Scale-Free Percolation in Continuum Space
SScale-Free Percolation in Continuum Space
Philippe Deprez ∗ Mario V. W¨uthrich ∗† April 23, 2018
Abstract
The study of real-life network modeling has become very popular in recent years. Anattractive model is the scale-free percolation model on the lattice Z d , d ≥
1, because itfulfills several stylized facts observed in large real-life networks. We adopt this model tocontinuum space which leads to a heterogeneous random-connection model on R d : particlesare generated by a homogeneous marked Poisson point process on R d , and the probabilityof an edge between two particles is determined by their marks and their distance. In thismodel we study several properties such as the degree distributions, percolation propertiesand graph distances. The study of real-life networks such as virtual social networks or financial networks has becomevery popular in recent years, see for example [20, 1, 7]. Such networks can be seen as sets ofparticles that are possibly linked to each other. Several stylized facts of large real-life networkshave been observed using large empirical data sets (see [20] and Section 1.3 in [13] for furtherdetails): • The minimal number of links that connect two particles, called the graph distance, istypically small for distant particles. This is called the “small-world effect”. There is theobservation that most particles in many real-life networks are connected by at most sixlinks, see [23]. • Particles that are linked tend to have common friends, which is called the “clusteringproperty”. • The number of links of a given particle, called the degree, has a heavy-tailed distributionwith (power law) tail parameter τ >
0. The tail parameter is often observed to be between1 and 2, i.e. the degree distribution has finite mean and infinite variance. We refer to [13]for explicit examples. ∗ RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland † Swiss Finance Institute SFI Professor, 8006 Zurich, Switzerland a r X i v : . [ m a t h . P R ] S e p ince it is too complicated to model large real-life networks particle by particle, many theoreticalrandom graph models have been developed and their geometrical properties studied. One ofthese models is the homogeneous long-range percolation model on Z d , d ≥
1, first introducedin [25] for d = 1. The set of particles is the lattice Z d and any two particles x, y ∈ Z d areindependently linked with probability p xy which behaves as λ | x − y | − α for | x − y | → ∞ , withfixed constants λ, α >
0. Since close particles are likely linked, this model has a local clusteringproperty. Moreover, depending on α , the graph distance of two connected particles is roughlyof logarithmic order as their separation tends to infinity, see [6]. This is a version of the small-world effect. However, this model does not fulfill the stylized fact of having heavy-tailed degreedistributions. Therefore, [11] extended the homogeneous long-range percolation model to a scale-free percolation model on Z d (also known as inhomogeneous long-range percolation model). Intheir model they consider a collection ( W x ) x ∈ Z d of i.i.d. positive weights that are heavy-tailedwith tail parameter β >
0, and they assign to each particle x ∈ Z d the random weight W x .Given these weights, any two particles x, y ∈ Z d are independently linked with probability p xy which is approximately λW x W y | x − y | − α for large | x − y | and given constants λ, α >
0. Notethat p xy is increasing in the weights W x and W y , and decreasing in the distance between x and y . This means that the weights make particles more or less attractive, i.e. particles withlarge weights play the role of hubs in this network. This extension of the homogeneous modelis very natural since the existence of hubs is often observed in real-life networks. Again, thismodel has a local clustering property. Depending on α and β , [11] showed that the degreedistribution is heavy-tailed, i.e. this model fulfills the stylized fact of having heavy-tailed degreedistributions. Moreover, they showed that whenever the degree distribution has finite mean butinfinite variance, the graph distance of two particles behaves doubly logarithmically as theirseparation tends to infinity. This is again a version of the small-world effect, sometimes calledthe ultra-small-world effect.In this article we adopt the scale-free percolation model on Z d to the continuum space R d asproposed in [11], which leads to a heterogeneous random-connection model (RCM) on R d whereparticles are no longer restricted to a lattice. Instead of taking the particles to be the vertices of Z d with assigned weights, we distribute particles randomly in space according to a homogeneousPoisson point process on R d , and to each particle x we attach (independently of its location) apositive random weight W x whose distribution is heavy-tailed with tail parameter β >
0. Giventhe Poisson cloud and the weights, two particles x and y are linked with probability p xy ( λ, α ) as inthe scale-free percolation model on Z d . This heterogeneous RCM can be seen as an extension ofthe homogeneous RCM on R d , which was introduced and studied in [21], while an applied versionalready appeared in [15]. The main reference for the homogeneous RCM and other continuumpercolation models is [18]. The goal of this article is to prove similar results as in [11, 12] for theheterogeneous RCM. In particular, depending on α and β , we show that in our heterogeneousRCM the degree distribution is heavy-tailed with tail parameter τ ( α, β ) >
0. Assuming thatthe weights follow a Pareto distribution with tail parameter β , we give an explicit expression2f the degree distribution as well as the expected degree of a given particle in terms of themodel parameters β , α , λ and the intensity of the Poisson point process. This result improvesthe bounds given in Proposition 2.3 in [11] for this particular choice of weight distribution.This explicit expression is also helpful to calibrate the model parameters for real-life networkapplications. Moreover, we show that there is a non-trivial phase transition depending on α and β , where λ plays the role of the percolation parameter. Above criticality there is a unique infiniteconnected component. For real-life network applications the interesting case is τ ( α, β ) ∈ (1 , λ >
0. In other words, in this lattercase the network contains infinitely many particles that are all connected through links. Wefurthermore study graph distances between particles that lie in the same connected component.Similar to [11, 12] we prove the existence of different asymptotic regimes, which are characterizedby α and the power law constant β of the marks. As key step in that proof, we show that thesize of the largest connected component restricted to a finite box is of the same order as thetotal number of particles in that box. This result is of independent interest and states that thenumber of particles belonging to the largest connected network in a finite box [0 , m ) d is of order m d . Moreover, we show that there is no percolation at criticality whenever p xy ( λ, α ) does notdecrease too fast in the distance of two particles.Compared to inhomogeneous long-range percolation on Z d , the heterogeneous RCM has theadvantage that some proofs of the results are more easy to handle since we can use standardintegration in R d . This also allows us for calculating several graph properties explicitly whichis of central interest for calibrating model parameters. On the other hand, some proofs aremore involved because one needs to make sure that the Poisson cloud is sufficiently regular.We also mention that the continuum space model, as an extension of the lattice model, has theadvantage that it can be extended to Poisson point processes with space dependent (random)intensity functions. This can be used to model networks that have more densely populated areasthan other areas.The paper is organized as follows. In the next section we introduce the model. In Section 3we state the main results on the degree distributions, the percolation properties, the absenceof percolation at criticality, the size of largest connected components in finite boxes and thegraph distances in the random graph. Section 4 gives the proofs of the results on the degreedistributions and in Section 5 we prove the percolation properties. In Section 6 we prove theabsence of percolation at criticality and the results on the size of largest connected componentsin finite boxes are given. Finally, Section 7 contains the proofs of the results on graph distances. We introduce a heterogeneous RCM which modifies the homogeneous RCM defined in [21]and which is a continuum space analogue to the inhomogeneous long-range percolation model3resented by [11]. The tuple (
X, ν, β, λ, α ) denotes a heterogeneous RCM on R d , d ≥
1, wherewe make the following assumptions:1. (
X, ν, β ) is a homogeneous marked Poisson point process, where X denotes the spatiallyhomogeneous Poisson point process on R d with fixed intensity ν >
0, and β > W x , x ∈ X . We assumethat W x , x ∈ X , has Pareto distribution with scale parameter 1, i.e. P [ W x > w ] = w − β , for w ≥ X and ( W x ) x ∈ X , we have an edge (link) between two distinct particles x (cid:54) = y ∈ X ,write x ⇔ y , independently of all other possible edges, with probability p xy = p xy ( λ, α ) = 1 − exp (cid:8) − λW x W y | x − y | − α (cid:9) , with constants λ > α >
0, and | · | denoting the Euclidean norm on R d .By replacing λ by θ λ we can extend the results to Pareto distributions with arbitrary scaleparameter θ >
0, but we use θ = 1 as normalization of the model. In [11] there is a moregeneral version for the choice of the distribution of the marks ( W x ) x , but since eventually onlythe choice of β > p xy is relevant, which is of order λW x W y | x − y | − α ; but we make the particular choice of p xy tosimplify calculations. We call X the Poisson cloud with particles x ∈ X . The marks ( W x ) x ∈ X are the weights in the particles x ∈ X that determine the edge probabilities p xy between thecorresponding particles x and y of X . It follows from [9] that the model is shift invariant andergodic. We define the degree D x of particle x ∈ X to be the number of particles y ∈ X such that x and y are linked, i.e. x ⇔ y . Observe that the distribution of D x is translation invariant in thesense that we may start at every particle x of the Poisson cloud X . Since D is only definedif the origin belongs to the Poisson cloud X , we consider D under the conditional probability P , conditionally given that the Poisson cloud has a particle at the origin. The probability P is the Palm measure of P , and the conditioning on the event of having a particle at the origindoes not influence the rest of the Poisson process, see for instance Chapter 12 in [10]. The firstresult describes the distribution of the degree D under P . Theorem 3.1
We obtain the following cases.(i) For min { α, βα } ≤ d we obtain P [ D = ∞ ] = 1 . ii) For min { α, βα } > d we obtain that D has (under P ) a mixed Poisson distribution withmixing distribution being the Pareto distribution with shape parameter τ = βα/d > andscale parameter c /τ , where c = c ( d, β, α, λ, ν ) = (cid:16) νv d Γ(1 − d/α ) ττ − (cid:17) τ λ β , and where v d denotes the volume of the unit ball in R d . That is, for k ≥ , P [ D = k ] = τ c k ! (cid:90) ∞ c /τ t k − τ − e − t dt. Moreover, the survival probability of this distribution fulfills lim n →∞ P [ D > n ] n − τ = c , and, hence, the degree distribution is heavy-tailed with tail parameter τ = βα/d > . Thefirst moment of this distribution is given by E [ D ] = νv d Γ(1 − d/α ) (cid:18) ττ − (cid:19) λ d/α . This theorem is the continuum space analogue to Theorems 2.1 and 2.2 in [11]; the explicitexpression of E [ D ] improves the bounds given in Proposition 2.3 in [11] for our choice of thedistribution of the weights ( W x ) x ∈ X . We observe that if βα/d ≤ | x | − α is tooslow for | x | → ∞ , namely if α ≤ d , then any given particle shares edges with infinitely manyother particles, a.s. This trivial case is, of course, not of interest for real-life network modeling.In the non-trivial case min { α, βα } > d the distribution of the degree of a given particle isheavy-tailed with tail parameter τ = βα/d >
1. Hence, in this latter case, the continuum spacemodel fulfills the stylized fact of having heavy-tailed degree distributions. This differs from thehomogeneous RCM, where the degree distribution always is light-tailed, see formula (6.1) in [18].According to the stylized facts the interesting case for real-life applications is τ = βα/d ∈ (1 , α > d , see also Section 1.4 in [13]. Note that, even if α > d , weight distributions havingan infinite variance ( β <
2) do not immediately imply degree distributions having an infinitevariance ( τ < α > d , if the weight distributionshave a finite variance ( β > τ >
2) as well.
In order to study the percolation properties of the heterogeneous RCM, denote the (maximal)connected component of x ∈ X by C ( x ) = { y ∈ X | there is a finite path of edges connecting x and y } , which is the set of all particles that can be reached from x within the network. The percolationprobability is defined by θ ( λ ) = P [ |C (0) | = ∞ ] , d βα τ = βα/d = 2 λ c = 0 λ c ∈ (0 , ∞ ) d ≥ λ c ∈ (0 , ∞ ) d = 1 : λ c = ∞ d d βατ = βα/d = 1 τ = βα/d = 2 d (0 , x ) ∼ | x | d (0 , x ) ∼ (log | x | ) ∆ d ( , x ) ∼ l og l og | x | d (0 , x ) bounded, a.s. Figure 1: The phase transition picture (lhs) and the graph distances (rhs) for the heterogeneousRCM on R d , d ≥
1, for different model parameters α and β . Recall that the degree distributionshave infinite variances if τ = βα/d > α > d . The existence of ∆ for the asymptoticbehavior of the graph distances in the case α ∈ ( d, d ) and τ = βα/d > d (0 , x ) ∼ log log | x | . Additionally, in the case min { α, βα } > d ,only the asymptotic linear lower bound on d (0 , x ) is known.where |C (0) | denotes the number of particles in the connected component of the origin. Thecritical percolation value is defined by λ c = inf { λ > | θ ( λ ) > } . By ergodicity it follows that there are only finite connected components, a.s., whenever λ < λ c ;and there exists an infinite connected component, a.s., if λ > λ c . By the uniqueness theorem forthe homogeneous RCM, see Theorem 6.3 of [18], and the fact that p xy ∈ (0 ,
1) for all particles x and y , a.s., such an infinite connected component is unique, a.s.; we denote it by C ∞ .We refer to [8, 16] for a general introduction to percolation theory. For min { α, βα } ≤ d it followsfrom Theorem 3.1 ( i ) that θ ( λ ) = 1 for all λ >
0, hence λ c = 0. The next theorem gives thepercolation properties in the non-trivial case min { α, βα } > d , see also Figure 1. Theorem 3.2
Assume min { α, βα } > d .(a) In the case d ≥ we obtain:(a1) if βα < d , then λ c = 0 ;(a2) if βα > d , then λ c ∈ (0 , ∞ ) .(b) In the case d = 1 we obtain: b1) if βα < , then λ c = 0 ;(b2) if βα > and α ∈ (1 , , then λ c ∈ (0 , ∞ ) ;(b3) if min { α, βα } > , then λ c = ∞ . This result also holds true in the discrete space model, see [11]. It shows the existence of anon-trivial phase transition if the degree distribution has finite variance ( τ = βα/d >
2) and α > d ( α ∈ (1 ,
2] in d = 1). Note that in the interesting case for real-life network applications( τ = βα/d ∈ (1 ,
2) with α > d ) there is a unique infinite connected component C ∞ for all λ > Z , where the probability of an edge between two sites x and y isgiven by 1 − exp {− λ | x − y | − α } , see [19, 22]. It is shown that for α ≤ λ >
0, for α ∈ (1 ,
2] percolation occurs only for λ sufficiently large, and for α > λ c = 0 whenever τ = βα/d ∈ (1 ,
2) and, therefore, there is trivially no infinite con-nected component at criticality λ c . The next theorem states that there is no infinite connectedcomponent at criticality λ c > α ∈ ( d, d ) and τ = βα/d >
2. This correspondsto Theorem 1.5 of [3] for homogeneous long-range percolation and to Corollary 4 of [12] forinhomogeneous long-range percolation on the lattice. The case α > d and τ = βα/d > d = 1 where there is never an infinite connected component. Theorem 3.3
Assume α ∈ ( d, d ) and τ = βα/d > . There is no infinite connected componentat criticality λ c > , a.s. For n ∈ (0 , ∞ ) we define the box Λ n = [ − n, n ) d and we denote by C n the largest connected com-ponent in Λ n (with a deterministic rule if there is more than one largest connected component).The next result shows that in case of percolation and α ∈ ( d, d ), the number of particles in C n presents with high probability at least a positive fraction of the Lebesgue measure of box Λ n .This result is the continuum space analogue to Theorem 6 of [12]. Theorem 3.4
Assume α ∈ ( d, d ) and τ = βα/d > . Choose λ ∈ (0 , ∞ ) such that θ ( λ, α ) > .Then, for all α (cid:48) ∈ ( α, d ) there exist ρ > and n < ∞ such that for all n ≥ n , P (cid:104) |C n | ≥ ρn d (cid:105) ≥ − exp {− ρn d − α (cid:48) } , where |C n | denotes the number of particles of the largest connected component in Λ n . For x ∈ R d and n ∈ (0 , ∞ ) we write Λ n ( x ) = x + [ − n, n ) d for the box of side length 2 n centeredat x . For x ∈ X we write C n ( x ) for the set of particles in Λ n ( x ) ∩ X that are connected to x within Λ n ( x ). For (cid:96) > ρ > x ∈ X a ( ρ, (cid:96) )-dense particle if the number of particles7n X belonging to C (cid:96) ( x ), denoted by |C (cid:96) ( x ) | , is at least ρ (2 (cid:96) ) d , see also Definition 4.1 of [6]. Theset of ( ρ, (cid:96) )-dense particles in Λ n = Λ n (0) is denoted by D ( ρ,(cid:96) ) n = (cid:110) x ∈ Λ n ∩ X (cid:12)(cid:12)(cid:12) |C (cid:96) ( x ) | ≥ ρ (2 (cid:96) ) d (cid:111) . Corollary 3.5 below shows that whenever a particle x ∈ X belongs to the infinite connectedcomponent C ∞ , the probability that it is ( ρ, (cid:96) )-dense converges to 1 as (cid:96) → ∞ for some ρ > n presents with highprobability at least a positive fraction of the Lebesgue measure of that box. Corollary 3.5 is theanalogue to Corollaries 3.3 and 3.4 of [6]. We use it to prove an estimate on the graph distancein the infinite connected component, below. Corollary 3.5
Assume α ∈ ( d, d ) and τ = βα/d > . Choose λ ∈ (0 , ∞ ) such that θ ( λ, α ) > .(i) There exists ρ > such that for x ∈ R d , lim (cid:96) →∞ P (cid:104) |C (cid:96) ( x ) | ≥ ρ (2 (cid:96) ) d (cid:12)(cid:12)(cid:12) x ∈ C ∞ (cid:105) = 1 . (ii) For all α (cid:48) ∈ ( α, d ) there exist ρ > and (cid:96) > such that for all n > (cid:96) and (cid:96) ∈ ( (cid:96) , n/(cid:96) ) , P (cid:104) |D ( ρ,(cid:96) ) n | ≥ ρ (2 n ) d (cid:105) ≥ − exp {− ρn d − α (cid:48) } , where |D ( ρ,(cid:96) ) n | denotes the number of ( ρ, (cid:96) ) -dense particles in Λ n . For x, y ∈ X we write d ( x, y ) for the graph distance or chemical distance between x and y , i.e. d ( x, y ) = inf (cid:8) n ∈ N (cid:12)(cid:12) ∃ x , . . . , x n ∈ X : x ⇔ x ⇔ . . . ⇔ x n − ⇔ x n = y (cid:9) , where we use the convention that d ( x, y ) = ∞ if x and y are not in the same connected compo-nent. In order to measure events involving d ( x, y ) for x, y ∈ R d , one needs to make sure that theparticles x and y lie in the Poisson cloud X . We therefore consider the 2-fold Palm measure P x,y of P which can be interpreted as the conditional distribution of the marked Poisson point processunder the condition that there are particles of the process in x and y , i.e. P x,y [ · ] = P [ · | x, y ∈ X ].Note that we have P [ · | x, y ∈ C ∞ ] = P x,y [ · | x, y ∈ C ∞ ]. The next theorem states bounds on thegraph distance in the case min { α, βα } > d , see also Figure 1 for an illustration. Theorem 3.6
Assume min { α, βα } > d .(a) Assume τ = βα/d ∈ (1 , and choose λ > λ c = 0 . There exists η > such that for all ε > , lim | x |→∞ P (cid:20) η | log( α ( β ∧ /d − | ≤ d (0 , x )log log | x | ≤ (1 + ε ) 2 | log( βα/d − | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ (cid:21) = 1 . b1) Assume α ∈ ( d, d ) , τ = βα/d > and choose λ > λ c . For all ε > , lim | x |→∞ P (cid:20) − ε ≤ log d (0 , x )log log | x | ≤ (1 + ε ) log 2log(2 d/α ) (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ (cid:21) = 1 . (b2) Assume min { α, βα } > d . There exists η > such that lim | x |→∞ P (cid:20) η < d (0 , x ) | x | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ X (cid:21) = 1 . This theorem is the continuum space analogue to the results in Section 5 of [11] and Theorem 8of [12]. We note that in homogeneous long-range percolation on Z d the picture about graphdistances in the case α > d (where the degree distribution has finite mean) has two differentregimes: the graph distances behave roughly logarithmically if α ∈ ( d, d ), while if α > d , thereis a linear lower bound on the graph distances, see Theorem 1.1 of [6] and Theorem 1 of [4],respectively. In our model we observe the same behavior if in addition the degree distributionhas finite variance ( τ = βα/d > d < min { α, βα } < d , we see that distant particles are connected by very short paths of edgeswhich is a version of the (ultra-) small-world effect.An upper bound on the graph distances in the case min { α, βα } > d is still open. As inhomogeneous long-range percolation with α > d it is believed that a linear upper bound shouldhold, see Conjecture 1 of [4]. For independent nearest-neighbor bond percolation on Z d , d ≥ (b1) states that d (0 , x ) is roughly (log | x | ) ∆ for large | x | and some constant ∆ >
0. The existence of ∆ is still unknown, even in homogeneous long-rangepercolation. Moreover, the optimal constants in all asymptotic behaviors are still open.
In this section we prove Theorem 3.1. We start with the following observation.
Lemma 4.1
The distribution of degree D , conditionally given W , is given by P [ D = k | W ] = exp (cid:26) − ν (cid:90) R d E [ p x | W ] dx (cid:27) (cid:0) ν (cid:82) R d E [ p x | W ] dx (cid:1) k k ! , for k ∈ N .Note that this distribution is trivial if the integral appearing twice on the right-hand side doesnot exist. Proof of Lemma 4.1.
Let X be a Poisson cloud with 0 ∈ X and denote by X ( A ) the number of particles in X ∩ A for A ⊂ R d . Conditionally given W , every particle x ∈ X \ { } is now independently of the others removed fromthe Poisson cloud with probability 1 − p x . By Proposition 1.3 of [18], the resulting process (cid:101) X is a thinned Poisson loud, conditionally given W , with intensity function x (cid:55)→ ν E [ p x | W ]. Since D = ˜ X ( R d \ { } ) in distribution,it follows that, conditionally given W , D has a Poisson distribution with parameter ν (cid:82) R d E [ p x | W ] dx . (cid:3) We now provide a necessary and sufficient condition for the existence of (cid:82) R d E [ p x | W ] dx interms of α and β . Proposition 4.2
The following two statements are equivalent:(i) min { α, βα } > d ;(ii) (cid:82) R d E [ p x | W ] dx < ∞ . Lemma 4.1 and Proposition 4.2 imply that the distribution of degree D , conditionally given W , has a Poisson distribution whenever min { α, βα } > d , and that D is infinite, a.s., otherwise. (i) of Theorem 3.1 is therefore a direct consequence of Lemma 4.1 and Proposition 4.2. Theproof of Proposition 4.2 is based on integral calculations. Proof of Proposition 4.2.
We obtain, using integration by parts in the second step, E [ p x | W ] = (cid:90) ∞ βw − β − (cid:0) − exp (cid:8) − λW | x | − α w (cid:9)(cid:1) dw = 1 − exp (cid:8) − λW | x | − α (cid:9) + λW | x | − α (cid:90) ∞ w − β exp (cid:8) − λW | x | − α w (cid:9) dw = 1 − exp (cid:8) − λW | x | − α (cid:9) + (cid:0) λW | x | − α (cid:1) β (cid:90) ∞ λW | x | − α z − β e − z dz. Note that, given W , 1 − exp (cid:8) − λW | x | − α (cid:9) is integrable over R d if and only if α > d . It therefore remains toconsider the integrability of | x | − βα (cid:82) ∞| x | − α z − β e − z dz . For | x | α ≥ | x | − βα (cid:90) ∞| x | − α z − β e − z dz ≥ | x | − βα (cid:90) ∞ z − β e − z dz, which is not integrable over R d for βα ≤ d . This finally shows that (ii) implies (i) . For | x | α ≥ β (cid:54) = 1, | x | − βα (cid:90) ∞| x | − α z − β e − z dz ≤ | x | − βα (cid:34)(cid:90) | x | − α z − β dz + (cid:90) ∞ e − z dz (cid:35) = | x | − βα (cid:20) − | x | βα − α − β + e − (cid:21) , which is integrable over R d for min { α, βα } > d . Similarly if β = 1. This finally shows that (i) implies (ii) . (cid:3) In order to prove part ( ii ) of Theorem 3.1 we first calculate ν (cid:82) R d E [ p x | W ] dx , which is finitefor min { α, βα } > d . Proposition 4.3
Assume min { α, βα } > d and set τ = βα/d > . We obtain ν (cid:90) R d E [ p x | W ] dx = c /τ W d/α , where c is defined in Theorem 3.1, which has a Pareto distribution with scale parameter c /τ and shape parameter τ . roof of Proposition 4.3. From Proposition 4.2 we obtain that we can apply Fubini’s theorem which provides ν (cid:90) R d E [ p x | W ] dx = ν (cid:90) R d (cid:18)(cid:90) ∞ βw − β − (cid:0) − exp (cid:8) − λW | x | − α w (cid:9)(cid:1) dw (cid:19) dx = ν (cid:90) ∞ βw − β − (cid:18)(cid:90) R d − exp (cid:8) − λW | x | − α w (cid:9) dx (cid:19) dw. We first calculate the inner integral. Using polar coordinates and integration by parts, we obtain for w ≥ v d denoting the volume of the unit ball in R d , (cid:90) R d − exp (cid:8) − λW | x | − α w (cid:9) dx = dv d (cid:90) ∞ (cid:0) − exp (cid:8) − λW wr − α (cid:9)(cid:1) r d − dr = dv d α (cid:90) ∞ (1 − exp {− λW wt } ) t − d/α − dt = − v d (1 − exp {− λW wt } ) t − d/α (cid:12)(cid:12)(cid:12) ∞ + v d λW w (cid:90) ∞ exp {− λW wt } t − d/α dt = v d λW w Γ(1 − d/α )( λW w ) − d/α (cid:90) ∞ ( λW w ) − d/α Γ(1 − d/α ) t − d/α − exp {− λW wt } dt. The latter is an integral over a gamma density for 1 − d/α >
0. Therefore, we obtain (cid:90) R d − exp (cid:8) − λW w | x | − α (cid:9) dx = v d Γ(1 − d/α ) ( λW w ) d/α . For βα > d this implies that ν (cid:90) R d E [ p x | W ] dx = νv d Γ(1 − d/α ) ( λW ) d/α (cid:90) ∞ βw − β − w d/α dw = νv d Γ(1 − d/α ) ββ − d/α ( λW ) d/α . Since β/ ( β − d/α ) = τ / ( τ − (cid:3) Proof of Theorem 3.1.
The proof of part ( i ) is a direct consequence of Lemma 4.1 and Proposition 4.2. Inorder to prove part ( ii ), assume min { α, βα } > d and set τ = βα/d >
1. Then, by Lemma 4.1 and Proposition4.3, D has (under P ) a mixed Poisson distribution with mixing distribution being the Pareto distribution withscale parameter c /τ and shape parameter τ . From this we obtain, since E [ W d/α ] = β/ ( β − d/α ) = τ / ( τ − E [ D ] = E (cid:104) c /τ W d/α (cid:105) = νv d Γ(1 − d/α ) (cid:18) ττ − (cid:19) λ d/α < ∞ , and for n ≥
0, see for instance Lemma 3.1.1 of [24], P [ D > n ] = 1 n ! (cid:90) ∞ x n e − x P (cid:104) c /τ W d/α > x (cid:105) dx = 1 n ! (cid:90) c /τ x n e − x dx + c n ! (cid:90) ∞ c /τ x n − τ e − x dx = e − c /τ ∞ (cid:88) j = n +1 c j/τ j ! + c Γ( n + 1) (cid:90) ∞ c /τ x n − τ e − x dx. Choose n ≥ n − τ + 1 >
0. Then, P [ D > n ] = P [ Z > n ] + c Γ( n + 1 − τ )Γ( n + 1) P (cid:104) Y n > c /τ (cid:105) , where Z has a Poisson distribution with parameter c /τ , and Y n has a gamma distribution with shape parameter n − τ + 1 > n τ P [ Z ≥ n ] = n τ P (cid:104) e Z − n ≥ (cid:105) ≤ n τ E (cid:104) e Z (cid:105) e − n , which converges to 0 as n → ∞ . Moreover, by Stirling’s formula, n τ Γ( n + 1 − τ ) / Γ( n + 1) tends to 1 as n → ∞ ,and so does P [ Y n > s ] for any s >
0. Therefore,lim n →∞ n τ P [ D > n ] = c . (cid:3) Phase transition
In this section we prove Theorem 3.2 which gives the phase transition picture of the hetero-geneous RCM in the case min { α, βα } > d . We first prove (a1) and (b1) , namely that in anydimension d ≥ λ c equals 0 whenever α > d and τ = βα/d ∈ (1 , Proof of (a1) and (b1) . The idea of the proof is similar to the proof of Theorem 4.4 in [11]. Assume α > d and τ = βα/d ∈ (1 , θ ( λ ) > λ >
0. Choose λ > <ε < min { d/β, α (2 /τ − } . Define for k ≥ k = [ − k , k ) d and for k ≥ R k = Λ k \ Λ k − . For k ≥ z k the particle with maximal weight in R k (if it exists). Using thatfor k ≥
1, the number of particles in R k , denoted by X ( R k ), has a Poisson distribution with parameter of order ν dk , one derives that the event { X ( R k ) ≥ W z k ≥ k ( d/β − ε ) for all k ≥ } has positive probability. Giventhis event, we obtain that z k − ⇔ z k for all k ≥ z = 0. This implies θ ( λ ) >
0. We refer to [11] for the details. (cid:3)
Next, we prove (a2) and (b2) , which provide the non-trivial phase transition for appropriatechoices of α and β . Proof of (a2) and (b2) . We first show that for any dimension d ≥ λ c > α > d and τ = βα/d > α > d and τ = βα/d >
2. Set x = 0. We say that( x , . . . , x n ) ∈ X n is a self-avoiding path in X of length n ∈ N starting from the origin, write ( x , . . . , x n ) s.a.,if for all i = 1 , . . . , n there is an edge between x i − and x i , and every particle x i in that path occurs at mostonce. Since the degree distribution has finite mean, see Theorem 3.1, we obtain that the degree of each particle insuch a path is bounded, a.s. Therefore, the event that the origin lies in an infinite connected component impliesthat for each n ∈ N there is a self-avoiding path in X of length n starting from the origin. Therefore, using1 − e − x ≤ x ∧ θ ( λ ) ≤ E (cid:88) ( x , . . . , x n ) s.a. n (cid:89) i =1 p x i − x i ≤ E (cid:88) ( x , . . . , x n ) s.a. E (cid:34) n (cid:89) i =1 (cid:18) λW x i − W x i | x i − − x i | α ∧ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) . For n even and distinct x , . . . , x n ∈ R d , Cauchy-Schwarz’ inequality implies that E (cid:34) n (cid:89) i =1 (cid:18) λW x i − W x i | x i − − x i | α ∧ (cid:19)(cid:35) = E n/ (cid:89) i =1 (cid:18) λW x i − W x i − | x i − − x i − | α ∧ (cid:19) n/ (cid:89) i =1 (cid:18) λW x i − W x i | x i − − x i | α ∧ (cid:19) ≤ n (cid:89) i =1 E (cid:34)(cid:18) λW x i − W x i | x i − − x i | α ∧ (cid:19) (cid:35) / , and similarly for n odd. It follows that for all n ∈ N , θ ( λ ) ≤ E (cid:88) ( x , . . . , x n ) s.a. n (cid:89) i =1 E (cid:34) (cid:18) λW x i − W x i | x i − − x i | α (cid:19) ∧ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:35) / . Similar to Lemma 4.3 of [11] we get for u ≥ W and W having a Paretodistribution with scale parameter 1 and shape parameter β , using integration by parts in the first step, E (cid:2) ( W W /u ) ∧ (cid:3) = 1 u + 2 u (cid:90) u v P [ W W > v ] dv = 1 u + 2 u (cid:90) u v − β (1 + β log v ) dv ≤ (1 + β log u ) (cid:18) u − ( β ∧ + 2 u (cid:90) u v − β dv (cid:19) ≤ (1 + 1 { β (cid:54) =2 } / | β − | ) (1 + max { , β } log u ) u − ( β ∧ , here the last step follows by considering the cases β < β = 2 and β > u ≥
0, set c = c ( β ) = (1 + 1 { β (cid:54) =2 } / | β − | ) / , E (cid:2) ( W W /u ) ∧ (cid:3) / ≤ { u< } + 1 { u ≥ } c (1 + max { , β } log u ) u − ( β/ ∧ = g ( u ) , (5.1)where the equality defines the function g . Using this bound, we obtain θ ( λ ) ≤ E (cid:88) ( x , . . . , x n ) s.a. n (cid:89) i =1 g ( λ − | x i − x i − | α ) ≤ ν n (cid:90) R d · · · (cid:90) R d n (cid:89) i =1 g ( λ − | x i − x i − | α ) dx · · · dx n = (cid:18) ν (cid:90) R d g ( λ − | x | α ) dx (cid:19) n . (5.2)Choose λ < (cid:0) { , β } log( λ − ) (cid:1) λ ( β/ ∧ (1 / ≤
1. This implies for all u ≥ g ( λ − u ) = 1 { u<λ } + 1 { u ≥ λ } c (cid:0) { , β } log( λ − u ) (cid:1) ( λ − u ) − ( β/ ∧ ≤ { u<λ } + 1 { u ≥ λ } λ ( β/ ∧ (1 / c (1 + max { , β } log u ) u − ( β/ ∧ . This provides upper bound (cid:90) R d g ( λ − | x | α ) dx ≤ v d λ d/α + λ ( β/ ∧ (1 / c (cid:90) | x |≥ λ /α (1 + max { , β } log( | x | α )) | x | − α ( β/ ∧ dx. Note that the latter integral is finite because α > d and βα/ > d . Therefore, we can choose λ > / (2 ν ). We finally obtain for all λ > θ ( λ ) ≤ − n → , as n → ∞ .This finally implies that for any dimension d ≥ λ c > α > d and τ = βα/d > (a2) and (b2) it remains to show that the critical percolation value λ c is finite in the case τ = βα/d > α > d ( α ∈ (1 ,
2] if d = 1). We adapt the proof of Theorem 3.1 of [11].Partition the space R d into cubes of side-length n and let r = r ( n, d ) be the maximal possible distance betweentwo particles in neighboring cubes. We call a cube Λ in the partition of R d good if X (Λ) ≥
1, i.e. at least oneparticle x of the Poisson cloud X falls into Λ. Note that a cube Λ is good with probability 1 − exp {− νn d } . Iftwo neighboring cubes are both good, then the probability that the two particles with maximal weight in therespective cubes are connected is bounded below by 1 − exp {− λr − α } , note that W ≥
1, a.s. We now considerthe site-bond percolation model on Z d where sites are alive independently with probability 1 − exp {− νn d } andedges are added independently between alive nearest-neighbor sites with probability 1 − exp {− λr − α } . Note that1 − exp {− νn d } can be chosen arbitrarily close to 1 by taking n large and then the nearest-neighbor edges can haveprobabilities arbitrarily close to 1 by choosing λ large. Therefore, in dimensions d ≥
2, the site-bond percolationmodel on Z d percolates for sufficiently large λ , see [17] and Theorem 3.2 of [11]. In the case d = 1 and α ∈ (1 ,
2] itfollows from Theorem 1.2 of [19] that the described site-bond percolation model percolates. But this immediatelyimplies that there exists an infinite connected component, a.s., in our model for sufficiently large λ . (cid:3) In order to complete the proof of Theorem 3.2 it remains to prove (b3) , namely that the criticalpercolation value λ c is infinite in dimension d = 1 whenever min { α, βα } >
2. The proof is similarto the one of part (c) in Theorem 3.1 of [11]. Since the Poisson cloud induces an additional levelof complexity, we prove (b3) in detail. 13 roof of (b3) . Assume d = 1 and min { α, βα } >
2. We describe the particles of a Poisson cloud X = ( x i ) i ∈ Z containing the origin as follows. x = 0; x i = inf { x ∈ X | x > x i − } , for i ∈ N ; x − i = sup { x ∈ X | x < x − i +1 } , for i ∈ N .Note that x i − x i − , i ∈ Z , are i.i.d. having an exponential distribution with parameter ν . The proof of (b3) issimpler in the case where the weights ( W x ) x ∈ X have finite mean. We start with this case. Case 1.
Assume β > E [ W ] < ∞ . Choose x ∈ R and define the event A x = { no particle y ≤ x shares an edge with any particle z > x } . The aim is to prove P [ A ] >
0. Stationarity and ergodicity then imply that A x occurs for infinitely many x ∈ R ,a.s., hence λ c = ∞ . We define for n ∈ N the event A ( n )0 = { x − n + k (cid:54)⇔ x k for all k = 1 , . . . , n } . We get, using independence of edges, P [ A ] = P (cid:34) (cid:92) n ∈ N A ( n )0 (cid:35) = E (cid:34) (cid:89) n ∈ N n (cid:89) k =1 exp (cid:8) − λW x − n + k W x k ( x k − x − n + k ) − α (cid:9)(cid:35) . Note that the weights ( W x ) x ∈ X are independent for different particles and also independent of their locations.Using Jensen’s inequality, we therefore get P [ A ] ≥ exp (cid:40) − λ (cid:88) n ∈ N n (cid:88) k =1 E (cid:2) W x − n + k (cid:3) E [ W x k ] E (cid:2) ( x k − x − n + k ) − α (cid:3)(cid:41) . Since for each n ∈ N and k = 1 , . . . , n , the difference x k − x − n + k is the sum of n i.i.d. random variables havingexponential distributions with parameter ν , x k − x − n + k has a gamma distribution with shape parameter n andscale parameter ν . Therefore, P [ A ] ≥ exp (cid:40) − λ E [ W ] (cid:88) n ∈ N n (cid:88) k =1 ν α Γ( n − α )Γ( n ) (cid:41) = exp (cid:40) − λ E [ W ] ν α (cid:88) n ∈ N n Γ( n − α ) n ! (cid:41) . By Stirling’s approximation and since α >
2, the latter sum is finite, which finishes the proof in the case β > Case 2.
Assume β ≤
1. Using independence of edges, we obtain P [ A ] = E exp − λ (cid:88) i,j ≥ , ( i,j ) (cid:54) =(0 , W x − i W x j ( x j − x − i ) − α . Since we condition on having a particle at the origin, we obtain x j − x − i = x j + | x − i | for all i, j ≥ x j + | x − i | ) ≥ x j | x − i | > i, j ≥
1. This implies P [ A ] = E exp − λW (cid:88) j ≥ W x j x − αj − λW (cid:88) i ≥ W x − i | x − i | − α − λ (cid:88) i,j ≥ W x − i W x j ( x j − x − i ) − α ≥ E exp − λW (cid:88) j ≥ W x j x − αj − λW (cid:88) i ≥ W x − i | x − i | − α − λ (cid:88) i ≥ W x − i | x − i | − α/ (cid:88) j ≥ W x j x − α/ j . In order to prove that all sums on the right-hand side are finite, a.s., it suffices to check that E (cid:104)(cid:80) j ≥ W x j x − α/ j (cid:105) < ∞ . Since β ≤ βα > ε > − α/ − β )(1 − ε ) /β < −
1, and we set a j = j (1+ ε ) /β for j ≥
1. Write (cid:88) j ≥ W x j x − α/ j = (cid:88) j ≥ W x j ∧ a j x α/ j + (cid:88) j ≥ ( W x j − a j ) + x α/ j . (5.3) ince P [ W > a j ] = a − βj = j − (1+ ε ) , the Borel-Cantelli lemma implies that ( W x j − a j ) + is positive for only finitelymany j ≥
1, a.s. Hence, the second sum in (5.3) is finite, a.s. For the first sum in (5.3) note that E (cid:34) W x j ∧ a j x α/ j (cid:35) = E (cid:104) x − α/ j (cid:105) E (cid:2) W x j ∧ a j (cid:3) ≤ ν α/ Γ( j − α/ j ) (cid:88) ≤ k ≤ a j P [ W > k ] = ν α/ j Γ( j − α/ j ! (cid:88) ≤ k ≤ a j k − β . For β < c >
0, using Stirling’s approximation, E (cid:34) W x j ∧ a j x α/ j (cid:35) ≤ c j Γ( j − α/ j ! a − βj = c Γ( j − α/ j ! j − β )(1+ ε ) /β = c j − α/ − β )(1+ ε ) /β (1 + o (1)) , as j → ∞ . By the choice of ε >
0, the right-hand side is summable in j . The same conclusion holds true in thecase β = 1. This completes the proof of (b3) . (cid:3) In this section we prove Theorems 3.3 and 3.4 and Corollary 3.5. The key result is Lemma 6.1below which corresponds to Lemma 2.3 of [3] in homogeneous long-range percolation on Z d . Lemma 6.1
Assume α ∈ ( d, d ) and τ = βα/d > , and choose λ ∈ (0 , ∞ ) with θ ( λ, α ) > .For every ε ∈ (0 , , ρ > and α (cid:48) < d there exists m (cid:48) ≥ such that for all m ≥ m (cid:48) , P (cid:104) |C m | ≥ ρm α (cid:48) / (cid:105) ≥ − ε, where |C m | denotes the number of particles of the largest connected component in Λ m . The proof of Lemma 6.1 is based on a renormalization technique introduced in [3]. We explainthis renormalization in detail because the Poisson cloud induces an additional level of complexity.For integers m ≥ k ≥ x ∈ m Z d we define the box with corner x and side length m andits k -enlargement by B m ( x ) = x + [0 , m ) d and B ( k ) m ( x ) = x + [ − k, m + k ) d , respectively. We write B m and B ( k ) m if x = 0. We call a set of at least (cid:96) ≥ B m ( x ) ∩ X an (cid:96) -semi-cluster if these particles are connected within its k -enlargement B ( k ) m ( x ).For an integer valued sequence ( a n ) n ∈ N with a n > n ≥
0, we define for n ∈ N the cubelengths m n = a n m n − = m n (cid:89) i =1 a i = n (cid:89) i =0 a i , with m = a .For x ∈ m n Z d we call B m n ( x ) an n -stage box . Note that each n -stage box contains a dn of ( n − B m n − ( z ) ⊂ B m n ( x ) with z ∈ m n − Z d , which we call children of B m n ( x ). In thefollowing we recursively define the aliveness of n -stage boxes. Definition 6.2
Let ( a n ) n ∈ N be an integer valued sequence with a n > , n ≥ , and define ( m n ) n ∈ N as above. Choose k ≥ and let ( θ n ) n ∈ N be a real valued sequence with θ n ∈ (0 , for n ≥ . For x ∈ m Z d we say that -stage box B m ( x ) is alive if it contains a ( θ a d ) -semi-cluster,i.e. it contains at least θ a d particles that are connected within B ( k ) m ( x ) . • For n ∈ N and x ∈ m n Z d we say that n -stage box B m n ( x ) is alive if the event A n,x = A ( a ) n,x ∩ A ( b ) n,x occurs, where A ( a ) n,x = { at least θ n a dn children of B m n ( x ) are alive } ; A ( b ) n,x = { all ( (cid:81) n − i =0 θ i a di ) -semi-clusters of all alive children of B m n ( x ) are connected within B ( k ) m n ( x ) } . For n ∈ N define u n = (cid:81) ni =0 θ i and note that every alive n -stage box B m n ( x ) contains at least (cid:81) ni =0 θ i a di = m dn u n particles that are connected within k -enlargement B ( k ) m n ( x ). The next lemmaprovides a recursive lower bound for p n = P [ A n,x ], n ∈ N and x ∈ m n Z d . Lemma 6.3
Assume α ∈ ( d, d ) and choose λ > . Let ξ ∈ ( α/d, and γ ∈ (0 , such that γ >
16 + ξ . Choose a real valued sequence ( θ n ) n ∈ N with θ n ∈ (0 , , n ∈ N , and an integervalued sequence ( a n ) n ∈ N with a n > , n ∈ N . Assume that there exists m (cid:48) ≥ such that for all a = m ≥ m (cid:48) , all n ∈ N and all s ∈ (2 e − νm dn − , eνm dn − ) , s γ < m dn − n − (cid:89) i =0 θ i = u n − m dn − and s − ξ < λ (cid:16) √ dm n (cid:17) − α . (6.1) There exist ϕ > and m (cid:48)(cid:48) ≥ m (cid:48) such that for every a = m ≥ m (cid:48)(cid:48) , k ≥ , and for all n ∈ N and x ∈ m n Z d , p n = P [ A n,x ] ≥ − − p n − − θ n − a dn (cid:18) e − νm dn − (1 − /e ) + (cid:16) e − νm dn − (cid:17) − ϕ (cid:19) . Proof of Lemma 6.3.
Let n ∈ N and x ∈ m n Z d . We obtain1 − p n = P (cid:2) A c n,x (cid:3) = P (cid:104) ( A ( a ) n,x ∩ A ( b ) n,x ) c (cid:105) ≤ P (cid:104) ( A ( a ) n,x ) c (cid:105) + P (cid:104) ( A ( b ) n,x ) c (cid:105) . (6.2)For the first term in (6.2), Markov’s inequality and translation invariance provide P (cid:104) ( A ( a ) n,x ) c (cid:105) = P (cid:88) B mn − ( z ) ⊂ B mn ( x ) A n − ,z < θ n a dn = P (cid:88) B mn − ( z ) ⊂ B mn ( x ) A c n − ,z > (1 − θ n ) a dn ≤ − θ n ) a dn (cid:88) B mn − ( z ) ⊂ B mn ( x ) P (cid:2) A c n − ,z (cid:3) = 11 − θ n P (cid:2) A c n − ,z (cid:3) = 1 − p n − − θ n . The second term in (6.2) is more involved due to possible dependence in the k -enlargements, k ≥
0. For twochildren B and B of B m n ( x ) let E ( B , B ) be the event that at least two ( u n − m dn − )-semi-clusters in B ∪ B are not connected within B ( k ) m n ( x ). We obtain P (cid:104) ( A ( b ) n,x ) c (cid:105) ≤ (cid:32) a dn (cid:33) sup ( B ,B ) P (cid:2) E ( B , B ) (cid:3) , (6.3) here the supremum is taken over all possible choices of two distinct children B and B of B m n ( x ). We fix twodifferent children B and B of B m n ( x ). We obtain P (cid:2) E ( B , B ) (cid:3) ≤ P (cid:104) E ( B , B ) , e − νm dn − < X ( B ∪ B ) < eνm dn − (cid:105) + P (cid:104) X ( B ∪ B ) ≥ eνm dn − (cid:105) + P (cid:104) X ( B ∪ B ) ≤ e − νm dn − (cid:105) , where X ( A ) denotes the number of particles of X in A ⊂ R d . A random variable Y having a Poisson distributionwith parameter µ > P (cid:2) Y ≤ e − µ (cid:3) ≤ e − µ (1 − /e ) and P [ Y ≥ eµ ] ≤ e − µ , (6.4)see for instance (A.12) of [14]. Using these bounds we obtain P (cid:2) E ( B , B ) (cid:3) ≤ P (cid:104) E ( B , B ) , e − νm dn − < X ( B ∪ B ) < eνm dn − (cid:105) + 2 e − νm dn − (1 − /e ) . To estimate the probability above we will condition on the Poisson cloud restricted to B ( k ) m n ( x ). We fix integers s ∈ (2 e − νm dn − , eνm dn − ), t ≥ x , . . . , x s + t ∈ B ( k ) m n ( x ) with x , . . . , x s ∈ B ∪ B . Assume that { x , . . . , x s } = ( B ∪ B ) ∩ X and { x , . . . , x t } = B ( k ) m n ( x ) ∩ X . Consider the edge probabilities (cid:101) p x i x j = 1 − exp {− λ | x i − x j | − α } ≤ − exp {− λW x i W x j | x i − x j | − α } = p x i x j , a.s., for every i (cid:54) = j ∈ { , . . . , s + t } . Denote by (cid:101) P X the probability measure of the resulting edge configurationsrestricted to { x , . . . , x s + t } induced by (cid:101) p x i x j , i (cid:54) = j ∈ { , . . . , s + t } . Note that on B ( k ) m n ( x ) ∩ X = { x , . . . , x s + t } , E ( B , B ) is determined by edges with end points in { x , . . . , x s + t } . We therefore get P (cid:20) E ( B , B ) (cid:12)(cid:12)(cid:12)(cid:12) X ∩ (cid:0) B ∪ B (cid:1) = { x , . . . , x s } , X ∩ B ( k ) m n ( x ) = { x , . . . , x s + t } (cid:21) ≤ (cid:101) P X [ E ( B , B )] . We can now argue as in the proof of Lemma 2.3 of [5], which we briefly recall. By an abuse of notation, assumethat B ∪ B = { x , . . . , x s } and B ( k ) m n ( x ) = { x , . . . , x s + t } . Using that sequence ( a n ) n ∈ N satisfies (6.1) and B ∪ B ⊂ B m n ( x ), we obtain (cid:101) p x i x j > − exp {− s − ξ } = ν n > x i (cid:54) = x j ∈ B ∪ B , where the equalitydefines ν n . For x i (cid:54) = x j ∈ B ∪ B we define (cid:101) q x i x j by (cid:101) p x i x j = (cid:101) q x i x j + ν n − ν n (cid:101) q x i x j . We now sample an edgeconfiguration ω on B ( k ) m n ( x ) induced by the (cid:101) p x i x j ’s in two steps. We sample ω (cid:48) on B ( k ) m n ( x ) with edge probabilities (cid:101) q x i x j if x i (cid:54) = x j ∈ B ∪ B and with edge probabilities (cid:101) p x i x j otherwise. ω (cid:48)(cid:48) is then independently sampled on B ∪ B with edge probabilities ν n . Note that ω = ω (cid:48) ∨ ω (cid:48)(cid:48) in distribution. Let S (cid:54) = S ⊂ B ∪ B be two disjointmaximal sets in B ∪ B that are each connected within B ( k ) m n ( x ) by ω (cid:48) -edges. Here, the maximality of set S means that there is no particle in B ∪ B which does not belong to S but is connected to S within B ( k ) m n ( x )by ω (cid:48) -edges. Given ω (cid:48) , the probability that there is an ω -edge between S and S is equal to the probability thatthere is an ω (cid:48)(cid:48) -edge between S and S , given ω (cid:48) , which follows by maximality of S and S . The latter probabilityis by definition of ω (cid:48)(cid:48) and ν n given by 1 − exp {−| S || S | s − ξ } , where | S i | denotes the number of particles of S i , i = 1 ,
2. Given ω (cid:48) , denoting by S , . . . , S l all disjoint maximal sets in B ∪ B that are connected within B ( k ) m n ( x )by ω (cid:48) -edges, the indices { , . . . , l } form an inhomogeneous random graph of size (cid:80) li =1 | S i | = s and parameter ξ ,as defined in [5]. Lemma 2.5 of [5] shows that there exists ϕ > M sufficiently large and allinhomogeneous random graphs of size M and parameter ξ , the probability that such a graph contains more thanone cluster of size at least M γ , is at most M − ϕ . This implies that there exist ϕ > m (cid:48)(cid:48) ≥ m (cid:48) such that forall m ≥ m (cid:48)(cid:48) , given ω (cid:48) , there is at most one cluster of size at least s γ formed by S , . . . , S l that are connected by ω -edges, with probability at most s − ϕ . Note that the existence of ϕ is uniform in s ∈ (2 e − νm dn − , eνm dn − ), t ≥
0, the locations of { x , . . . , x s + t } , ω (cid:48) , n ≥ k ≥ m ≥ m (cid:48)(cid:48) . Using that sequences ( a n ) n ∈ N and ( θ n ) n ∈ N satisfy (6.1), i.e. s γ < u n − m dn − , we conclude that there exist ϕ > m (cid:48)(cid:48) ≥ m (cid:48) such that for every m ≥ m (cid:48)(cid:48) , k ≥ n ∈ N , s ∈ (2 e − νm dn − , eνm dn − ), t ≥
0, and x , . . . , x s + t ∈ B ( k ) m n ( x ) with x , . . . , x s ∈ B ∪ B , (cid:101) P X [ E ( B , B )] ≤ s − ϕ . ntegrating over the particles in B ( k ) m n ( x ) \ ( B ∪ B ) and B ∪ B we then get for all m ≥ m (cid:48)(cid:48) , P (cid:2) E ( B , B ) (cid:3) ≤ e − νm dn − (1 − /e ) + (cid:100) eνm dn − − (cid:101) (cid:88) s = (cid:98) e − νm dn − +1 (cid:99) P (cid:2) X ( B ∪ B ) = s (cid:3) s − ϕ ≤ e − νm dn − (1 − /e ) + (cid:16) e − νm dn − (cid:17) − ϕ (cid:100) eνm dn − − (cid:101) (cid:88) s = (cid:98) e − νm dn − +1 (cid:99) P (cid:2) X ( B ∪ B ) = s (cid:3) ≤ e − νm dn − (1 − /e ) + (cid:16) e − νm dn − (cid:17) − ϕ , which together with (6.3) implies for all m ≥ m (cid:48)(cid:48) , k ≥ n ∈ N , P (cid:104) ( A ( b ) n,x ) c (cid:105) ≤ a dn (cid:18) e − νm dn − (1 − /e ) + (cid:16) e − νm dn − (cid:17) − ϕ (cid:19) . This finishes the proof of Lemma 6.3. (cid:3)
Using the recursion in Lemma 6.3 and sequences ( a n ) n ∈ N = ( n a ) n ∈ N and ( θ n ) n ∈ N = ( n − b ) n ∈ N for appropriate constants 1 < a < b , for explicit choices see (5) in [5], 1 − p n can be boundedby a multiple (independent of n ) of 1 − p . In order to prove that p n is arbitrarily close to 1,it remains to show that 0-stage boxes are alive with arbitrarily high probability for sufficientlylarge a = m and well chosen θ ∈ (0 , Proof of Lemma 6.1.
Note that α ∈ ( d, d ) and τ = βα/d > λ c < ∞ , see Theorem 3.2. Hence, thereexists λ ∈ (0 , ∞ ) with θ = θ ( λ, α ) > C ∞ ⊂ X , a.s. In order to prove Lemma 6.1 it suffices to check that for any ε (cid:48) ∈ (0 ,
1) there exists θ ∈ (0 ,
1) suchthat p = P [ A , ] > − ε (cid:48) for all m sufficiently large. The remainder of the proof then follows from the one ofLemma 2.3 of [5] using Lemma 6.3 above with appropriate sequences ( a n ) n ∈ N and ( θ n ) n ∈ N . Choose ε (cid:48) ∈ (0 , θ = νv d − d θ/
2, where v d denotes the Lebesgue measure of the unit ball in R d . For m ≥ |C ∞ ∩ B m | the number of particles in B m that belong to the infinite connected component C ∞ . For m ≥ P (cid:104) |C ∞ ∩ B m | ≥ θ m d (cid:105) ≥ P (cid:88) x ∈ X ∩ B ( m/ { x ∈C ∞ } ≥ θ m d , where B ( m/
2) denotes the ball of radius m/ . .
3) of [10] it followsthat 1 νv d ( m/ d (cid:88) x ∈ X ∩ B ( m/ { x ∈C ∞ } −→ P [0 ∈ C ∞ ] = θ, as m → ∞ , a.s.Hence, for all m sufficiently large we obtain P (cid:104) |C ∞ ∩ B m | ≥ θ m d (cid:105) ≥ − ε (cid:48) / . Since the infinite connected component C ∞ is unique, a.s., there exists k = k ( m ) ≥ C ∞ ∩ B m isconnected within k -enlargement B ( k ) m . Choose k = k ( m ) ≥ P (cid:104) C ∞ ∩ B m is connected within B ( k ) m (cid:105) > − ε (cid:48) / . This implies that for all m sufficiently large, p = P [ A , ] ≥ P (cid:104) |C ∞ ∩ B m | ≥ θ m d and C ∞ ∩ B m is connected within B ( k ) m (cid:105) ≥ − ε (cid:48) . (cid:3) roof of Theorems 3.3 and 3.4. Note that by Lemma 6.1, the number of particles of the largest connectedcomponent in Λ m is at least ρm α (cid:48) for ρ > α (cid:48) < d , with high probability. In order to prove that this numberis proportional to m d , with high probability, we apply a second renormalization based on site-bond percolation,as in the proof of Theorem 6 in [12], where the bound on the probability that a site is alive is given by Lemma 6.1above. Theorem 3.4 then follows directly from the results in [12]. This is immediately clear because the boundson attachedness of alive sites derived in Lemma 10 (a) in [12] also apply to the Poisson case, see in particularestimate (2) in [12]. In the same way, using a site-bond percolation model, Theorem 3.3 follows from Theorem 3in [12]. (cid:3) Proof of Corollary 3.5.
Using Theorem 3.4, the statements ( i ) and ( ii ) of Corollary 3.5 follow from Corollaries3.3 and 3.4 of [6], respectively. Note that Lemma 3.5 therein corresponds to Lemma 10 (a) in [12]. We refer to[6] for the details. (cid:3) In order to prove Theorem 3.6 we first prove the upper bound of statement (a) which we recallin the next proposition.
Proposition 7.1
Assume α > d and τ = βα/d ∈ (1 , . For every λ > λ c = 0 and ε > weobtain lim | x |→∞ P (cid:20) d (0 , x ) ≤ (1 + ε ) 2 log log | x || log( βα/d − | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ (cid:21) = 1 . For the proof of this proposition we use the following technical lemma.
Lemma 7.2
Assume α > d and τ = βα/d ∈ (1 , . If there exists a particle in Λ r = [ − r, r ) d , r > , let z ∈ X ∩ Λ r be the particle with maximal weight W z in Λ r . We obtain for anyconstant c ≥ , lim w →∞ P (cid:104) |C ( z ) | = ∞ (cid:12)(cid:12)(cid:12) W z ≥ w, X (Λ r ) ≥ c (cid:105) = 1 , where |C ( z ) | denotes the number of particles of the connected component of z , and where X (Λ r ) denotes the number of particles in box Λ r . Note that this result corresponds to Lemma 5.2 of [11]. The only difference lies in the additionalcondition that Λ r contains at least c ≥ z with maximalweight in Λ r exists. Proof of Lemma 7.2.
Let α > d and τ = βα/d ∈ (1 , b > d/β − bα > ε ∈ ( α/ , d/β ). Choose r > w ≥ R = Λ w /α \ Λ r , R = Λ w b \ Λ w /α and R k = Λ w b k \ Λ w b k − for k ≥
3. If X (Λ r ) ≥ c , let z ∈ X ∩ Λ r be the particlewith maximal weight W z in Λ r . For k ≥
1, if X ( R k ) ≥
1, let z k ∈ X ∩ R k be the particle with maximalweight W z k in R k . By the choices of b and ε , and since X ( R k ) is Poisson distributed, we obtain that the event { X ( R k ) ≥ W z k ≥ kε w d/β for all k ≥ } has probability arbitrarily close to 1 for w sufficiently large.Given this event and the event { X (Λ r ) ≥ c, W z ≥ w } , we obtain that z ⇔ z and z k +1 ⇔ z k for all k ≥ robability arbitrarily close to 1 for w sufficiently large, which implies the claim. We refer to the correspondingproof in [11] for the details. (cid:3) Proof of Proposition 7.1.
Let α > d and τ = βα/d ∈ (1 , ε > b ∈ (0 ,
1) such that d (1 + b ) /β > α and 1 + ε/ | log b | ≤ ε | log( βα/d − | . Fix m > max { e, / (1 − b ) , c − /d , m , m } , with c = (1 − /e ) ν − d and where m = m ( b ) ≥ m = m ( b ) ≥ x ∈ R d with | x | ≥ e (log m ) /b and set k = k ( x ) = (cid:22) log log | x | − log log m | log b | (cid:23) ≥ . Note that m ≤ | x | b k ≤ m /b for all x ∈ R d . For i = 0 , , . . . , k write Λ( x, b i ) for the box with side length | x | b i / | x | b i / x . If there is aparticle in Λ( x, b i ), let z i ∈ Λ( x, b i ) ∩ X be the particle with maximal weight W z i in Λ( x, b i ). Moreover, writeΛ (cid:48) ( x, b i ) for the box with side length | x | b i / | x | b i / x (instead of the origin)on the segment with end points 0 and x . If there is a particle in Λ (cid:48) ( x, b i ), let z (cid:48) i ∈ Λ (cid:48) ( x, b i ) ∩ X be the particlewith maximal weight W z (cid:48) i in Λ (cid:48) ( x, b i ), this choice is similar to the one in the proof of Theorem 5.1 of [11]. Inorder to make sure that particles z i and z (cid:48) i exist and that boxes Λ( x, b i ) and Λ (cid:48) ( x, b i ) contain sufficiently manyparticles for all i = 0 , . . . , k we consider the probability measure P k [ · ] = P (cid:104) · (cid:12)(cid:12)(cid:12) X (Λ( x, b i )) ≥ c | x | db i and X (Λ (cid:48) ( x, b i )) ≥ c | x | db i for all i = 0 , . . . , k (cid:105) , which is the conditional probability given that there are at least c | x | db i ≥ c m d ≥ x, b i ) and Λ (cid:48) ( x, b i ) for i = 0 , . . . , k . Using Chernoff’s bound, see (6.4), we obtain for each i = 0 , . . . , k , since X (Λ( x, b i )) has a Poisson distribution with parameter ν − d | x | db i , P (cid:104) X (Λ( x, b i )) < c | x | db i (cid:105) ≤ P (cid:104) X (Λ( x, b i )) ≤ e − ν − d | x | db i (cid:105) ≤ e − c | x | dbi , for each i = 0 , . . . , k . Note that m > / (1 − b ) and m ≤ | x | b k imply | x | b i − | x | b i +1 > i = 0 , . . . , k − x, b i ) are disjoint for i = 0 , . . . , k . It follows that P (cid:104) X (Λ( x, b i )) ≥ c | x | db i and X (Λ (cid:48) ( x, b i )) ≥ c | x | db i for all i = 0 , . . . , k (cid:105) ≥ k (cid:89) i =0 (cid:18) − e − c | x | dbi (cid:19) , where the inequality comes from the fact that Λ( x,
1) = Λ (cid:48) ( x, k = k ( x ) = (cid:98) N ( x ) − M (cid:99) with N = N ( x ) = (log log | x | ) / | log b | and M = (log log m ) / | log b | , and we obtain, note that | log b | = − log b (because b ∈ (0 , | x |→∞ k (cid:88) i =0 exp (cid:110) − c | x | db i (cid:111) = lim N →∞ (cid:98) N − M (cid:99) (cid:88) i =0 exp (cid:110) − c e b − N db i (cid:111) ≤ lim N →∞ (cid:98) N − M (cid:99) (cid:88) i =0 exp (cid:110) − c e db − ( (cid:98) N − M (cid:99)− i ) − M (cid:111) = lim N →∞ (cid:98) N − M (cid:99) (cid:88) j =0 exp (cid:110) − c e db − j − M (cid:111) = (cid:88) j ≥ exp (cid:110) − c e db − j − M (cid:111) ∈ (0 , ∞ ) . (7.1)For fixed ε (cid:48) > m ≥ m > m ) so large that for any sufficientlylarge | x | , the first equality defines event N k = N k ( x, b ), P [ N k ] = P (cid:104) X (Λ( x, b i )) ≥ c | x | db i and X (Λ (cid:48) ( x, b i )) ≥ c | x | db i for all i = 0 , . . . , k (cid:105) ≥ − ε (cid:48) . (7.2)Note that for every δ ∈ (0 ,
1) and i = 0 , . . . , k , P k (cid:20) W z i ≤ (cid:16) c | x | db i (cid:17) (1 − δ ) /β (cid:21) ≤ P (cid:104) W z i ≤ X (Λ( x, b i )) (1 − δ ) /β (cid:12)(cid:12)(cid:12) X (Λ( x, b i )) ≥ c | x | db i (cid:105) = P (cid:20) max z ∈ Λ( x,b i ) ∩ X W z ≤ X (Λ( x, b i )) (1 − δ ) /β (cid:12)(cid:12)(cid:12)(cid:12) X (Λ( x, b i )) ≥ c | x | db i (cid:21) = E (cid:20) (cid:16) − X (Λ( x, b i )) δ − (cid:17) X (Λ( x,b i )) (cid:12)(cid:12)(cid:12)(cid:12) X (Λ( x, b i )) ≥ c | x | db i (cid:21) . sing 1 − x ≤ e − x , we obtain P k (cid:20) W z i ≤ (cid:16) c | x | db i (cid:17) (1 − δ ) /β (cid:21) ≤ E (cid:104) e − X (Λ( x,b i )) δ (cid:12)(cid:12)(cid:12) X (Λ( x, b i )) ≥ c | x | db i (cid:105) ≤ e − c δ | x | dδbi . (7.3)Since | z i − z i +1 | < d | x | b i for each i = 0 , . . . , k −
1, we therefore obtain P k (cid:34) k − (cid:91) i =0 { z i (cid:54)⇔ z i +1 } (cid:35) ≤ k − (cid:88) i =0 E k (cid:20) e − λd − α W zi W zi +1 | x | − αbi (cid:21) ≤ k − (cid:88) i =0 E k (cid:34) e − λd − α W zi W zi +1 | x | − αbi (cid:26) W zj > (cid:16) c | x | dbj (cid:17) (1 − δ ) /β ; j = i,i +1 (cid:27) (cid:35) + 2 e − c δ | x | dδbi +1 ≤ k − (cid:88) i =0 exp (cid:110) − λd − α c − δ ) /β | x | b i d (1 − δ )(1+ b ) /β | x | − αb i (cid:111) + 2 e − c δ | x | dδbi +1 = k − (cid:88) i =0 exp (cid:110) − λd − α c − δ ) /β | x | b i ( d (1 − δ )(1+ b ) /β − α ) (cid:111) + 2 e − c δ | x | dδbi +1 . Since b was chosen such that d (1+ b ) /β > α , we can choose δ = δ ( b ) ∈ (0 ,
1) so small that d (1 − δ )(1+ b ) /β − α > m ≥ m > m ) so large that forany sufficiently large | x | , P k (cid:104)(cid:83) k − i =0 { z i (cid:54)⇔ z i +1 } (cid:105) ≤ ε (cid:48) . Using symmetry we therefore obtain for sufficiently large | x | , P k (cid:2) z i ⇔ z i +1 and z (cid:48) i ⇔ z (cid:48) i +1 for all i = 0 , . . . , k − (cid:3) ≥ − ε (cid:48) . (7.4)Recall that z is the Poisson particle with maximal weight W z in box Λ( x,
1) with side length | x | / x . For every w ≥ | x | , seealso (7.3) with i = 0, P k [ W z ≤ w ] ≤ ε (cid:48) . Moreover, using that box Λ( x,
1) does not intersect boxes Λ( x, b i ) and Λ (cid:48) ( x, b i ) for all i = 1 , . . . , k , we obtain for | x | sufficiently large, P k (cid:104) |C ( z ) | < ∞ (cid:12)(cid:12)(cid:12) W z ≥ w (cid:105) ≤ P (cid:104) |C ( z ) | < ∞ (cid:12)(cid:12)(cid:12) W z ≥ w, X (Λ( x, ≥ c | x | d (cid:105) P (cid:2) X (Λ( x, b i )) ≥ c | x | db i and X (Λ (cid:48) ( x, b i )) ≥ c | x | db i for all i = 1 , . . . , k (cid:3) ≤ P (cid:104) |C ( z ) | < ∞ (cid:12)(cid:12)(cid:12) W z ≥ w, X (Λ( x, ≥ c | x | d (cid:105) − ε (cid:48) , where we used (7.2) for the second inequality. Using Lemma 7.2 with r = | x | / ε (cid:48) for sufficiently large w (note that convergence in Lemma 7.2 is uniform in r and c ≥ | x | , P k [ |C ( z ) | = ∞ ] ≥ P k (cid:104) |C ( z ) | = ∞ (cid:12)(cid:12)(cid:12) W z ≥ w (cid:105) P k [ W z ≥ w ] ≥ − ε (cid:48) − ε (cid:48) (1 − ε (cid:48) ) = 1 − ε (cid:48) . Together with (7.4), this implies that for sufficiently large | x | , the event E = { z i ⇔ z i +1 and z (cid:48) i ⇔ z (cid:48) i +1 for all i = 0 , . . . , k − } ∩ {|C ( z ) | = ∞} satisfies P k [ E ] > − ε (cid:48) . It follows that, using (7.2), P k [ E | , x ∈ C ∞ ] ≥ − ε (cid:48) / P k ,x [0 , x ∈ C ∞ ]= 1 − ε (cid:48) P [ N k ] P (cid:2) N k (cid:12)(cid:12) , x ∈ C ∞ (cid:3) P ,x [0 , x ∈ C ∞ ] ≥ − ε (cid:48) (1 − ε (cid:48) / P ,x [0 , x ∈ C ∞ ]) P ,x [0 , x ∈ C ∞ ] = 1 − ε (cid:48) P ,x [0 , x ∈ C ∞ ] − ε (cid:48) = 1 − ε (cid:48)(cid:48) , here the last equality defines ε (cid:48)(cid:48) >
0. Note that on event E , because of the choice of k and since z = z (cid:48) , d (0 , x ) ≤ d (0 , z k ) + k (cid:88) i =1 (cid:0) d ( z i , z i − ) + d ( z (cid:48) i − , z (cid:48) i ) (cid:1) + d ( z (cid:48) k , x )= d (0 , z k ) + 2 k + d ( z (cid:48) k , x ) ≤ d (0 , z k ) + 2 log log | x || log b | + d ( z (cid:48) k , x ) . Moreover, on event E , because z = z (cid:48) , we have that z k and z (cid:48) k are both in the infinite connected component C ∞ .If, in addition, we assume that 0 ∈ C ∞ , then 0 and z k are in the same component C ∞ . Since | z k | ≤ | x | b k / ≤ m /b and the infinite connected component C ∞ is unique, a.s., it follows that on E ∩ { ∈ C ∞ } , 0 and z k are connectedwithin box Λ (cid:101) m with probability arbitrarily close to 1, for some (cid:101) m = (cid:101) m ( m ) < ∞ . Note that (cid:101) m is independent of x . This implies for any κ > | x | , P k (cid:104) d (0 , z k ) ≤ κ log log | x | (cid:12)(cid:12)(cid:12) ∈ C ∞ , E (cid:105) ≥ − ε (cid:48) . By symmetry we therefore obtain for | x | sufficiently large, P k (cid:104) d (0 , z k ) + d ( z (cid:48) k , x ) ≤ κ log log | x | (cid:12)(cid:12)(cid:12) , x ∈ C ∞ , E (cid:105) ≥ − ε (cid:48) . Therefore, if we choose κ = ε/ (2 | log b | ) and | x | sufficiently large, P k (cid:20) d (0 , x ) ≤ ε/
2) log log | x || log b | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ (cid:21) ≥ P k (cid:20) d (0 , x ) ≤ ε/
2) log log | x || log b | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ , E (cid:21) P k [ E | , x ∈ C ∞ ] ≥ P k (cid:20) d (0 , z k ) + d ( z (cid:48) k , x ) ≤ ε log log | x || log b | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ , E (cid:21) (1 − ε (cid:48)(cid:48) ) ≥ (1 − ε (cid:48) )(1 − ε (cid:48)(cid:48) ) . It follows that for sufficiently large | x | , using (7.2) in the last step, P (cid:20) d (0 , x ) ≤ ε/
2) log log | x || log b | (cid:12)(cid:12)(cid:12)(cid:12) , x ∈ C ∞ (cid:21) ≥ (1 − ε (cid:48) )(1 − ε (cid:48)(cid:48) ) P (cid:2) N k (cid:12)(cid:12) , x ∈ C ∞ (cid:3) ≥ (1 − ε (cid:48) )(1 − ε (cid:48)(cid:48) )(1 − ε (cid:48) / P ,x [0 , x ∈ C ∞ ]) . This finishes the proof of Proposition 7.1. (cid:3)
Next, we give the proof of the lower bound of statement (a) of Theorem 3.6 which we recall inthe following proposition. Note that this proposition differs from the corresponding Theorem5.3 of [11] in the discrete space model.
Proposition 7.3
Assume α > d and τ = βα/d ∈ (1 , . For every λ > λ c = 0 there exists η > such that lim | x |→∞ P ,x (cid:20) d (0 , x ) ≥ η | x || log κ | (cid:21) = 1 , with κ = α ( β ∧ /d − ∈ (0 , . Proof of Proposition 7.3.
We modify the proof of Theorem 5.3 of [11] to our situation. Choose ϑ > µ > d/ϑ − dκ + µ < µ < dκ. ote that this choice is possible since the above constraints require ϑ > /κ and µ ∈ (0 , d ( κ − /ϑ )). For x ∈ X and n ∈ N we define the random variable S n ( x ) = sup y ∈ X : d ( x,y ) ≤ n | x − y | , which represents the Euclidean distance between x and the furthest particle that can be reached from x usingat most n edges. For r > B ( r ) the ball of (Euclidean) radius r around the origin. For t > − e − x ≤ x ∧ P ,x (cid:104) S n − (0) < t /ϑ , S n (0) ≥ t (cid:105) ≤ P ,x (cid:104) ∃ z ∈ B ( t /ϑ ) ∩ X, z (cid:48) ∈ B ( t ) c ∩ X such that z ⇔ z (cid:48) (cid:105) ≤ E ,x (cid:88) z ∈ B ( t /ϑ ) ∩ X (cid:88) z (cid:48) ∈ B ( t ) c ∩ X E (cid:20) λW z W z (cid:48) | z − z (cid:48) | α ∧ (cid:12)(cid:12)(cid:12)(cid:12) X (cid:21) . For two i.i.d. random variables W and W having a Pareto distribution with scale parameter 1 and shapeparameter β we obtain for u ≥
1, using integration by parts in the first step, E (cid:20) W W u ∧ (cid:21) = 1 u + 1 u (cid:90) u P [ W W > v ] dv = 1 u + 1 u (cid:90) u v − β (1 + β log v ) dv ≤ (1 + β log u ) (cid:18) u − ( β ∧ + 1 u (cid:90) u v − β dv (cid:19) ≤ max { u, { β (cid:54) =1 } / | β − |} (1 + β log u ) u − ( β ∧ , where the last step follows by distinguishing between the cases β = 1, β > β <
1. This provides for u ≥ E (cid:20) W W u ∧ (cid:21) ≤ (1 + 1 { β (cid:54) =1 } / | β − | ) (1 + max { , β } log u ) u − ( β ∧ . (7.5)Choose t so large that λ − ( t − t /ϑ ) α ≥ P ,x (cid:104) S n − (0) < t /ϑ , S n (0) ≥ t (cid:105) ≤ E ,x (cid:88) z ∈ B ( t /ϑ ) ∩ X (cid:88) z (cid:48) ∈ B ( t ) c ∩ X E (cid:20) W z W z (cid:48) λ − | z − z (cid:48) | α ∧ (cid:12)(cid:12)(cid:12)(cid:12) X (cid:21) ≤ (1 + 1 { β (cid:54) =1 } / | β − | ) E ,x (cid:88) z ∈ B ( t /ϑ ) ∩ X (cid:88) z (cid:48) ∈ B ( t ) c ∩ X (cid:0) { , β } log (cid:0) λ − | z − z (cid:48) | α (cid:1)(cid:1) (cid:0) λ − | z − z (cid:48) | α (cid:1) − ( β ∧ . Choose t so large that (1 + 1 { β (cid:54) =1 } / | β − | ) (cid:0) { , β } log (cid:0) λ − | z − z (cid:48) | α (cid:1)(cid:1) λ β ∧ ≤ | z − z (cid:48) | µ for all z ∈ B ( t /ϑ )and z (cid:48) ∈ B ( t ) c . It follows that for sufficiently large t , note that d ( κ + 1) = α ( β ∧ P ,x (cid:104) S n − (0) < t /ϑ , S n (0) ≥ t (cid:105) ≤ E ,x (cid:88) z ∈ B ( t /ϑ ) ∩ X (cid:88) z (cid:48) ∈ B ( t ) c ∩ X | z − z (cid:48) | − α ( β ∧ µ = E ,x (cid:88) z ∈ B ( t /ϑ ) ∩ X E ,x (cid:88) z (cid:48) ∈ B ( t ) c ∩ X | z − z (cid:48) | − d ( κ +1)+ µ . We estimate the right-hand side under the unconditional measure P instead of P ,x . Note that the tail behavioris the same under both measures. We obtain E (cid:88) z ∈ B ( t /ϑ ) ∩ X E (cid:88) z (cid:48) ∈ B ( t ) c ∩ X | z − z (cid:48) | − d ( κ +1)+ µ = E (cid:88) z ∈ B ( t /ϑ ) ∩ X ν (cid:90) z (cid:48) ∈ B ( t ) c | z − z (cid:48) | − d ( κ +1)+ µ dz (cid:48) ≤ E (cid:88) z ∈ B ( t /ϑ ) ∩ X ν (cid:90) | z (cid:48) |≥ t − t /ϑ | z (cid:48) | − d ( κ +1)+ µ dz (cid:48) = ν v d t d/ϑ (cid:90) | z (cid:48) |≥ t − t /ϑ | z (cid:48) | − d ( κ +1)+ µ dz (cid:48) . We therefore obtain for an appropriate constant c > t sufficiently large, P ,x (cid:104) S n − (0) < t /ϑ , S n (0) ≥ t (cid:105) ≤ c t d/ϑ − dκ + µ . (7.6)Define f : N → (0 , ∞ ) by f ( m, n ) = m ϑ n for all m, n ∈ N . Observe ) (cid:80) ∞ k =2 f (2 , k ) d/ϑ − dκ + µ < ∞ because ϑ > d/ϑ − dκ + µ < m ≥ η = η ( m ) > f (cid:16) m, (cid:108) η | x || log κ | (cid:109)(cid:17) ≤ | x | / | x | .We choose m ≥ t = f ( m, n ) with m ≥ m and n ≥
2. Using (7.6) andinduction we obtain for each n ≥ m ≥ m , note that f ( m, n ) /ϑ = f ( m, n − P ,x [ S n (0) ≥ f ( m, n )] ≤ P ,x [ S n − (0) ≥ f ( m, n − P ,x [ S n − (0) < f ( m, n − , S n (0) ≥ f ( m, n )] ≤ P ,x [ S n − (0) ≥ f ( m, n − c f ( m, n ) d/ϑ − dκ + µ ≤ P ,x [ S (0) ≥ f ( m, c n (cid:88) k =2 f ( m, k ) d/ϑ − dκ + µ ≤ P ,x [ ∃ y ∈ X with | y | ≥ f ( m,
1) and 0 ⇔ y ] + c ∞ (cid:88) k =2 f ( m, k ) d/ϑ − dκ + µ . Note that the right-hand side is independent of n ≥ m ≥
2. Since f ( m, k ) d/ϑ − dκ + µ isdecreasing in m ≥
2, there exists m ≥ m such that the right-hand side is less than ε for fixed ε >
0. We finallyobtain for sufficiently small η = η ( m ) > | x | , set n ( x ) = (cid:108) η | x || log κ | (cid:109) ≥ P ,x (cid:20) d (0 , x ) ≤ η | x || log κ | (cid:21) ≤ P ,x [ d (0 , x ) ≤ n ( x )] ≤ P ,x (cid:2) S n ( x ) ( x ) ≥ | x | / (cid:3) + P ,x (cid:2) S n ( x ) (0) ≥ | x | / (cid:3) = 2 P ,x (cid:2) S n ( x ) (0) ≥ | x | / (cid:3) ≤ P ,x (cid:20) S n ( x ) (0) ≥ f (cid:18) m, (cid:24) η log log | x || log κ | (cid:25)(cid:19)(cid:21) ≤ ε, which finishes the proof of Proposition 7.3. (cid:3) In the following we give the proof of part (b1) of Theorem 3.6. We first prove the lower boundwhich follows from the following proposition.
Proposition 7.4
Assume α > d and τ = βα/d > . For every λ > λ c there exists η (cid:48) > suchthat P ,x (cid:2) d (0 , x ) ≥ η (cid:48) log | x | (cid:3) = 1 . Proof of Proposition 7.4.
Choose n ∈ N , 0 , x ∈ X and set x = 0 and x n = x . As in (5.2) we obtain, the firstsum is over all self-avoiding paths of length n starting from 0, note that x n = x is now fixed, P ,x [ d (0 , x ) = n ] ≤ E ,x (cid:88) ( x , . . . , x n ) s.a. n (cid:89) i =1 p x i − x i ≤ ν n − (cid:90) R d · · · (cid:90) R d n (cid:89) i =1 h ( x i − x i − ) dx · · · dx n − , where for y ∈ R d we define function h by, recall function g defined in (5.1), h ( y ) = g ( λ − | y | α ) = 1 { | y | <λ /α } + 1 { | y |≥ λ /α } c λ ( β/ ∧ (cid:0) { , β } log( λ − | y | α ) (cid:1) | y | − α ( β/ ∧ . Note that h is integrable because α > d and τ = βα/d >
2. Using x = 0 and x n = x , and substituting inductively x i by x i − (cid:80) i − l =1 x l for i = 1 , . . . , n −
1, it follows that P ,x [ d (0 , x ) = n ] ≤ ν n − (cid:90) R d · · · (cid:90) R d (cid:32) n − (cid:89) i =1 h ( x i ) (cid:33) h (cid:32) x − n − (cid:88) i =1 x i (cid:33) dx · · · dx n − . e condition on 1 {| x i | < | x | /n } and 1 {| x i |≥| x | /n } for all i = 1 , . . . , n −
1. Note that if | x i | < | x | /n for all i = 1 , . . . , n − | x − (cid:80) n − i =1 x i | ≥ | x | /n , and we bound the corresponding factor in the integral by sup y ∈ R d : | y |≥| x | /n h ( y ).Otherwise, by exchangeability of the x i ’s, there are n − x i ’s satisfies | x i | ≥ | x | /n . In each of these n − y ∈ R d : | y |≥| x | /n h ( y ).Note that the restriction on x − (cid:80) n − i =1 x i then drops and we obtain P ,x [ d (0 , x ) = n ] ≤ n (cid:32) sup y ∈ R d : | y |≥| x | /n h ( y ) (cid:33) (cid:18) ν (cid:90) R d h ( y ) dy (cid:19) n − , (7.7)where ν (cid:82) R d h ( y ) dy < ∞ since h is integrable. Next, we bound the supremum on the right-hand side of (7.7).Choose η > | x | be so large that η log | x | ≥
1. Choose n ∈ N with n ≤ η log | x | . Let µ ∈ (0 , α ( β/ ∧ | x | so large that any y ∈ R d with | y | ≥ | x | /n satisfies c λ ( β/ ∧ (cid:0) { , β } log( λ − | y | α ) (cid:1) ≤ | y | µ . If, in addition, | x | is so large that | x | /n > λ /α , then for any y ∈ R d with | y | ≥ | x | /n > λ /α , h ( y ) = c λ ( β/ ∧ (cid:0) { , β } log( λ − | y | α ) (cid:1) | y | − α ( β/ ∧ ≤ | y | − α ( β/ ∧ µ ≤ n α ( β/ ∧ − µ | x | − α ( β/ ∧ µ . We finally obtain for all η > ≤ n ≤ η log | x | with | x | sufficiently large,sup y ∈ R d : | y |≥| x | /n h ( y ) ≤ η α ( β/ ∧ − µ (log | x | ) α ( β/ ∧ − µ | x | − α ( β/ ∧ µ . Together with (7.7) we obtain for any η > ≤ n ≤ η log | x | with | x | sufficiently large, P ,x [ d (0 , x ) = n ] ≤ n (cid:18) ν (cid:90) R d h ( y ) dy (cid:19) n − η α ( β/ ∧ − µ (log | x | ) α ( β/ ∧ − µ | x | − α ( β/ ∧ µ ≤ (cid:18) ν (cid:90) R d h ( y ) dy (cid:19) η log | x | η α ( β/ ∧ − µ +1 (log | x | ) α ( β/ ∧ − µ +1 | x | − α ( β/ ∧ µ = η α ( β/ ∧ − µ +1 (log | x | ) α ( β/ ∧ − µ +1 | x | − α ( β/ ∧ µ + η log ( ν (cid:82) R d h ( y ) dy ) ≤ | x | − δ , where the last inequality holds for some δ > | x | is sufficiently large and η > − α ( β/ ∧
1) + µ + η log (cid:0) ν (cid:82) R d h ( y ) dy (cid:1) <
0. We conclude that there exist η (cid:48) > δ > | x | sufficiently large, P ,x (cid:2) d (0 , x ) ≤ η (cid:48) log | x | (cid:3) = (cid:88) ≤ n ≤ η (cid:48) log | x | P ,x [ d (0 , x ) = n ] ≤ η (cid:48) (log | x | ) | x | − δ , which converges to 0 as | x | → ∞ . (cid:3) In order to finish the proof of statement (b1) of Theorem 3.6 it remains to show the correspondingupper bound on the graph distances. The result follows from the following proposition, see alsoProposition 4.1 of [6].
Proposition 7.5
Assume α ∈ ( d, d ) and τ = βα/d > , and choose λ > λ c . For each ε > and ∆ (cid:48) > ∆ = log(2) / log(2 d/α ) there exists N < ∞ such that for all x, y ∈ R d with | x − y | ≥ N , P x,y (cid:104) d ( x, y ) ≥ (log | x − y | ) ∆ (cid:48) , x, y ∈ C ∞ (cid:105) ≤ ε. P (cid:104) d ( x, y ) ≤ (log | x − y | ) ∆ (cid:48) (cid:12)(cid:12)(cid:12) x, y ∈ C ∞ (cid:105) = 1 − P x,y (cid:104) d ( x, y ) > (log | x − y | ) ∆ (cid:48) , x, y ∈ C ∞ (cid:105) P x,y [ x, y ∈ C ∞ ] ≥ − ε P x,y [ x, y ∈ C ∞ ] . To prove Proposition 7.5 we use the concept of hierarchies of particles. For k ≥ σ ∈ { , } k , such as σ = 01110001, a hierarchical index. If k = 0, σ ∈ { , } k denotes theempty string. For σ ∈ { , } k and σ ∈ { , } l , k, l ≥
1, we denote by σ σ the concatenationof σ and σ . Then [6] provides the following definition of a hierarchy. Definition 7.6
For m ≥ and two distinct particles x, y ∈ X we say that the set of particles H m ( x, y ) = (cid:110) z σ ∈ X (cid:12)(cid:12)(cid:12) σ ∈ { , } k , k = 1 , . . . , m (cid:111) ⊂ X is a hierarchy of depth m connecting x and y if1. z = x and z = y ;2. z σ = z σ and z σ = z σ for all σ ∈ { , } k and k = 0 , . . . , m − ;3. for all σ ∈ { , } k and k = 0 , . . . , m − there is an edge between z σ and z σ as long as z σ (cid:54) = z σ ;4. each edge as in 3. appears only once in H m ( x, y ) .For σ ∈ { , } m − we call the pairs of particles ( z σ , z σ ) and ( z σ , z σ ) “gaps”. Recall that for x ∈ R d and n ∈ (0 , ∞ ) we write Λ n ( x ) = x + [ − n, n ) d for the box with center x and side length 2 n , and for x ∈ X we write C n ( x ) for the set of particles in Λ n ( x ) ∩ X that areconnected to x within Λ n ( x ). For any L > x ∈ R d we set annulus R L ( x ) = Λ L ( x ) \ Λ L/ ( x ).Moreover, for (cid:96) ∈ (0 , L ) and ρ ∈ (0 ,
1) we define by D ( ρ,(cid:96) ) L ( x ) = (cid:110) z ∈ R L ( x ) ∩ X (cid:12)(cid:12)(cid:12) |C (cid:96) ( z ) | ≥ ρ (2 (cid:96) ) d (cid:111) , the set of ( ρ, (cid:96) )-dense particles in R L ( x ). Note that this definition differs from D ( ρ,(cid:96) ) L defined inSection 3.3 because we now consider the particles in annulus R L ( x ) instead of the particles inbox Λ L . For x, y ∈ R d , α ∈ ( d, d ) and γ ∈ ( α/ (2 d ) ,
1) we define for m ≥ N m = N γ m with N = | x − y | . We denote by B m = B ( ρ,(cid:96) ) m,γ ( x, y ) the event that there exists a hierarchy H m ( x, y ) of depth m connecting x and y such that for all k = 0 , . . . , m − σ ∈ { , } k , z σ ∈ D ( ρ,(cid:96) ) N k +1 ( z σ ) and z σ ∈ D ( ρ,(cid:96) ) N k +1 ( z σ ) , N k +1 ( z σ ) R N k +2 ( z σ ) b R N k +2 ( z σ ) b z σ b z σ R N k +1 ( z σ ) R N k +2 ( z σ ) b R N k +2 ( z σ ) b z σ b z σ b z σ b z σ z σ z σ Figure 2:
Illustration of a hierarchy in B m . Assume we are given z σ and z σ for some σ ∈ { , } k and k = 0 , . . . , m −
3. We consider an edge between R N k +1 ( z σ ) and R N k +1 ( z σ ) and call the end points z σ and z σ ,respectively. Then, set z σ = z σ and consider an edge between R N k +2 ( z σ ) and R N k +2 ( z σ ) and call the endpoints z σ and z σ , respectively. Similarly, we define z σ and z σ . To get a path of edges from z σ to z σ itremains to connect the two particles of each gap ( z, z (cid:48) ). (If we assume m = 3, the pairs ( z σ , z σ ), ( z σ , z σ ),( z σ , z σ ) and ( z σ , z σ ) in the figure are the gaps of the illustrated hierarchy in B .) Such particles are likelyconnected by a short path of edges because they are ( ρ, (cid:96) )-dense and close to each other if m is sufficiently large,conditional on B m . see also (4.5) of [6] and Figure 2 for an illustration. Moreover, we denote by T = T ( ρ,(cid:96) ) ( x, y ) theevent that x and y are ( ρ, (cid:96) )-dense. Note that the event B m ∩ T ensures that there is a hierarchy H m ( x, y ) of depth m connecting x and y such that all particles in this hierarchy are ( ρ, (cid:96) )-denseand lie in the corresponding annulus R N k ( z σ ). Finally, given B m , we denote by S = S ( (cid:96) ) theevent that all gaps ( z, z (cid:48) ) in a hierarchy in B m satisfy X (Λ (cid:96) ( z )) ≤ eν (2 (cid:96) ) d and X (Λ (cid:96) ( z (cid:48) )) ≤ eν (2 (cid:96) ) d . (7.8)The event S ensures that we do not have too many particles in boxes Λ (cid:96) ( z ) and Λ (cid:96) ( z (cid:48) ), inparticular, the graph distance of connected paths within such boxes is bounded by eν (2 (cid:96) ) d . Forthe proof of Proposition 7.5 we use Lemma 7.7 below. Part (i) of Lemma 7.7 corresponds toLemma 4.2 of [6]. The only difference lies in the additional event S defined by (7.8). Part (ii) of Lemma 7.7 is the continuum space analogue to Lemma 4.3 of [6] and it shows that the event B m occurs with sufficiently high probability. Lemma 7.7
Choose α ∈ ( d, d ) . For all ε ∈ (0 , , γ ∈ ( α/ (2 d ) , , ∆ (cid:48) > log(2) / log(1 /γ ) and α (cid:48) ∈ ( α, dγ ) there exist N (cid:48) = N (cid:48) ( ε, γ, ∆ (cid:48) ) < ∞ , ρ ∈ (0 , and a constant c > such that thefollowing holds true: for all x, y ∈ R d with N = | x − y | ≥ N (cid:48) let m ∈ N be the maximal integersuch that m log(1 /γ ) ≤ log log N − ε log log log N. (7.9) For all ρ ∈ (0 , ρ ) and (cid:96) ∈ ( N m , N m ) in the definitions of B m , T and S we obtain i) P x,y (cid:104)(cid:110) d ( x, y ) ≥ (log N ) ∆ (cid:48) (cid:111) ∩ B m ∩ T ∩ S (cid:105) ≤ ε ; (ii) P x,y [ B cm ] ≤ m +1 e − c N dγ − α (cid:48) m ; (iii) P x,y [ B m ∩ S c ] ≤ ε. Note that choice (7.9) implies, see also (4.9) of [6],2 m ≤ (log N ) log 2 / log(1 /γ ) and e (log log N ) ε ≤ N m ≤ e (1 /γ )(log log N ) ε . (7.10) Proof of Lemma 7.7.
We first prove (i) . Let ρ ∈ (0 , S we argue as in the proof ofLemma 4.2 of [6] to see that it remains to estimate the probability of the event A m ∩ B m ∩ T ∩ S ⊂ A m ∩ B m ∩ T for N sufficiently large, where A m is the event that for any hierarchy in B m there exists a gap ( z, z (cid:48) ) such thatthere is no edge between the sets C (cid:96) ( z ) and C (cid:96) ( z (cid:48) ). Let z ∈ R d be such that the event A m ∩ B m ∩ T only dependson the particles in Λ N ( z ) and edges with end points in Λ N ( z ) (which exists if N is sufficiently large). For l ≥ x = x , y = x , x , . . . , x l ∈ Λ N ( z ) we consider the edge probabilities (cid:101) p x i x j = 1 − exp {− λ | x i − x j | − α } ≤ − exp {− λW x i W x j | x i − x j | − α } = p x i x j , a.s., for every i (cid:54) = j ∈ { , . . . , l } , and denote by (cid:101) P X the probability measure of the resulting edge configurations.We obtain P x,y (cid:104) A m ∩ B m ∩ T (cid:12)(cid:12)(cid:12) X ∩ Λ N ( z ) = { x, y, x , . . . , x l } (cid:105) ≤ (cid:101) P X [ A m ∩ B m ∩ T ] . Using the same arguments as in [6] we obtain for all N sufficiently large, (cid:101) P X [ A m ∩ B m ∩ T ] ≤ ε, (7.11)note that the left-hand side is zero for l too small. This proves ( i ) by additionally integrating over the Poissoncloud restricted to Λ N ( z ) (note that the right-hand side of (7.11) does not depend on the Poisson cloud).To prove (ii) we note that P x,y [ B cm ] ≤ P x,y [ B m − ∩ B cm ] + P x,y [ B cm − ] ≤ m − (cid:88) k =1 P x,y [ B k ∩ B ck +1 ] + P x,y [ B c ] = m − (cid:88) k =1 P x,y [ B k ∩ B ck +1 ] . Hence, it is sufficient to show that there exist ρ ∈ (0 ,
1) and a constant c > ρ ∈ (0 , ρ ) andfor (cid:96) ∈ ( N m , N m ) in the definition of B m we have for sufficiently large N , P x,y [ B k ∩ B ck +1 ] ≤ k +1 e − c N dγ − α (cid:48) m , for all k = 1 , . . . , m −
1, with m as in (7.9). For k = 1 , . . . , m − B (cid:48) k the event that there is ahierarchy H k ( x, y ) of depth k connecting x and y such that for each j = 0 , . . . , k − σ ∈ { , } j , z σ ∈ R N j +1 ( z σ ) and z σ ∈ R N j +1 ( z σ ) . By definition we obtain B k ⊂ B (cid:48) k . For all (cid:96) ∈ ( N m , N m ) and ρ ∈ (0 , ρ ), where ρ will be chosen below, we definethe events A = (cid:8) for any hierarchy in B (cid:48) k there exists σ ∈ { , } k such that |D ( ρ,(cid:96) ) N k ( z σ ) | ≤ ρ (2 N k ) d (cid:9) , A = (cid:8) for any hierarchy in B (cid:48) k there exists a gap ( z, z (cid:48) ) such that there is no edge betweenthe sets D ( ρ,(cid:96) ) N k ( z ) and D ( ρ,(cid:96) ) N k ( z (cid:48) ) (cid:9) . k ∩ B ck +1 implies that there is a hierarchy in B (cid:48) k but, by of Definition 7.6, there is no edge between D ( ρ,(cid:96) ) N k ( z )and D ( ρ,(cid:96) ) N k ( z (cid:48) ) for any pair ( z, z (cid:48) ) as in the definition of A . Hence, B k ∩ B ck +1 ⊂ B (cid:48) k ∩ A , and therefore we obtain P x,y [ B k ∩ B ck +1 ] ≤ P x,y (cid:2) B (cid:48) k ∩ A (cid:3) ≤ P x,y (cid:2) B (cid:48) k ∩ A (cid:3) + P x,y (cid:2) B (cid:48) k ∩ A c ∩ A (cid:3) , (7.12)and it remains to bound the two terms on the right-hand side. To bound the first term we note that for N sufficiently large it holds that for any hierarchy in B (cid:48) k and σ, σ (cid:48) ∈ { , } k with z σ (cid:54) = z σ (cid:48) , R N k + (cid:96) ( z σ ) ∩ R N k + (cid:96) ( z σ (cid:48) ) = ∅ for all (cid:96) ∈ ( N m , N m ). The latter implies that the events (cid:110) |D ( ρ,(cid:96) ) N k ( z σ ) | ≤ ρ (2 N k ) d (cid:111) are independent for different σ ∈ { , } k . It follows that for N sufficiently large, P x,y (cid:2) B (cid:48) k ∩ A (cid:3) ≤ k P (cid:104) |D ( ρ,(cid:96) ) N k (0) | ≤ ρ (2 N k ) d (cid:105) . By Corollary 3.5 ( ii ) there exist ρ ∈ (0 ,
1) and (cid:96) < ∞ such that for all (cid:96) ∈ ( (cid:96) , N k /(cid:96) ), P (cid:104) |D ( ρ ,(cid:96) ) N k (0) | ≤ ρ (2 N k ) d (cid:105) ≤ e − ρ N d − α (cid:48) k (although the definition of D ( ρ ,(cid:96) ) N k (0) differs from D ( ρ ,(cid:96) ) N k in Corollary 3.5 ( ii ) we can still use the result because R N k (0) contains a box of side length N k / N so large that N m ≥ (cid:96) and N k /(cid:96) ≥ N m for all k = 1 , . . . , m −
1, we finally obtain for all N sufficiently large, ρ ∈ (0 , ρ ) and (cid:96) ∈ ( N m , N m ), P x,y (cid:2) B (cid:48) k ∩ A (cid:3) ≤ k e − ρ N d − α (cid:48) k ≤ k e − ρ N d − α (cid:48) m ≤ k e − ρ N dγ − α (cid:48) m . (7.13)We proceed as in the derivation of (7.11), using the same arguments as in [6], to see that the second term in(7.12) satisfies P x,y (cid:2) B (cid:48) k ∩ A c ∩ A (cid:3) ≤ k exp (cid:110) − λρ d N dk (5 dN m − ) − α (cid:111) ≤ k exp (cid:110) − λρ d (5 d ) − α N dγ − αm − (cid:111) , for all N sufficiently large. Using (7.12), (7.13) and that N dγ − αm − ≥ N dγ − α (cid:48) m , it follows that for all N sufficientlylarge, P x,y [ B k ∩ B ck +1 ] ≤ k e − ρ N dγ − α (cid:48) m + 2 k exp (cid:110) − λρ d (5 d ) − α N dγ − αm − (cid:111) ≤ k +1 e − c N dγ − α (cid:48) m , with c = min { ρ , λρ d (5 d ) − α } .To prove (iii) we note that any hierarchy in B m has 2 m − gaps whose 2 m particles { v = x, v = y, v , . . . , v m } satisfy Λ (cid:96) ( v i ) ∩ Λ (cid:96) ( v j ) = ∅ for all i (cid:54) = j . We therefore obtain for N sufficiently large, P x,y [ B m ∩ S c ] ≤ m P x (cid:104) X (Λ (cid:96) ( x )) > eν (2 (cid:96) ) d (cid:105) . Using Chernoff’s bound, see (6.4), and (7.10) we obtain for (cid:96) ≥ N m ,2 m P (cid:104) X (Λ (cid:96) (0)) > eν (2 (cid:96) ) d (cid:105) ≤ m e − ν (2 (cid:96) ) d ≤ (log N ) log 2 / log(1 /γ ) exp (cid:110) − ν d e d (log log N ) ε (cid:111) , which converges to 0 as N → ∞ . (cid:3) Proof of Proposition 7.5.
Using part ( i ) of Corollary 3.5 (to bound the probability of event T c ) and Lemma7.7, Proposition 7.5 is proven by using the same arguments as in [6]. (cid:3) In order to finish the proof of Theorem 3.6 it remains to show statement (b2) . We start withthe following lemma.
Lemma 7.8
Assume min { α, βα } > d . For all δ ∈ (0 , α ( β ∧ − d ) there exist t ≥ and aconstant c > such that for all s ≥ and t ≥ t , P [ there is an edge in Λ s with size at least t ] ≤ c s d t d − α ( β ∧ δ . roof of Lemma 7.8. Using (7.5) we obtain for all | x − y | ≥ t with t sufficiently large, E (cid:20)(cid:18) λ W x W y | x − y | α (cid:19) ∧ (cid:21) ≤ (1 + 1 { β (cid:54) =1 } / | β − | ) λ β ∧ (cid:0) { , β } log( λ − | x − y | α ) (cid:1) | x − y | − α ( β ∧ ≤ | x − y | − α ( β ∧ δ . It follows that for all t ≥ t , using 1 − e − x ≤ x ∧ P [there is an edge in Λ s with size at least t ] ≤ E (cid:34) (cid:88) x,y ∈ X ∩ Λ s {| x − y | >t } | x − y | − α ( β ∧ δ (cid:35) = (cid:88) k ≥ P [ X (Λ s ) = k ](2 s ) dk (cid:90) Λ s · · · (cid:90) Λ s k (cid:88) i =1 k (cid:88) j =1 {| x i − x j | >t } | x i − x j | − α ( β ∧ δ dx · · · dx k = (cid:88) k ≥ P [ X (Λ s ) = k ](2 s ) d k ( k − (cid:90) Λ s (cid:90) Λ s {| x − y | >t } | x − y | − α ( β ∧ δ dx dy ≤ (cid:88) k ≥ P [ X (Λ s ) = k ](2 s ) d k ( k − (cid:90) | y | >t | y | − α ( β ∧ δ dy = ν (2 s ) d (cid:90) | y | >t | y | − α ( β ∧ δ dy, where in the last step we used that X (Λ s ) has a Poisson distribution with parameter ν (2 s ) d . It follows that foran appropriate constant c > t ≥ t with t sufficiently large, P [there is an edge in Λ s with size at least t ] ≤ c s d t d − α ( β ∧ δ , which finishes the proof of Lemma 7.8. (cid:3) Proof of (b2) of Theorem 3.6.
Once the proof of Lemma 7.8 is established, the proof of (b2) of Theorem3.6 follows one-to-one from the proof of Theorem 1 of [4] and Theorem 8 (b2) of [12]. There, a renormalizationis applied to see that we have a linear lower bound on the graph distances within “good” finite boxes, see alsoDefinition 2 and Lemma 2 of [4]. Lemma 7.8 is then used to prove that all centered boxes of sufficiently large sidelengths are “good”, a.s., see Lemma 14 of [12], which then implies (b2) of Theorem 3.6. We refer to [12] for thedetails of the proof. (cid:3)
References [1] Amini, H., Cont, R., Minca, A. (2012). Stress testing the resilience of financial networks.Int. J. Theor. Appl. Finance 5(1), 1250006-1250020.[2] Antal, P., Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation.Ann. Probab. 24(2), 1036-1048.[3] Berger, N. (2002). Transience, recurrence and critical behavior for long-range percolation. Com-mun. Math. Phys. 226(3), 531-558.[4] Berger, N. (2008). A lower bound for the chemical distance in sparse long-range percolation models.arXiv:math/0409021v1.[5] Berger, N. (2014). Transience, recurrence and critical behavior for long-range percolation.arXiv:math/0110296v3.[6] Biskup, M. (2004). On the scaling of the chemical distance in long-range percolation models.Ann. Probab. 32, 2938-2977.
7] Cont, R., Moussa, A., Santos, E.B. (2010). Network structure and systemic risk in banking system.SSRN Server, Manuscript ID 1733528.[8] Bollob´as, B., Riordan, O. (2006). Percolation. Cambridge University Press.[9] Van de Brug, T., Meester, R. (2004). On central limit theorems in the random connection model.Physica A 332, 263-278.[10] Daley, D.J., Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer.[11] Deijfen, M., van der Hofstad, R., Hooghiemstra, G. (2013). Scale-free percolation. Annales IHPProbab. Statist. 49(3), 817-838.[12] Deprez, P., Hazra, R.S., W¨uthrich, M.V. (2015). Inhomogeneous long-range percolation for real-lifenetwork modeling. Risks 3(1), 1-23.[13] Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.[14] Franceschetti, M., Meester, R. (2007). Random Networks for Communication. Cambridge UniversityPress.[15] Gilbert, E.N. (1961). Random plane networks. J. Soc. Indust. Appl. Math. 9, 533-543.[16] Grimmett, G. (1999). Percolation. Springer.[17] Liggett, T.M., Schonmann, R.H., Stacey, A.M. (1997). Domination by product measures.Ann. Probab. 25(1), 71-95.[18] Meester, R., Roy, R. (1996). Continuum Percolation. Cambridge University Press.[19] Newman, C.M., Schulman, L.S. (1986). One dimensional 1 / | j − i | s percolation models: the existenceof a transition for s ≤
2. Commun. Math. Phys. 104, 547-571.[20] Newman, M.E.J., Strogatz, S.H., Watts, D.J. (2002). Random graph models of social networks.Proc. Natl. Acad. Sci. 99(suppl 1), 2566-2572.[21] Penrose, M.D. (1991). On a continuum percolation model. Adv. Appl. Prob. 23, 536-556.[22] Schulman, L.S. (1983). Long range percolation in one dimension. J. Phys. A 16, L639-L641.[23] Watts, D.J. (2003). Six Degrees: The Science of a Connected Age. W.W. Norton.[24] Willmot, G., Lin, X. (2012). Lundberg Approximations for Compound Distributions with InsuranceApplications. Lecture Notes in Statistics. Springer New York.[25] Zhang, Z.Q., Pu, F.C., Li, B.Z. (1983). Long-range percolation in one dimension.J. Phys. A: Math. Gen. 16(3), L85-L90.2. Commun. Math. Phys. 104, 547-571.[20] Newman, M.E.J., Strogatz, S.H., Watts, D.J. (2002). Random graph models of social networks.Proc. Natl. Acad. Sci. 99(suppl 1), 2566-2572.[21] Penrose, M.D. (1991). On a continuum percolation model. Adv. Appl. Prob. 23, 536-556.[22] Schulman, L.S. (1983). Long range percolation in one dimension. J. Phys. A 16, L639-L641.[23] Watts, D.J. (2003). Six Degrees: The Science of a Connected Age. W.W. Norton.[24] Willmot, G., Lin, X. (2012). Lundberg Approximations for Compound Distributions with InsuranceApplications. Lecture Notes in Statistics. Springer New York.[25] Zhang, Z.Q., Pu, F.C., Li, B.Z. (1983). Long-range percolation in one dimension.J. Phys. A: Math. Gen. 16(3), L85-L90.