Inhomogeneous Long-Range Percolation for Real-Life Network Modeling

The study of random graphs has become very popular for real-life network modeling, such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice Z, d ≥ 1, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model, various geometric properties, such as the percolation behavior, the degree distribution and graph distances, have been analyzed. In the present paper, we complement the picture of graph distances and we prove continuity of the percolation probability in the phase transition point. We also provide an illustration of the model connected to financial networks.


Introduction
The inhomogeneous long-range percolation model (also known as scale-free percolation model) was introduced in [12] to accommodate features of inhomogeneous random graphs. Inhomogeneous random graphs arise as natural objects in real life network modeling where the edge connection between vertices depends on weights assigned to the vertices of the underlying graph. This feature is, for example, observed in social networks or in financial networks.
In homogeneous long-range percolation on Z d , an edge (x, y) for x, y ∈ Z d is occupied with probability p xy which behaves as λ|x − y| −α for |x − y| → ∞, where λ > 0 and α > 0 are fixed constants. It is known that there is a critical value λ c = λ c (α, d) such that there is an infinite connected component of occupied edges for λ > λ c and there is no such component for λ < λ c . This phase transition picture in homogeneous long-range percolation can be traced back to the work of [2,17,19]. Later work concentrated more on the geometrical properties of percolation like chemical graph distances, see [4,8,9,11,20]. A good overview of the literature for long range percolation is provided in [8,10]. In [6] it is proved for homogeneous long-range percolation that p xy = 1 − exp − λW x W y |x − y| α , for fixed given parameters α, λ ∈ (0, ∞).
For | · | we choose the l 1 -norm. If there is an occupied edge between x and y we write x ⇔ y; if there is a finite path of occupied edges between x and y we write x ↔ y and we say that x and y are connected. Clearly {x ⇔ y} ⊂ {x ↔ y}. We define the cluster of x ∈ Z d to be the connected component Our aim is to study the size of the cluster C(x) and to investigate percolation properties as a function of λ > 0 and α > 0, that is, as a function of the edge probabilities (λ, α) → p xy = p xy (λ, α). The percolation probability is defined by This is increasing in λ and decreasing in α. For given α > 0, the critical value λ c (α) is defined as Trivial case. For min{α, βα} ≤ d, we have λ c = 0. This comes from the fact that for any λ > 0 P {y ∈ Z d ; 0 ⇔ y} = ∞ = 1, see Theorem 2.1 in [12]. This says that the degree distribution of a given vertex is infinite, a.s. For this reason we only consider the case min{α, βα} > d (where a phase transition may occur). (c) If d = 1 and min{α, βα} > 2, then λ c = ∞.
Since W x ≥ 1 , a.s., the edge probability stochastically dominates a configuration with independent edges being occupied with probability 1 − exp(−λ|x − y| −α ). The latter is the homogeneous long-range percolation model on Z d and it is well known that this model percolates (for d ≥ 2 see [6]; for d = 1 and α ∈ (1, 2] see [17]). For part (c) of the theorem we refer to Theorem 3.1 of [12]. The next theorem follows from Theorems 4.2 and 4.4 of [12].
Theorems 1 and 2 give the phase transition pictures for d ≥ 1. They differ for d = 1 and d ≥ 2 in that the former has a region where λ c = ∞ and the latter does not.

Continuity of percolation probability
We say that there exists an infinite cluster C if there is an infinite cluster C(x) for some x ∈ Z d . Since the model is translation invariant and ergodic, the event of having an infinite cluster C is a zero-one event. Thus, for λ > λ c there exists an infinite cluster, a.s. Moreover, from Theorem 1.3 in [6] we know that an infinite cluster is unique, a.s. This justifies the notation C for the infinite cluster in the case of percolation θ(λ, α) > 0.
An immediate consequence is the following corollary.
Next we prove continuity of the percolation probability in λ which was conjectured in [12].

Percolation on finite boxes
For integers m ≥ 1 we define the box of size m by B m = [0, m − 1] d , and by C m we denoted the largest connected component in box B m (with a fixed deterministic rule if there are two largest connected components).
This statement says that in case of percolation the largest connected components in finite boxes cover a positive fraction of these box sizes. This is the analog to the statement in homogeneous long-range percolation, see Theorem 3.2 in [8]. For integers n ≥ 1 and x ∈ Z d define the box centered at x with total side length 2n + 1 by Λ n (x) = x + [−n, n] d and let C n (x) be the vertices in Λ n (x) that are connected to x within Λ n (x).
Corollaries 7 and 8 are the analog to Corollaries 3.3 and 3.4 in [8]. Once the proofs of Theorem 6 and Lemma 13 (a), below, are established they follow from the derivations in [8].

Chemical graph distances
For x, y ∈ Z d we define d(x, y) to be the minimal number of occupied edges which connect x and y, we set d(x, y) = ∞ for y / ∈ C(x). The value d(x, y) is called chemical graph distance between x and y. Theorem 9. Assume min{α, βα} > d.

Discussion and outlook
In percolation theory one important problem is to understand the behavior of the model at criticality. In nearest-neighbor Bernoulli bond percolation on Z d , where nearest-neighbor edges are vacant or occupied with probability p, it is known that for d = 2 and for d ≥ 19 there is no percolation at criticality and hence the percolation function is continuous at the critical value (see [15] and [16] for more details). For the cases, 3 ≤ d ≤ 18, the question is still open. In the homogeneous long-range percolation model it was shown by [6] that there is no percolation at criticality for α ∈ (d, 2d). It is believed that the long-range percolation model behaves similarly to the nearest-neighbor Bernoulli percolation when α > 2d and, thus, showing continuity for such values remains a difficult problem. Another thing which remains to be answered in both homogeneous and inhomogeneous longrange models is the continuity of the critical parameter λ c (α) as a function of α and also as a function of parameter β, the exponent of the power law in weights (in case of the inhomogeneous model). Moreover, for real life network applications it will be important to (at least) get reasonable bounds on the critical value λ c (α) and the percolation probability θ(λ, α).
After the work of [4], which proposed the use of long-range percolation models for social network and small world phenomenon modeling, there was quite some work done to understand the geometry of the homogeneous long-range percolation model. In general, there are five different behaviors depending on α < d, α = d, α ∈ (d, 2d), α = 2d and α > 2d, for a review of existing results see discussion in [10]. In some of the cases, like α = 2d (for d ≥ 1) and α > 2d, the results are not yet fully known. The case d = 1 and α = 2 was resolved recently in [14]. It is clear that in the case of inhomogeneous long-range percolation the complexity even increases due to having more parameters and, hence, degrees of freedom. For instance, the understanding of the chemical graph distance behavior is still poor for min{α, βα} > 2d, though we believe that it should behave similarly to nearest-neighbor Bernoulli bond percolation, see [3,7]. Moreover, the optimal constants in the asymptotic behaviors of Theorem 9 are still open. However, we would like to emphasize that the inhomogeneous long-range percolation model fulfills the stylized fact that the degree distribution is heavy-tailed in many real network application, see [12,18], which is not the case for the homogeneous long-range percolation model. Therefore, the inhomogeneous model is very appealing for real life network modeling.

Bounds on percolation on finite boxes
The basis for all the proofs of the previous statements is Lemma 12, below, which determines large connected components on finite boxes. Its proof is based on renormalization arguments which we are going to analyze in the next lemma. These renormalization arguments were also used in the homogeneous long-range percolation model, see Lemma 2.3 of [6]. We provide a detailed proof because the random weights (W x ) x∈Z d induce additional dependencies compared to [6] and because several steps become rather subtle.
For integers m ≥ 1 we denote the m-box with corner x ∈ mZ d and its k-environment, k ≥ 0, by We drop the argument if x = 0. For every box B m (x) a semi-cluster is a set of vertices in B m (x) which are connected within the corresponding k-environment B , with a deterministic rule if there is more than one largest semi-cluster (for simplicity this deterministic rule should not depend on the location of the corner x ∈ mZ d ).
Now we define the renormalization argument recursively. We choose a sequence θ n ∈ (0, 1), n ∈ N 0 , of densities, an integer valued sequence a n > 1, n ∈ N 0 , and k ≥ 0 fixed. Define the box lengths (m n ) n∈N 0 by m 0 = a 0 and for n ∈ N m n = a n m n−1 = m 0 For x ∈ m n Z d , we call B mn (x) an n-stage box. Every n-stage box has volume m d n = n i=0 a d i and contains a d n disjoint (n − 1)-stage boxes B m n−1 (y) ⊂ B mn (x), for y ∈ m n−1 Z d . We define aliveness of n-stage boxes recursively.
Definition 10 (living n-stage boxes). Choose n ∈ N 0 and x ∈ m n Z d fixed.
• 0-stage box B m 0 (x) is alive if the following event occurs Define the semi-cluster n,x occurs, where we define where we say that B m n−1 (y) and B m n−1 (z) are pairwise attached for y, z ∈ m n−1 Z d if there exists an occupied edge between the semi-clusters U n−1,y and U n−1,z , write B m n−1 (y) ⇔ B m n−1 (z). We then define for a living n-stage box B mn (x) the semi-cluster Assume n-stage box B mn (x) is alive. This living n-stage box contains at least θ n a d n living (n−1)stage boxes that are all pairwise attached, and by iteration it contains at least m 0 (y). The attachedness property then implies that for a living n-stage box B mn (x) mn (x). Thus, the renormalization given in Definition 10 builds up (large) connected components and we would like to calculate the probabilities p n = P[A n,x ] for the occurrence of such large connected components. Lemma 11. Choose a sequence θ n ∈ (0, 1), n ∈ N 0 , of densities, an integer valued sequence a n > 1, n ∈ N 0 , and k ≥ 0 fixed. Define the sequence (m n ) n∈N 0 by (2) and set v n = m d n n i=0 θ i . For any n ∈ N we have Proof of Lemma 11. For n ≥ 1 we have For the first term in (3) we have, using Chebychev's inequality and translation invariance, The second term in (3) is more involved due to dependence in the k-environments. There are at most a d n living (n − 1)-stage boxes in B mn (x). This provides upper bound We analyze this last probability. Using the tower property for conditional expectation, we obtain for the conditional probability. Our aim is to analyze the event of non-attachedness {B m n−1 (y) ⇔ B m n−1 (z)}, conditional on A n−1,y ∩ A n−1,z . If A n−1,y occurs, then B m n−1 (y) is alive and contains a semi-cluster U n−1,y which has at least size v n−1 and which is connected within the k-environment B (k) m n−1 (y). This semi-cluster U n−1,y is a union of largest 0-stage semi-clusters U (k) m 0 (z) ⊂ B m 0 (z), and since the possible shapes of U (k) m 0 (z) do not depend on the particular choice of z (because the deterministic rule of choice in case of several largest semi-clusters is independent of z) we can define the (deterministic) sets of possible shapes by A n−1,0 = {U n−1,0 ; A n−1,0 occurs for some edge configuration} , This now allows for the calculation of the above conditional probability For U y ∈ A n−1,y and U z ∈ A n−1,z we denote by E Uy,Uz all edges (y 1 , z 1 ) with y 1 ∈ U y and z 1 ∈ U z . Observe |E Uy,Uz | ≥ v 2 n−1 . This allows to analyze the event in the above probability Observe that the above events are independent because they need to occur on the disjoint edges in (B ) \ E Uy,Uz and E Uy,Uz . Therefore we obtain Since y 1 , z 1 ∈ B mn (x) we have |y 1 − z 1 | ≤ dm n , and therefore we have, a.s., Therefore we obtain for n ≥ 1 This completes the proof of Lemma 11.
For simplicity, we assume that δ is an integer which implies that also a n is integer valued. Note that these sequences are exactly the ones used in Lemma 11 except that we still need to define the initial values θ 0 and a 0 . Observe that for κ > 1 we have for all n ≥ 1 and define constant Choose This immediately implies for any a 0 = m 0 ≥ M 0 and k = k(m 0 ) ≥ 0 as above: for 0-stage boxes Moreover, Lemma 11 directly applies to p n , n ≥ 1. Next we analyze the explicit choices (4), these provide m n = m 0 ((n + 1)!) δ and v n = θ 0 m d 0 ((n + 1)!) −κ+dδ . The second term in the statement of Lemma 11 reads as follows Note that this term decays faster than exponentially as n → ∞. Therefore it decays faster than θ n = (n + 1) −κ . Since m 0 ≥ M 0 is arbitrary and because c 3 (m 0 ) → ∞ as m 0 → ∞ (for α < 2d) there exists M 1 ≥ M 0 such that for all m 0 ≥ M 1 a 2d n exp −λd −α m −α n v 2 n−1 ≤ ε ′ θ n , for all n ≥ 1.
We have just proved that there exists M 1 > 0 such that for any m 0 ≥ M 1 and k = k(m 0 ) ≥ 0 (as above) we have 1 − p 0 ≤ ε ′ and for all n ≥ 1 we obtain with Lemma 11 Note that this is now as in Lemma 2.3 of [6]. Applying induction we obtain for all n ≥ 1 where c 1 , c 2 ∈ (1, ∞) were defined in (5) and (6) and ε ′ > 0 was chosen such that last inequality holds in the above statement. Thus, we obtain for all m 0 ≥ M 1 , k = k(m 0 ) and all n ≥ 1 P |U n,0 | ≥ v n and U n,0 is connected within The box length of the k-environment B Note that the choices of δ and κ are such that d−κ/δ > α ′ /2 > 0. This implies that ρ 0 m d−κ/δ−α ′ /2 ≥ ρ for all n sufficiently large, which proves the claim on the grid m ∈ {m(m 0 , n), m(m 0 , n + 1), . . .} with m(m 0 , n + 1) = (n + 2) δ m(m 0 , n).
for all n sufficiently large, where the latter again follows from d−κ/δ > α ′ /2 > 0 and the definition of m(m 0 , n). This finishes the proof of Lemma 12.
Although the above lemma does not allow the connected component to have size proportional to the size of the box it is useful because it allows to start a new renormalization scheme to improve these bounds. This results in our Theorem 6 and is done similar as in Section 3 of [8].
For the proof of Theorem 6 we use the following lemma which has two parts. The first one gives the initial step of the renormalization and the second one gives a standard site-bond percolation model result. Once the lemma is established the proof of Theorem 6 becomes a routine task.
Let C m (x) denote the largest connected component in box B m (x). For x, y ∈ mZ d , we say that boxes B m (x) and B m (y) are pairwise attached, write B m (x) ⇔ B m (y), if there is an occupied edge between a vertex in C m (x) and a vertex in C m (y). Note that if we choose the semi-cluster U (k) m for k = 0, then this semi-cluster coincides with C m (x) and the pairwise attachedness property is identical to the one of Definition 10 on the 0-stage level.

Lemma 13.
(a) Assume min{α, βα} > d and α < 2d. Choose λ ∈ (0, ∞) such that θ(λ, α) > 0. For each β < ∞ and r ∈ (0, 1) there exist δ > 0 and an integer m < ∞ such that for all x = y ∈ mZ d . Proof of Lemma 13(a). We adapt the proof of Lemma 3.5 of [8] to our model. Let α and λ be as in Lemma 12. Choose α ′ ∈ (α, 2d) and ε = 1 − r ∈ (0, 1). Lemma 12 then provides that for any ρ > 0 there exists N 0 ≥ 1 such that for all m ≥ N 0 Let Z d be written as a disjoint union of boxes B m (x) for x ∈ mZ d . We call a box B m (x) alive if it contains a connected component C m (x) of at least size ρm α ′ /2 . Note that this aliveness property occurs independently with probability at least r for disjoint boxes. For the choice δ = ρm α ′ /2−d the first part of the result follows. Note that we still have freedom in the choice of ρ > 0: for β > 0 we choose ρ so large that λ(2d + 1) −α ρ 2 ≥ β, note that this differs from choice (3.13) in [8]. For the second part we then observe that on the event {|C m (x)| ≥ δ|B m (x)|, |C m (y)| ≥ δ|B m (y)|} the two clusters C m (x) and C m (y) contain at least ρm α ′ /2 vertices each (for our choice of δ). Choose x ′ ∈ B m (x) and y ′ ∈ B m (y) for x = y ∈ mZ d . Conditionally on (W z ) z∈Z d , we have the upper bound, a.s., where the latter no longer depends on the weights (W z ) z∈Z d . This implies for our choice of ρ, note α ′ > α and m ≥ 1, This shows the second inequality of part (a). For part (b) see Lemma 3.6 in [8].
Proof of Theorem 6. The proof follows as in Theorem 3.
This finishes the proof of Theorem 6.

Proof of continuity of the percolation probability
The key to the proofs of the continuity statements is again Lemma 12.
The proof is similar to the proof of Theorem 6. Choose ρ > 0 such that λ(2d + 1) −α ′ ρ 2 ≥ κ. From Lemma 12 we know that for all m sufficiently large and any x ∈ mZ d , C m (x) denotes the largest connected component in B m (x), The latter events define alive vertices x on the lattice mZ d (which due to scaling is equivalent to the above aliveness in the site-bond percolation model on Z d ). Note that this aliveness property is independent between different vertices x ∈ mZ d . Attachedness B m (x) ⇔ B m (y), for x, y ∈ mZ d , is then used as in the proof of Theorem 6 and we obtain in complete analogy to the proof of the latter theorem P x, y ∈ mZ d are alive and attached we get percolation and there exists an infinite cluster C, a.s., which implies θ(λ, α) > 0. Of course, this is no surprise because of the choice λ > λ c with θ(λ, α) > 0.
Note that the probability of a vertex x ∈ mZ d being alive depends only on finitely many edges of maximal distance dm (they all lie in the box B m (x)) and therefore this probability is a continuous function of λ and α. This implies that we can choose δ ∈ (0, χλ) and γ ∈ (0, α ′ − α) so small that where P λ−δ,α+γ is the measure where for occupied edges we replace parameters λ by λ − δ ∈ (0, λ) and α by α + γ ∈ (α, α ′ ). As in (9) we obtain P λ−δ,α+γ x, y ∈ mZ d are alive and attached .

Proof of Theorem 5. We need to modify Proposition 1.3 of [1] because in our model edges are not occupied independently induced by the random choices of weights
To prove this we couple all percolation realization as λ varies. This is achieved by randomizing the percolation constant λ, see [1] and [5]. Conditionally given the i.i.d. weights (W x ) x∈Z d , define a collection of independent exponentially distributed random variables φ (x,y) , indexed by the edges (x, y), which have conditional distribution compare to (1). We denote the probability measure of (φ (x,y) ) x,y∈Z d by P in order to distinguish this coupling model. We say that an edge (x, y) is ℓ-open if φ (x,y) < ℓ, and we define the connected cluster C ℓ (0) of the origin to be the set of all vertices x ∈ Z d which are connected to the origin by an ℓ-open path. Note that we have a natural ordering in ℓ, i.e. for ℓ 1 < ℓ 2 we obtain C ℓ 1 (0) ⊂ C ℓ 2 (0). Moreover for ℓ = λ > 0, the λ-open edges are exactly the occupied edges in this coupling (note that the exponential distribution (11) is absolutely continuous). This implies for ℓ = λ By countable subadditivity of P and the increasing property of C ℓ (0) in ℓ we have Moreover, the increasing property of C ℓ (0) in ℓ provides {|C λ ′ (0)| = ∞ for some λ ′ < λ} ⊂ {|C λ (0)| = ∞}. Therefore, to prove (10) it suffices to show that Choose λ 0 ∈ (λ c , λ). Since there is a unique infinite cluster for λ 0 > λ c , a.s., there exists an infinite cluster C λ 0 ⊂ C λ (0) on the set {|C λ (0)| = ∞}. If the origin belongs to C λ 0 then the proof is done. Otherwise, because C λ 0 is a subgraph of C λ (0), there exists a finite path π of λ-open edges connecting the origin with an edge in C λ 0 . By the definition of λ-open edges we have φ (x,y) < λ for all edges (x, y) ∈ π. Since π is finite we obtain the strict inequality λ 1 = max (x,y)∈π φ (x,y) < λ. Choose λ ′ ∈ (λ 0 ∨ λ 1 , λ) and it follows that |C λ ′ (0)| = ∞. This completes the proof for the leftcontinuity in λ.
(iii) Finally, we need to prove right-continuity of λ → θ(λ, α) on λ ≥ λ c . For integers n > 1 we consider the boxes Λ n = [−n, n] d centered at the origin, see also Corollary 7. We define the events A n = {C(0)∩Λ c n = ∅}, i.e. the connected component C(0) of the origin leaves the box Λ n = [−n, n] d . Note that θ(λ, α) is the decreasing limit of P[A n ] as n → ∞. Therefore, it suffices to show that P[A n ] is a continuous function in λ. We write P λ = P to indicate on which parameter λ the probability law depends. We again denote by C n (0) the connected component of the origin connected within box Λ n , see Corollary 7. Then, we have Choose δ 0 ∈ (0, λ), then we have for all λ ′ ∈ (λ − δ 0 , λ + δ 0 ) and all n ′ > n We bound the two terms on the right-hand side of (12).
(a) First we prove that for all ε > 0 there exists n ′ > n such that for all x ∈ Λ n This is done as follows. For m > n we define the following events This implies for n ′ > n that Moreover, note that E n ′ is decreasing in n ′ and therefore lim sup We prove (13) by contradiction. Assume that (13) does not hold true, i.e. lim sup n ′ →∞ P [E n ′ ] > 0.
Then the first lemma of Borel-Cantelli implies The latter implies that the degree distribution D x = |{y ∈ Z d ; x ⇔ y}| has an infinite mean. This is a contradiction to Theorem 2.2 of [12] saying that for min{α, βα} > d the survival function of the degree distribution has a power-law decay with rate αβ/d > 1 which provides a finite mean. Therefore, claim (13) holds true.

Proofs of the chemical graph distances
In this section we prove Theorem 9. Statement (a) of Theorem 9 is proved in Theorem 5.1 and 5.3 of [12], the lower bound of statement (b1) is proved in Theorem 5.5 of [12]. Therefore, there remain the proofs of the upper bound in (b1) and of the lower bound in (b2) of Theorem 9.
The proof of the upper bound in Theorem 9 (b1) follows from the following proposition and the fact that α → ∆(α, 2d) = log 2/ log(2d/α) is a continuous function. The following proposition corresponds to Proposition 4.1 in [8] in the homogeneous long-range percolation model.
The pairs of vertices (z σ00 , z σ01 ) and (z σ10 , z σ11 ) are called gaps. The proof is then based on the fact that for large distances |x − y| the event B m of the existence of a hierarchy H m (x, y) of depth m that connects x and y through points z σ which are dense is very likely (m appropriately chosen), see Lemma 4.3 in [8], in particular formula (4.18) in [8] (where the key is Corollary 8). On this likely event B m Lemma 4.2 of [8] then proves that the chemical distance cannot be too large, see (4.8) in [8]. We can now almost literally translate Lemmas 4.2 and 4.3 of [8] to our situation. The only changes are that in formulas (4.16) and (4.21) of [8] we need to replace β > 0 of [8]'s notation by λ in our notation and we need to use that the weights W x ≥ 1, a.s. We refrain from giving more details.
There remains the proof of the lower bound in (b2) of Theorem 9. Therefore, we use again a renormalization technique which is based on a scheme introduced by [7]. We start with a technical lemma. Denote by · the Euclidean distance in R d . Then we have a similar statement as in the Poisson inhomogeneous long-range percolation model, see remark after Proposition 4.1 in [13], which basically says that the degree distribution is finite for min{α, βα} > d.
Proof of Lemma 15. Applying Fubini's theorem we have We first calculate the inner integral. We have using polar coordinates and integration by parts for The latter is an integral over a gamma density for shape parameter 1 − d/α > 0. Therefore we obtain This implies that we need to calculate for min{α, βα} > d. This proves the claim.
As in Section 5.1 we define a renormalization scheme based on n-stage boxes with m n = n i=0 a i for a given integer valued sequence (a n ) n∈N 0 with a n ≥ 1. For n ≥ 1, the children of the n-stage box B mn (x) are the a d n disjoint (n − 1)-stage boxes Similar to Definition 2 of [7], we are going to define good n-stage boxes B mn (·).
Definition 16 (good n-stage boxes). Choose n ∈ N 0 and x ∈ Z d fixed.
• 0-stage box B m 0 (x) is good under a given edge configuration if there is no occupied edge starting in B m 0 (x) with size larger than m 0 /100.
There exists t 0 ≥ 1 such that for all t ≥ t 0 and all s ≥ 1 P there is an occupied edge starting in [0, s − 1] d with size larger than t + 1 ≤ c 4 s d t d−min{α,β ′ α} .
In the proof of the lemma we will see that β ′ is only needed for the case β ≤ 1, for the case β > 1 we may set β ′ = 1.
Applying Fubini's theorem and the Pareto distributional assumptions we need to calculate Using polar coordinates we obtain identity, v d denotes the volume of the unit ball in R d , Thus, we need to study the asymptotic behavior as t → ∞ of the following integral We distinguish two cases: Case (i). We assume that β > 1, i.e. the Pareto distribution has finite mean. Using integration by parts and 1 − e −x ≤ x we obtain For β > 1 we have the proof in case (i) then follows for t ≥ 1 because t −α ≤ t − min{α,β ′ α} . Case (ii). We assume that β ≤ 1. This is more sophisticated because w −β is not integrable on [1, ∞). Using 1 − e −x ≤ (x ∧ 1), we rewrite (15) as follows, where W 1 and W 2 are two independent copies of the Pareto random variable W 0 . If H is the distribution function of the product of W 1 and W 2 , then it satisfies for v ≥ 1, see (4.3) in [12] and Section 5 in [13], Thus, we obtain for V ∼ H for λu < 1 where we need to assume β < 1. For β = 1, we obtain E [((λuV ) ∧ 1)] ≤ λu + (1 − β log(λu)) (− log(λu)) (λu) β ≤ 2 (1 − log(λu)) 2 (λu) β .
Therefore, for any β ′ ∈ (d/α, β) and any λ > 0 there exists t 0 ≥ 1 such that for any This implies for (15) and for all t ≥ t 0 , note that β ′ α > d and β ′ < 1, This finishes the proof of the lemma also in case (ii).
This lemma is the analog in our model to Lemma 1 of [7] and provides a Borel-Cantelli type of results that eventually the boxes B mn (0) are good, a.s., for all n sufficiently large.
Note that this slightly differs from the proof of Lemma 1 in [7]. We prove (18) by induction. For n = 0 the prove follows from (16) and the choice of ε < 2 −4d−1 e −2 . So we assume that (18)  This proves (18). Observe that the bound on the right-hand side of (18) is summable and hence the claim of the lemma follows.
The following lemma is the analogue of Proposition 3 of [7] and it depends on Lemma 2 of [7] and Lemma 18. Since its proof is completely similar to the one of Proposition 3 of [7] once Lemma 18 has been established we skip this proof.

Lemma 19 (Proposition 3 of [7]
). Choose a n = n 2 for n ≥ 1. There exists a constant c 5 > 0 such that for n ≥ 16 · 3 d , if for every j ∈ {−1, 0, 1} d the n-stage box B mn j mn 2 is good and for every l > n the l-stage boxes B m l centered at B mn (0) are good, then if x, y ∈ B mn (0) satisfy |x − y| > m n /8 then d(x, y) ≥ c 5 |x − y|.
Proof of Theorem 9 (b2). Lemma 18 says that, a.s., the l-stage boxes B m l are eventually good for all l ≥ n. Moreover, from Lemma 19 we obtain the linearity in the distance for these good boxes which says that, a.s., for n sufficiently large and |x| > m n /8 we have d(0, x) ≥ c 5 |x|.