where V ( r ) denotes the volume of the ball of radius r in M . This implies that | P o ∩ B r n | = (1 + o ( n )) λV ( r n ).Finally, since V ( r ) ≤ e br for all r large enough one obtains that log( | P o ∩ B r n | ) = O ( r n ) = o ( n ).For the second term on the right hand side above we use the previously establishedfact that n log (cid:0) | B Gn | (cid:1) = O (1), which immediately implies (since 1 − p n goes to 0)that the term is o ( n ).Hence we have shown that h = 0 almost surely from which (cid:96) G = 0 almost surelyas claimed. Part V
Hyperbolic Poisson-Delaunayrandom walks
The purpose of this part of the article is to establish an estimate for the speed ofPoisson-Delaunay random walks in hyperbolic space when the intensity of the pointprocess is small.In what follows H d denotes d -dimensional hyperbolic space, and o some fixedbase point. We assume that we have, defined on the same probability space, for28ach λ > P λ on H d with intensity λ . One way to do this isto let P be a unit intensity Poisson process on R × H d and let P λ be the projectiononto H d of P ∩ [0 , λ ] × H d (this is a ‘Poisson rain’ process as discussed in [Kin93,pg. 57]).For each λ we let P λ be the degree biased probability defined by P λ , and use E λ ( · )to denote expectation relative to this probability.We assume that, on the same probability space, there are defined for each λ aDelaunay random walk ( x n,λ ) n ∈ Z on P λ ∪ { o } starting at o . And that there exists arandom variable u which is uniformly distributed on [0 ,
1] and independent from allthe previously defined random objects.Let (cid:96) λ denote the speed of ( x n,λ ) n ∈ Z and (cid:96) G,λ its graph speed. By Corollary 3both speeds are almost surely constant.We will fix from now on for each λ a random horofunction ξ λ given by theFurstenberg type formula for speed established in Theorem 3 applied to the se-quence ( x n,λ ) n ∈ Z . We will now state our main result and give the proof asuming some results whichwill be established later on.
Theorem 6 (Speed asymptotics for low intensity) . The speed of the Poisson-Delaunayrandom walk on H d is almost surely constant for each λ and satisfies the followingasymptotic: lim λ → (cid:96) λ d − log( λ − ) = 1 . Proof.
By Corollary 3 the speed (cid:96) λ is almost surely constant.By the Furstenberg type formula for speed established in Theorem 3 we have, foreach λ , a random horofunction ξ λ such that (cid:96) λ = E λ ( (cid:96) λ ) = − E λ ( ξ λ ( x ,λ )) . Since all horofunctions are 1-Lipschitz one obtains | E λ ( ξ λ ( x ,λ )) | ≤ E λ ( d ( o, x ,λ )) . We will show in Theorem 7 that the right hand side is equivalent to d − log( λ − )when λ →
0. 29o show that this upper bound is nearly optimal when λ is small we write − E λ ( ξ λ ( x ,λ )) = E λ ( d ( o, x ,λ )) − E λ ( ξ λ ( x ,λ ) + d ( o, x ,λ )) . It remains to show that the second term on the right hand side is small relativeto the first one.For this purpose first notice that, since ξ ( x ) + d ( o, x ) ≥ ξ one has 0 ≤ E λ ( ξ λ ( x ,λ ) + d ( o, x ,λ )) . To obtain an upper bound for this expected value, we must first show that ξ λ isalmost surely a boundary horofunction for each fixed λ .To see this notice that, since the Delaunay graph of P λ ∪ { o } is connected andinfinite, the simple random walk on it cannot be positively recurrent (i.e. spenda positive fraction of its time in a finite set of vertices). Also, since this graphembedded in H d with only finitely many vertices in each bounded subset the claimfollows from Proposition 1.Setting f ξ ( x ) = ξ ( x ) + d ( o, x ), it is now possible to use the following worst casebound E λ ( ξ λ ( x ,λ ) + d ( o, x ,λ )) ≤ E ( | N λ | ) − E (cid:32) max ξ (cid:88) x ∈ N λ f ξ ( x ) (cid:33) where the maximum is over all boundary horofunctions ξ , and N λ is the set ofneighbors of o in P λ ∪ { o } .We will show in Lemma 6 that E ( | N λ | ) − ≤ Cλ for some constant C > λ small enough. We will also prove later on in Theorem 8 that E (cid:32) max ξ (cid:88) x ∈ N λ f ξ ( x ) (cid:33) = o (cid:0) λ − log( λ − ) (cid:1) , when λ →
0. Combining these two results completes the proof.Combined with the comparison of graph and ambient speeds (Proposition 4),the theorem above yields an alternate proof that the hyperbolic Poisson-Delaunayrandom walk has positive graph and ambient speed almost surely if the intensity issmall enough.A more general result (in particular valid for all intensities) is established in[Paq17] by showing that the Delaunay graph is invariantly non-amenable and usingthe theory of unimodular random graphs. Here we rely instead on the distancestationarity of the random walk and the Furstenberg type formula for speed. Theoverlap between the two proofs is the need for some estimates on the number ofneighbors of the root, the distance to the neighbors, and some exponential boundon the growth of the Delaunay graph. 30 orollary 4 (Positive speed for low intensities) . For all λ small enough both (cid:96) λ and (cid:96) G,λ are almost surely positive.
Theorem 6 also allows one to show that the graph speed goes to 1 (its maximumpossible value) as the intensity goes to zero. This answers a question posed in[BPP14].
Corollary 5 (Graph speed for small intensities) . For the Poisson-Delaunay randomwalk on H d , one has (cid:96) G,λ → as λ → .Proof. Recall that by Corollary 3 the graph speed (cid:96)
G,λ is almost surely constant foreach λ .By [Paq17, Proposition 4.1] for each λ > t λ > n centered at o in P λ ∪ { o } is contained in the metricball B t λ n for all n large enough. From this one obtains t λ (cid:96) G,λ ≥ (cid:96) λ almost surely.Notice that the isoperimetric constant of H d is d −
1. Therefore, by [Paq17,Proposition 4.1], for any positive α < d − t λ = α log( λ − )+ o (log( λ − )) when λ → (cid:96) G,λ → The purpose of this section is to prove the following result, which was used in theproof of Theorem 6 (we use f ∼ g to mean that f /g converges to 1), Theorem 7.
For each α > one has E λ ( d ( o, x ,λ ) α ) ∼ (cid:18) d − λ − ) (cid:19) α when λ → . As a first step towards the proof of Theorem 7 we will estimate the connectionprobability of two points in the Poisson-Delaunay triangulation. This will allow usto obtain the asymptotic behavior of the number of Delaunay neighbors of the root o when the intensity goes to 0 (which appears implicitly whenever one calculates anexpected value with respect to the degree biased probability).In what follows we use x ∼ y to mean that x and y are Delaunay neighbors insome discrete set under consideration, and V ( r ) to denote the volume of the ball ofradius r in H d (we use the convention that V ( r ) = 0 if r is negative).31 emma 4. There is a positive constant r such that for all λ > one has e − λV ( r/ − r ) ≥ P ( x ∼ o in P λ ∪ { o, x } ) ≥ e − λV ( r/ , where r = d ( o, x ) . The lower bound follows from the observation that if the open ball W , withdiameter given by the geodesic segment [0 , x ] , contains no points of P λ , then x ∼ o .The probability that W ∩ P λ = ∅ is e − λV ( r/ giving the lower bound.In order to bound from above the probability that a given point x is a Delaunayneighbor of o in P λ ∪{ o, x } , we will use the fact that this implies that the intersectionof all balls containing both o and x is disjoint from P λ .Basic hyperbolic geometry implies that the volume of this set is of order V ( r/ r = d ( o, x ). The result could be obtained by applying [CN07, Proposition14] from which one obtains immediately that the set contains a ball of radius r − δ where δ > H d . We give an independent (moreelementary) proof here. Lemma 5 (The intersection of balls containing two points is thick) . There existsa positive constant r such that for all p, q in H d the ball of radius d ( p, q ) / − r centered at the midpoint m of p and q is contained in all open balls having p and q on their boundary. Before embarking on this proof, we note that this will complete the proof of Lemma 4,since the fact that o ∼ x in P λ ∪ { o, x } implies that a ball of volume V ( r/ − r ) isdisjoint from P λ . Proof.
We will show that the proposition is valid for any r such that tanh( r ) r ≥ log(8) (for example r = 3 will suffice). In what follows E ( p, q ) denotes the inter-section of all open balls having p and q on their boundary.First, observe that it is enough to prove the two dimensional case. In fact, supposethere is a point z in a ball B r ( m ) which is not in E ( p, q ). Consider the embeddedhyperbolic plane H passing through p, q and z , and let E H ( p, q ) be the intersectionof all hyperbolic disks of H having p and q on their boundary. Since z is not in E ( p, q ), there exists a ball B having p, q on its boundary that does not contain z .The intersection B ∩ H is an Euclidean disk, and therefore, a hyperbolic disk of H having p and q on its boundary that does not contain z . Also, the intersection B Hr ( m ) = B r ( m ) ∩ H is a hyperbolic disk centered at m of radius r in H . This showsthat z belongs to B Hr ( m ) \ E H ( p, q ). Therefore, if the statement hold for some r in H , it also holds for the same r in H d .In the upper half plane model of the hyperbolic plane we assume from now on thatthe points are p = ( − x/ ,
1) and q = ( x/ , r = d ( p, q ) statisfies32igure 5: The intersection of all balls containing the points p and q on theirboundary, with d ( p, q ) ≥ r , contains a ball of radius d ( p, q ) / − r .cosh( r ) = 1 + x /
2. Using the fact that hyperbolic disks in the upper half planemodel are simply Euclidean disks which do not meet the boundary, one obtains that E ( p, q ) is the set of points with y ≥ D passingthrough (0 , p and q . See Figure 5.Let (0 , e t ) be the center of D . The Euclidean disk B whose diameter is the segmentjoining (0 ,
1) to (0 , e t ) is contained in E ( p, q ). But B is also a hyperbolic disk, withthe same segment as a diameter and centered at m . This follows since the inversionwith respect to C , the circle passing through p and q which is ortogonal to theboundary of H , exchanges D and the horizontal line { y = 1 } , and thus d ((0 , e t ) , m ) = log (cid:0) e t /r (cid:1) = log ( r /
1) = d ( m, (0 , , where r is the Euclidean radius of C . The hyperbolic radius of B is (log(2) + t ) / t ≥ r − r . From the fact that (0 , e t ) is equidistant (withrespect to the Euclidean distance) to (0 ,
0) and ( x/ , e t = 1 + x . From this it follows that cosh( r ) / cosh( t ) ≤
8. But using the intermediate valuetheorem for log(cosh( t )), we havelog(cosh( r )) − log(cosh( r − r )) ≥ tanh( r − r ) r ≥ tanh( r ) r ≥ log(8) . This implies, since cosh is increasing on the positive reals, that t ≥ r − r asclaimed. 33 .2 Expected number of neighbors As a first application of Lemma 4 we obtain the asymptotic behavior of the expecteddegree of the root in the Poisson-Delaunay graph as the intensity goes to zero.Before proving the result, let us record some estimates on the behavior of thevolume growth function V ( r ) and its derivative v ( r ) = V (cid:48) ( r ).First, observe that both v ( r ) e − r ( d − and V ( r ) e − r ( d − converge to positive con-stants as r → ∞ . Further, both are continuous functions on [0 , ∞ ) which vanishonly at 0 as r d and r d − respectively.Hence, v ( r ) is bounded from below by a positive multiple of V ( r/ v ( r/
2) on allits domain.From now on, when comparing positive functions f and g we will write f (cid:46) g tomean that f is bounded from above by a positive multiple of g in the domain underconsideration. We write f ≈ g when f (cid:46) g (cid:46) f .In particular, we have established above that v ( r ) (cid:38) V ( r/ v ( r/
2) on [0 , + ∞ ]. Lemma 6.
For all λ small enough one has E ( | N λ | ) ≈ λ − where N λ is the set of Delaunay neighbors of o in P λ ∪ { o } .Proof. By Slivnyak’s formula and Lemma 4 one has E ( | N λ | ) = (cid:90) P ( x ∼ o in P λ ∪ { o, x } ) dx ≥ λ (cid:90) ∞ e − λV ( r/ · v ( r ) dr (cid:38) λ (cid:90) ∞ e − λV ( r/ · V ( r/ v ( r/ d ( r/ λ − (cid:90) ∞ ue − u du = λ − . This establishes to the lower bound.For the upper bound we begin again using Slivnyak’s formula and Lemma 4 toobtain E ( | N λ | ) ≤ λ (cid:90) ∞ e − λV ( r/ − r ) · v ( r ) dr = λ (cid:90) r v ( r ) dr + λ (cid:90) + ∞ r e − λV ( r/ − r ) v ( r ) dr = λV (4 r ) + λ (cid:90) + ∞ r e − λV ( r/ v ( r + 2 r ) dr. λ . For the second term weuse the fact that v ( r + 2 r ) (cid:46) V ( r/ v ( r/
2) on [2 r , + ∞ ) to obtain λ (cid:90) + ∞ r e − λV ( r/ v ( r + 2 r ) dr (cid:46) λ (cid:90) + ∞ r e − λV ( r/ V ( r/ v ( r/ dr/ λ (cid:90) + ∞ V ( r ) e − λu udu ≈ λ − , which concludes the proof. The last tool we will need in order to prove Theorem 7 is an estimate showingthat we can ignore neighbors outside of a neighborhood of the sphere of radius R λ = d − log( λ − ).Recall that N λ denotes the set of Delaunay neighbors of o in P λ ∪ { o } . In whatfollows, given M >
0, we define N λ,M = { x ∈ N λ : | d ( o, x ) − R λ | < M } . Lemma 7.
For each α > and (cid:15) > there exists M > such that E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) α ≤ (cid:15)λ − log( λ − ) α for all λ small enough.Proof. By Slivnyak’s formula and Lemma 4 one has E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) α = λ (cid:90) | d ( o,x ) − R λ |≥ M d ( o, x ) α P ( x ∼ o in P λ ∪ { o, x } ) dx ≤ λ (cid:90) | r − R λ |≥ M r α e − λV ( r/ − r ) v ( r ) dr ≤ λ (4 r ) α V (4 r ) + λ (cid:90) | r − R λ |≥ Mr ≥ r r α e − λV ( r/ − r ) v ( r ) dr. The first term above goes to zero and therefore can be ignored.35o control the second term we first observe, repeating the argument from Lemma6, that λ (cid:90) R λ − M r r α e − λV ( r/ − r ) v ( r ) dr ≤ λR αλ (cid:90) R λ − M − r r e − λV ( r/ v ( r + 2 r ) dr (cid:46) λR αλ (cid:90) R λ − M − r r e − λV ( r/ V ( r/ v ( r/ dr/ (cid:46) λR αλ (cid:90) a λ e − λu udr (cid:46) λ − log( λ − ) α (cid:90) λa λ e − u udr, where a λ = V (cid:0) R λ − M − r (cid:1) .Notice that lim λ → λa λ exists and goes to 0 when M → + ∞ . Hence, given (cid:15) > M sufficiently large so that λ (cid:90) R λ − M r r α e − λV ( r/ − r ) v ( r ) dr ≤ (cid:15)λ − log( λ − )for all λ small enough.Similarly one has, using that r (cid:46) log( V ( r/ λ (cid:90) + ∞ R λ + M r α e − λV ( r/ − r ) v ( r ) dr (cid:46) λ (cid:90) + ∞ R λ + M − r log( V ( r/ α e − λV ( r/ V ( r/ v ( r ) dr/ λ (cid:90) + ∞ b λ log( u ) α e − λu udu = λ − (cid:90) + ∞ λb λ log( u/λ ) α e − u udu (cid:46) λ − (cid:90) + ∞ λb λ (log( u ) α + log( λ − ) α ) e − u udu, where b λ = V (cid:0) R λ + M − r (cid:1) , and in the final step we have used that ( a + b ) α ≤ C α ( a α + b α ) for all a, b > C α depending only on α .Once again, lim λ → λb λ exists, but it goes to + ∞ when M → + ∞ . Using this oneobtains that, given (cid:15) >
0, there exists M large enough so that λ (cid:90) + ∞ R λ + M r α e − λV ( r/ − r ) v ( r ) dr ≤ (cid:15)λ − log( λ − ) α for all λ small enough. 36ombining the results above we have shown that, given (cid:15) >
0, one may choose M large enough so that E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) α ≤ (cid:15)λ − log( λ − )for all λ small enough, as claimed. To conlude the section we prove Theorem 7.Recall that R λ = d − log( λ − ), and that by Lemma 6, there exists a constant C > C − λ − ≤ E ( | N λ | ) ≤ Cλ − . Also, given M > N λ,M to be the set of neighbors of o which are at distance between R λ − M and R λ + M from o .By definition of the degree biased probability one has E λ ( d ( o, x ,λ ) α ) = E ( | N λ | ) − E (cid:32) (cid:88) x ∈ N λ d ( o, x ) α (cid:33) . For the lower bound notice at least | N λ | − | P λ ∩ B R λ | neighbors of o are at distancegreater than R λ from o . Combined with the bounds on E ( | N λ ) above one obtains E ( | N λ | ) − E (cid:32) (cid:88) x ∈ N λ d ( o, x ) α (cid:33) ≥ R αλ − E ( | N λ | ) − E ( | P λ ∩ B R λ | ) ≥ R αλ − C − λ V ( R λ )= R αλ + O (1)For the upper bound, notice that for all M > E ( | N λ | ) − E (cid:32) (cid:88) x ∈ N λ d ( o, x ) α (cid:33) ≤ ( R λ + M ) α + E ( | N λ | ) − E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) α ≤ ( R λ + M ) α + C − λ E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) α By Lemma 7, given (cid:15) > M so that C − λ E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) α ≤ C − (cid:15) log( λ − ) α λ small enough.This shows that lim sup λ → E λ ( d ( o, x ,λ ) α ) R αλ ≤ C − ( d − α α (cid:15), for all (cid:15) >
0. From which the theorem follows immediately.
The purpose of this section is to complete the proof of Theorem 6 by establishingthe following (recall that f = o ( g ) means that f /g converges to 0): Theorem 8.
One has E (cid:32) max ξ (cid:88) x ∈ N λ f ξ ( x ) (cid:33) = o (cid:0) λ − log( λ − ) (cid:1) when λ → , where the maximum is over all boundary horofunctions and f ξ ( x ) = ξ ( x ) + d ( o, x ) .Proof. Recall N λ denotes the set of Delaunay neighbors of o in P λ ∪ { o } and that foreach M > N λ,M as the set of neighbors x ∈ N λ with | d ( o, x ) − R λ |
0, we may split the sum and use f ξ ( x ) ≤ d ( o, x ) toobtain E (cid:32) max ξ (cid:88) x ∈ N λ f ξ ( x ) (cid:33) ≤ E (cid:88) x ∈ N λ \ N λ,M d ( o, x ) + E max ξ (cid:88) x ∈ N λ,M f ξ ( x ) Let (cid:15) > M such that the firstterm on the right hand side above is bounded by (cid:15)λ − log( λ − ).To bound the second term we will split it into a sum on points belonging to asmall cone, and points where f ξ is small.To make this precise denote by D θ ( v ) the geodesic cone with radius θ > o and direction v , where v is a unit tangent vector at o . By this we meanthe set of points of the form exp o ( tw ) for some t > w at o forming an angle less than θ with v (here exp o denoting the Riemannianexponential map at o ). 38n Lemma 9 we will show there exists a function r (cid:55)→ θ r , satisfying θ r ≈ e − r/ when r → + ∞ , such that for each boundary horofunction ξ the set of points where f ξ > r is contained in a some cone of the form D θ r ( v ξ ).Applying this to r = (cid:15)R λ , and splitting the sum among points where f ξ > r andthe rest, one obtains E max ξ (cid:88) x ∈ N λ,M f ξ ( x ) ≤ (cid:15)R λ E ( | N λ | )+2( R λ + M ) E (cid:18) max v ∈ S d − | P λ ∩ D θ r ( v ) ∩ B R λ + M | (cid:19) , where abusing notation slightly S d − denotes the unit tangent sphere at o , and recallthat B r denotes the ball of radius r centered at o .By the definition of R λ and Lemma 6, the first term is bounded by Cd − (cid:15)λ − log( λ − )for all λ small enough, where the constant C does not depend on (cid:15) .Notice that projecting the points of P λ ∩ B R λ + M onto the unit sphere S d − at o along geodisic rays one obtains a Poisson point process on S d − with intensity µ = λV ( R λ + M ) times the normalized volume. This means that the quantitymax v ∈ S d − | P λ ∩ D θ r ( v ) ∩ B R λ + M | can be interpreted as the maximum number of pointsof such a Poisson process which can be found in a metric ball with radius α = θ r .In this situation we will show in Lemma 10 that, as long as µα d − remainsbounded away from zero, the expected number of such points is bounded by aconstant multiple of µα d − when µ → + ∞ . In our case this applies if (cid:15) < / E (cid:18) max v ∈ S d − | P λ ∩ D θ r ( v ) ∩ B R λ + M | (cid:19) (cid:46) λ − (1 − (cid:15) ) when λ → λ → λ − log( λ − ) E (cid:32) max ξ (cid:88) x ∈ N λ f ξ ( x ) (cid:33) ≤ (cid:18) Cd − (cid:19) (cid:15) for all (cid:15) >
0. Which concludes the proof.
Recall that given a boundary horofunction ξ we have defined f ξ ( x ) = ξ ( x ) + d ( o, x ).In this section we analize the level sets of f ξ to obtain a result needed in the proofof Theorem 8 above.We recall that the upper half plane model of the hyperbolic plane is obtainedidentifying H with { x + iy ∈ C : y > } with the metric y ( dx + dy ). In this39odel we will set the base point o = i . We will use explicit formulas for the distancefunction and horofunctions in this model, as well as the correspondence betweenthe horofunction boundary and points on the extended real line (see for example[Bon09, Excersices 2.2, 6.10, 6.11]). Lemma 8.
In the upper half plane model of the hyperbolic plane the function f ξ asociated to the boundary point at ∞ is given by f ( z ) = 2 log (cid:18) | z − i | + | z + i | (cid:19) . In particular f extends to all of C as a continuous function whose level sets areellipses with foci at ± i .Proof. The proof is by direct calculation. The horofunction asociated to the bound-ary point at ∞ is ξ ( x + iy ) = log( y ). The distance d ( o, x + iy ) can be calculatedexplicitely and is given by d ( o, x + iy ) = 2 log (cid:18) | z − i | + | z + i | √ y (cid:19) . The above calculation allows us to estimate the angular size of the level sets of f ξ as viewed from o . Lemma 9.
For each r > there exists θ r such that for all boundary horofunctions ξ on H d there exists a unit tangent vector v ξ at o such that the set { f ξ > r } iscontained in the cone D θ r ( v ξ ) . Futhermore, θ r ≈ e − r/ when r → + ∞ Proof.
Given ξ there is a unique unit speed geodesic α ( t ) = exp o ( tv ) such that ξ ( α ( t )) = t . The function f ξ is invariant under all rotations in H d fixing the points of α ( t ). Hence it suffices to prove the result in H (by considering the planes containing α ).In the upper half plane model of H we may assume that the geodesic α ( t ) dis-cussed above is α ( t ) = e t i . And therefore that ξ ( x + iy ) = log( y ). By Lemma 8 onehas f ξ ( z ) = f ( z ) = 2 log (cid:18) | z − i | + | z + i | (cid:19) . Let x r > f ( x r ) = r . It suffices to calculate the angle θ r at o between the geodesic ray α and the geodesic ray β starting at o whose endpoint is x r .For this purpose we use the conformal transformation z (cid:55)→ z − iz + i which maps theupper half plane to the unit disk. Notice that α goes to the segment [0 ,
1] under this40 − i { f = r } x r βαθ r Figure 6: Illustration of the proof of Lemma 9.transformation. On the other hand β goes to another radius of the unit disk. Hence,the angle θ r is the absolute value of the smallest argument of x r − ix r + i from which oneobtains θ r = 2 arctan (cid:18) x r (cid:19) . To conclude the proof one calculates from the equation f ( x r ) = r obtaining x r = √ e r − . We consider the unit sphere S d − in R d with its normalized volume measure m . Tocomplete the proof of Theorem 8 we need the following application of the Vapnik-Chervonenkis inequality (see [V ˇC71]) to Poisson point processes on S d − . Lemma 10.
Suppose that for each µ > one has a Poisson point process A µ withintensity µ on S d − , and for each µ a radius α µ is chosen such that α d − µ µ remainsbounded away from when µ → + ∞ . Then E (cid:18) max p ∈ S d − | A µ ∩ B α µ ( p ) | (cid:19) ≈ α d − µ µ when µ → + ∞ .Proof. The fact that µα d − µ (cid:46) E (cid:18) max p | A µ ∩ B α µ ( p ) | (cid:19) for all µ large enough is trivialsince one can pick a fixed ball of radius α µ for each µ and the number of points init bounds the maximum from below. 41o prove the upper bound notice that, by the Cauchy-Schwarz inequality, for all µ one has E (cid:18) max p | A µ ∩ B α µ ( p ) | (cid:19) ≤ E (cid:0) | A µ | (cid:1) E (cid:18) max p | A µ ∩ B α µ ( p ) | | A µ | (cid:19) (cid:46) µ E (cid:18) max p | A µ ∩ B α µ ( p ) | | A µ | (cid:19) . = µ E (cid:0) X µ (cid:1) , where X µ is the maximal proportion of points of A µ to be found in a ball of radius α µ .To bound the second moment of X µ we use the fact that, conditioned on | A µ | = n , the distribution of A µ is that of n i.i.d. uniform points. Hence, the Vapnik-Chervonenkis inequality implies P (cid:18)(cid:12)(cid:12) X µ − m ( B α µ ) (cid:12)(cid:12) > t (cid:12)(cid:12)(cid:12)(cid:12) | A µ | = n (cid:19) ≤ Ce − c nt for all t ≥
0, where c and C are positive constants.Using this, the explicit formula for E (cid:0) e tX (cid:1) when X is Poisson, and the inequality1 − e − x ≥ e − c x for x ∈ [0 , c ], one obtains that P (cid:0)(cid:12)(cid:12) X µ − m ( B α µ ( p )) (cid:12)(cid:12) > t (cid:1) ≤ E (cid:16) Ce − c | A µ | t (cid:17) = Ce − µ (1 − e − c t ) ≤ Ce − cµt , for all t ∈ [0 ,
2] where c = e − c c .Since one has a Gaussian tail bound (notice that for t / ∈ [0 ,
2] the probability onthe left hand side above is clearly 0) one obtains that for some C (cid:48) > E (cid:0) X µ (cid:1) = m ( B α µ ) + E (cid:0) | X µ − m ( B α µ ) | (cid:1) ≤ m ( B α µ ) + C (cid:48) √ µ = m ( B α µ ) (cid:32) C (cid:48) (cid:112) m ( B α µ ) µ (cid:33) , from which the desired upper bound follows immediately. Part VI
Dimension drop phenomena
The term dimension drop refers to the fact, that in many situations, the distributionof the first exit point or limit point on a boundary at infinity, associated to a random42alk has been observed to have smaller dimension than may be expected.As an example consider a Brownian motion X t starting at an interior point ofa simply connected domain bounded by a Jordan curve in the plane, and let τ bethe first time X t hits the boundary curve. The distribution of X τ is always onedimensional as shown by Makarov in [Mak85]. Hence, when the Jordan curve hasdimension larger than one, the dimension drop phenomena occurs.A second example is given by the simple random walk on certain rooted Galton-Watson trees. In this case the limit point of the walk on the boundary at infinity(which is the set of infinite rays starting at the root with a natural metric) alsoexhibits the dimension drop phenomena for almost every realization of the randomtree (see [LPP95], and also [Rou17] and the references therein).The purpose of this part of the article is to establish that, conditioned on thePoisson process, the limit point on the visual boundary of low intensity hyperbolicPoisson-Delaunay random walks exhibits the dimension drop phenomena almostsurely.With the same techinique we will also give some examples of co-compact Fuchsiangroups for which the limit point at infinity of the simple random walk exhibits thedimension drop phenomena. It might be the case that this type of dimension dropoccurs for all co-compact Fuchsian groups, but our proof does not adapt easily toshow this.
10 Tools for proving dimension drop
In this section we prove two results (Lemma 11 and Lemma 12) which allow one toshow that dimension drop occurs for certain measures on the boundary of hyperbolicspace.Recall that the dimension of a probability measure ν is the smallest exponent α such that there exists a set of ν full measure with dimension less than α . Equivalentlyit is the smallest exponent such that for ν almost every x one haslim inf r → log( ν ( B r ( x )))log( r ) ≤ α. In this subsection we will give a general result which is useful to bound the dimensionof the distribution of a limit of random variables from above.We assume fixed in this subsection a complete separable metric space, we use43 ( x, y ) to denote the distance between two points x, y , and B r ( x ) to denote theopen ball of radius r centered at x .Assume one has a sequence of random variables X n taking values in the givenmetric space and converging almost surely to a random variable X when n → + ∞ .Let ν n be the distribution of X n and ν that of X .The lemma below allows one to transfer information on the measures ν n to esti-mate the dimension of ν from above in certain circumstances.Elementary examples where the lemma is applicable are obtained by setting X n = (cid:88) k ≤ n σ k − k or X n = (cid:88) k ≤ n σ k − k where the σ k are i.i.d. with P σ k = 0 = P σ k = 1 = 1 /
2. In these examples, taking r n = 2 − n and r n = 3 − n respectively, the lemma gives the optimal upper boundsfor the dimensions of the distribution of the limit distributions (which are 1 andlog(2) / log(3) respectively). Lemma 11 (Dimension upper bound) . Assume there exists a positive exponent α ,positive random variables C and D ≤ , and a deterministic sequence of positiveradii r n , n = 1 , , . . . which converges to , such that almost surely d ( X n , X ) ≤ Cr n and ν n ( B r n ( X n )) ≥ Dr αn for all n. Then the dimension of ν is at most α .Proof. It suffices to prove the result in the case where the random variable C isconstant. Assuming this the general case follows by noticing that for each constant K the dimension of the distribution of X conditioned on the event C ≤ K is atmost α . Hence, there exists a set of ν measure P ( C ≤ K ) with dimension at most α . Since this is valid for all K one obtains that the dimension of ν is at most α asclaimed.We will now prove the result in the case where C is constant.For this purpose consider random variables X (cid:48) n , n = 1 , , . . . and X (cid:48) with the samejoint distribution as X n , n = 1 , , . . . and X , and independent from them.Using the independence of these two sets of random variables, and the fact that d ( X, X (cid:48) ) ≤ d ( X n , X (cid:48) n ) + 2 Cr n , one obtains that almost surely ν ( B (1+2 C ) r n ( X )) = P (cid:0) X (cid:48) ∈ B (1+2 C ) r n ( X ) | X (cid:1) ≥ P ( X (cid:48) n ∈ B r n ( X n ) | X )= E ( P ( X (cid:48) n ∈ B r n ( X n ) | X n , X ) | X )= E ( ν n ( B r n ( X n )) | X ) ≥ E ( D | X ) r αn . R n = (1 + 2 C ) r n , for ν almost every x there exists a positiveconstant (cid:15) x such that ν ( B R n ( x )) ≥ (cid:15) x R αn for all n .This shows that the dimension of ν is at most α as claimed. We will now prove a geometric result which allows one to apply Lemma 11 tohyperbolic random walks.
Lemma 12 (Speed of angular convergence) . Let ( x n ) n ≥ be a sequence in H d and ( θ n ) n ≥ the corresponding sequence of projections onto the unit tangent sphere at abase point o . If d ( x n , x n +1 ) = o ( n ) and d ( x , x n ) = (cid:96)n + o ( n ) for some (cid:96) > when n → + ∞ then the limit θ ∞ = lim θ n exists and furthermore for all (cid:96) (cid:48) < (cid:96) one has d ( θ n , θ ∞ ) < e − (cid:96) (cid:48) n for all n large enough.Proof. Given (cid:96) (cid:48) < (cid:96) and (cid:15) > d ( x , x n ) ≥ (cid:96) (cid:48) n and d ( x n , x n +1 ) ≤ (cid:15)n for all n large enough.Hence the geodesic joining x n and x n +1 does not intersect the ball of radius ( (cid:96) (cid:48) − (cid:15) ) n centered at the base point.The hyperbolic metric in polar coordinates is given by dr + sinh( r ) dθ where dθ is the metric on the unit tangent sphere at the base point. Since sinh( r ) ≥ ce r forsome c > r large enough one obtains ce ( (cid:96) (cid:48) − (cid:15) ) n d ( θ n , θ n +1 ) ≤ d ( x n , x n +1 ) ≤ (cid:15)n. Hence one obtains d ( θ n , θ n +1 ) ≤ e − ( (cid:96) (cid:48) − (cid:15) ) n for all n large enough. In particular θ ∞ = lim θ n exists, and d ( θ n , θ ∞ ) ≤ (cid:88) k ≥ n d ( θ k , θ k +1 ) ≤ e − ( (cid:96) (cid:48) − (cid:15) ) n − e − ( (cid:96) (cid:48) − (cid:15) ) ≤ e − ( (cid:96) (cid:48) − (cid:15) ) n for all n large enough.Since this holds for all (cid:15) > (cid:96) (cid:48) < (cid:96) this concludes the proof.45igure 7: A tesselation by regular p -gons with q -meeting at each vertex in thePoincar´e disk model (here p = 3 and q = 10).
11 Dimension drop for some co-compact Fuchsiangroups
Given natural numbers p, q ≥ p + q < there exists an essentiallyunique tessellation of the hyperbolic plane by regular p -gons with q meeting at eachvertex. We fix from now on, for each suitable choice of p and q , such a tesselation inthe upper half plane model containing the base point i as a vertex. Let N p,q denotethe set vertices which are neighbors of i in the tesselation, r p,q denote the lengthof the sides of the polygons in the tesselation (which is also the distance from eachpoint in N p,q to i ).For each suitable p, q one may consider the simple random walk on the verticesof the tessellation starting at i . Let (cid:96) p,q be the speed of this random walk.The random walk may be realized in the form x n = g · · · g n ( i ) where the g i are i.i.d. and chosen uniformly from a finite symmetric generator of a co-compactFuchsian group. From Furstenberg’s theory of Lyapunov exponents (see [Gru08]and Section 4.1), one obtains that (cid:96) p,q is positive and almost surely constant. Hence46y Lemma 12 letting θ n be the projection of x n onto the unit tangent sphere at i (orequivalently onto the extended real line equiped with the visual metric at i ), thereexists a limit θ ∞ = lim θ n almost surely. Let ν p,q denote the distribution of the limitpoint, we call this the exit measure (or harmonic measure) of the random walk.We will prove that the dimension drop phenomena occurs for the exit measure ofthese simple random walks when q is large (the number of sides p plays no role inour estimates, in particular we show dimension drop for regular triangulations withsufficiently many triangles per vertex). Theorem 9 (Dimension drop for some co-compact Fuchsian groups) . The dimen-sion of the exit measure of the simple random walk on the tessellation of the hy-perbolic plane by regular p -gons with q meeting at each vertex satisfies the followingestimate uniformly in p : lim sup q → + ∞ dim ( ν p,q ) ≤ . The proof depends on obtaining good estimates for (cid:96) p,q and does not seem toextend easily to all co-compact Fuchsian groups.
Lemma 13.
The speed (cid:96) p,q satisfies (cid:96) p,q = 2 log( q ) + O (log(log( q ))) uniformly in p when q → + ∞ .Proof of Lemma 13. Consider a triangle joining a vertex, the center, and the mid-point of a side, of a regular p -gon with interior angle 2 π/q . The interior angles ofthis triangle are π/p, π/q and π/
2, and the side opposite to the angle π/p has length r p,q /
2. By the hyperbolic law of cosines one obtains r p,q = 2 acosh (cid:18) cos( π/p )sin( π/q ) (cid:19) . We set r ∞ ,q = lim p → + ∞ r p,q . Since r ∞ ,q − r ,q is uniformly bounded one obtains r p,q = 2 log( q ) + O (1)uniformly in p when q → + ∞ .Next observe that by postivity of the speed one obtains from the Furstenbergtype formula (Theorem 3) and Proposition 1 that there exists a random boundaryhorofunction ξ p,q such that0 ≤ r p,q − (cid:96) p,q = E q (cid:88) x ∈ N p,q d ( i, x ) + ξ p,q ( x ) . ξ q (cid:88) x ∈ N p,q f ξ ( x ) = O (log(log( q )))uniformly in p when q → + ∞ where f ξ ( x ) = ξ ( x ) + d ( i, x ) and the maximum is overall boundary horofunctions.By Lemma 9 the set of points where f ξ is larger than 2 log(log( q )) is containedin a cone with angle C log( q ) − for some constant C independent of q . Hence thereare at most O ( q/ log( q )) points of N p,q in this set. Bounding the value of f ξ at thosepoints by 2 r p,q = 4 log( q ) + O (1) one obtainsmax ξ q (cid:88) x ∈ N p,q f ξ ( x ) ≤ q O ( q/ log( q ))(4 log( q ) + O (1)) + 2 log(log( q )) = O (log(log( q )))which establishes the lemma.We will now prove the main theorem in this section. The proof below may besimplified somewhat by using the expression for the dimension of the harmonicmeasure on a Fuchsian group given for example in [Tan17]. Instead we will give anargument closer to that which will be applied later on to study establish dimensiondrop for hyperbolic Poisson-Delaunay random walks. Proof of Theorem 9.
As before, let θ n be the projection of x n onto the unit tangentsphere at i , and θ ∞ = lim θ n . Applying Lemma 12 one obtains, given (cid:96) (cid:48) < (cid:96) apositive random variable C such that d ( θ n , θ ∞ ) ≤ Ce − (cid:96) (cid:48) n for all n. Recall that the asymptotic entropy of the random walk on the tessellation isdefined by h p,q = lim n → + ∞ − n log( p n ( x , x n ))where p n ( x, y ) is the n -step transtition probability between the vertices x and y .By subadditivity one has the estimate h p,q ≤ log( q ) for all p and q . In fact, sincethere are q neighbors at each step one has p n ( x , x n ) ≥ q − n almost surely, but wewill ignore this observation in order to illustrate the argument to be used later onfor Poisson-Delaunay random walks.Letting ν n be the distribution of θ n notice that, given h (cid:48) > h p,q there exists apositive random variable D ≤ ν n ( B r n ( θ n )) ≥ De − h (cid:48) n for all n, r n = e − (cid:96) (cid:48) n .Hence by Lemma 11 one obtains that the dimension of ν p,q is at most h (cid:48) /(cid:96) (cid:48) . Sincethis holds for all h (cid:48) > h p,q and (cid:96) (cid:48) < (cid:96) one has the following inequality (in fact equalityholds as is shown in [Tan17]): dim( ν p,q ) ≤ h(cid:96) . Applying Lemma 13 one obtainsdim( ν p,q ) ≤ h p,q (cid:96) p,q ≤ log( q )2 log( q ) + O (log(log( q ))) = 12 + o (1)which concludes the proof.
12 Dimension drop for low intensity hyperbolicPoisson-Delaunay random walks
We return from now on to the notation of Part V. In particular let P λ be a Poissonpoint process in H d , o a fixed base point.Recall that the speed (cid:96) λ is defined as (cid:96) λ = lim n → + ∞ n d ( x ,λ , x n,λ )where, conditioned on P λ , ( x n,λ , n ≥
0) is a simple random walk on the Delaunaygraph of P λ ∪ { o } starting at o .By the results of [Paq17] one obtains that both (cid:96) λ and the corresponding speedmeasured in the graph distance are almost surely positive (we have also given anindependent proof of this for all small enough λ ).Recall also that the asymptotic entropy h λ is the limit h λ = lim n →∞ − n log( p n ( x ,λ , x n,λ ))where p n ( x, y ) denotes the n -step transition probability between x, y ∈ P λ ∪ { o } conditioned on P λ . This limit is guaranteed to exist and be positive almost surelyfor the Poisson-Delaunay graph by the positivity of the graph speed (see [CPL16,Lemma 5.1]).By distance stationarity d ( x n,λ , x n +1 ,λ ) = o ( n ) almost surely. Hence, letting θ n,λ denote the projection of x n,λ onto the unit tangent sphere at o one obtains that thelimit θ ∞ ,λ = lim θ n,λ exists almost surely by Lemma 12.Notice that by rotational invariance of the Poisson point process the distributionof θ ∞ ,λ is uniform on the unit tangent sphere at o . However, we will show that thedistribution ν λ of θ ∞ ,λ conditioned on P λ typically has dimension smaller than d − Lemma 14 (Dimension upper bound) . For each λ , the speed (cid:96) λ , asymptotic entropy h λ , and dimension dim ( ν λ ) are almost surely constant and dim ( ν λ ) ≤ h λ /(cid:96) λ .Proof. The first part of the statement follows immediately from Theorem 5 (see alsoCorollary 3). We will now prove the claimed inequality.Suppose λ is fixed in what follows, and set (cid:96) = (cid:96) λ , x n = x n,λ , and h = h λ .By Lemma 12, given (cid:96) (cid:48) < (cid:96) , there exists a positive random variable C such that d ( θ n,λ , θ ∞ ,λ ) ≤ Ce − (cid:96) (cid:48) n for all n. On the other hand, by the definition of the asymptotic entropy h , one has thatfor any h (cid:48) > h there exists a positive random variable D (which one may choose tobe bounded from above by 1) such that p n ( x , x n ) ≥ De − h (cid:48) n for all n, where p n ( x, y ) is the n -th step transition probability for the simple random walk onthe Delaunay graph of P λ ∪ { o } conditioned on P λ .Set r n = e − (cid:96) (cid:48) n and notice that if ν n is the distribution of θ n,λ then, trivially, ν n ( B r n ( θ n,λ )) ≥ p n ( x , x n ) (since ν n has a point mass of at least this amount at θ n,λ ). Hence, applying Lemma 11 one obtainsdim( ν λ ) ≤ h (cid:48) (cid:96) (cid:48) . Since this is valid for all (cid:96) (cid:48) < (cid:96) and h (cid:48) > h , the proof is complete. We will now prove the main result of this section, establishing dimension drop forlow intensity Poisson-Delaunay random walks.
Theorem 10 (Dimension drop for low intensity hyperbolic Poisson-Delaunay ran-dom walks) . In the notation above one has lim sup λ → dim ( ν λ ) ≤ d − .Proof. By Lemma 14 and Theorem 6 one haslim sup λ → dim( ν λ ) ≤ lim sup λ → h λ (cid:96) λ = d −
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