SINR percolation for Cox point processes with random powers
aa r X i v : . [ m a t h . P R ] O c t SINR PERCOLATION FOR COX POINT PROCESSESWITH RANDOM POWERS
Benedikt Jahnel and András Tóbiás (5 October 2020) Abstract:
Signal-to-interference plus noise ratio (SINR) percolation is an infinite-rangedependent variant of continuum percolation modeling connections in a telecommunica-tion network. Unlike in earlier works, in the present paper the transmitted signal powersof the devices of the network are assumed random, i.i.d. and possibly unbounded. Addi-tionally, we assume that the devices form a stationary Cox point process, i.e., a Poissonpoint process with stationary random intensity measure, in two or higher dimensions.We present the following main results. First, under suitable moment conditions onthe signal powers and the intensity measure, there is percolation in the SINR graphgiven that the device density is high and interferences are sufficiently reduced, but notvanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ / (2 τ ) , then there is no percolation for any intensity parameter. Third, inthe case of a Poisson point process with constant powers, for any intensity parameterthat is supercritical for the underlying Gilbert graph, the SINR graph also percolateswith some small but positive interference cancellation factor. MSC 2010.
Primary 82B43, 60G55, 60K35; secondary 90B18.
Keywords and phrases.
Signal-to-interference ratio, Cox point process, Poisson point process, con-tinuum percolation, SINR percolation, Gilbert graph, Boolean model, stabilization, random power,degree bound. 1.
Introduction
The study of percolation properties of infinite random graphs traces back many decades andmany of the classical results are already available in textbook form, see for example the mono-graphs [MR96, Gri99, BR06]. The first results for percolation in the continuum R d were presentedin the landmark paper by Gilbert [Gil61], where non-trivial regimes of existence and absence of perco-lation (i.e., existence of an unbounded connected component) were established for the Poisson–Gilbertgraph , i.e., where the set of nodes is given by a homogeneous
Poisson point process (PPP) and edgesare drawn between two nodes whenever their distance is below a certain fixed positive connectivitythreshold. The context of telecommunication systems was already mentioned there.In order to make the model more flexible, instead of using a fixed connectivity threshold, the nodesin the PPP can also be equipped with independent random radii, drawn from a common distribution,and any two nodes are connected by an edge whenever their distance is below the sum of the associatedradii. In view of our topic in this manuscript, we call the resulting model a
Poisson–Gilbert graph with Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
[email protected] Berlin Mathematical School, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
[email protected] random radii , whose percolation properties can equivalently be expressed in terms of the corresponding
Poisson–Boolean model with random radii , and note that a wide range of results for percolation for thismodel are available, see for example [MR96, ATT16, Gou08, Gou09, GT18] and references therein.In view of applications in wireless telecommunication systems, the extension of Poisson–Gilbertgraphs towards Gilbert graphs based on
Cox point processes (CPP), i.e., PPPs with random intensitymeasure, allows to study long-range communication properties in device-to-device networks wheredevices are placed according to a PPP in random environment that is represented by the intensitymeasure of the CPP. Recently continuum percolation and further properties of such
Cox–Gilbert graphs were studied under certain conditions of stabilization and connectedness, see [HJC19, CGH +
18] andbelow. However, the edge-drawing rule remained, as in the classical case of the Poisson–Gilbert graph,based on a fixed connectivity threshold.In the very recent manuscript [JTC20], first continuum percolation results are presented for Cox–Gilbert graphs where, as in the Poisson–Gilbert graph with random radii, each node is equipped with arandom radius, and edges are placed between any two nodes whenever their distance is below the sumof the radii. In this case, again under certain stabilization and connectedness assumptions, most of thepercolation properties of the Poisson–Gilbert graph with random radii can be reproduced also for this
Cox–Gilbert graph with random radii . We note that here, similarly to [MR96], percolation propertiesof the Gilbert graphs are again expressed in terms of the underlying
Cox–Boolean models with randomradii .Moving beyond a setting where edges are placed between pairs of points based on their mutualdistance and their associated radii, another line of research aims towards a different kind of extensionof Gilbert graphs with respect to the edges. Starting with the papers [DBT05, DFM + R , the edge-drawing mechanism is replaced by a highly non-local rule usingthe signal-to-interference ratio (SINR), which we describe precisely and more generally in (2.1). Inwords, very roughly, a pair of Poisson points is connected by an edge only if the weighted distancebetween the points is sufficiently small compared to the accumulated weighted distances of all theother points, the so-called interference . This definition is very much inspired by applications in wirelesstelecommunication networks, where the success of a transmission between network components is highlydependent on the relative signal strength between the components compared to the other (unwanted)signals present in the medium. In the simplest case, only the relative distances between points enterthe SINR, giving rise to the SINR graph on PPPs, or the Poisson-SINR graph . Let us note that thisdefinition introduces long-range, or even infinite-range dependencies for the construction of edges intothe system. This is the setting considered in [DBT05, DFM + Poisson–SINR graph with random powers ,the paper [KY07] presents first results similar to the assertions presented in [DBT05, DFM +
06] undervery strong boundedness assumptions on the powers. Let us note that the definition of an SINRgraph with random powers already occurs in [DBT05], but the only proven result of this paper for
INR PERCOLATION WITH RANDOM POWERS 3 this setting is about degree bounds (cf. Section 4.2). The first steps towards understanding the caseof unbounded powers in the Poisson–SINR graph with random powers were made recently in [Löf19].In this master’s thesis, supervised by the authors, results in the spirit of one of our main theorems,Theorem 2.1, are presented under much stronger assumptions and only for the case of an underlyinghomogeneous PPP. [Löf19] also provides sufficient conditions for the absence of percolation for smallintensities of the PPP. Before [Löf19], no positive assertions about percolation in an SINR graph withunbounded random powers had been known in the literature; regarding the case of bounded randompowers, see also Section 4.On the other hand, in [Tób20], the two extensions described above were for the first time consideredjointly, giving rise to the SINR graph based on CPPs, the
Cox-SINR graph , however without ran-dom powers. There it was established that for sufficiently large intensities and sufficiently connectedenvironments, the Cox-SINR graph percolates almost surely at least for non-vanishing interference.In [Tób19, Section 4.2.3.4] it was anticipated that the case of random but bounded powers can easilybe handled via the same methods, see Section 4.1 for further details.The present manuscript now completes this line of research by analyzing the
Cox-SINR graph withrandom powers , which are also not necessarily bounded. More precisely, in our first main result,Theorem 2.1, we present sufficient conditions for the existence of a supercritical percolation phase,i.e., a non-trivial regime for the intensity of the underlying CPP and non-vanishing interference, suchthat the Cox-SINR graph with random powers percolates. This substantially extends the results of[DBT05, DFM +
06, FM07] from the case of a homogeneous PPP in R with constant powers to theone of a CPP in R d , d ≥ , with random and possibly unbounded powers, combining the methods of[Tób20] for the case of a CPP with constant powers and the ones of [Löf19] about the case of a PPPwith random powers (both in dimensions 2 or higher). We will discuss the relation of Theorem 2.1 tothese results in detail in Section 4.In our second main result, Theorem 2.2, we establish a uniform bound on the strength of theinterference, above which no percolation is possible. In essence, this theorem claims that there is nopercolation in the SINR graph whenever degrees of its vertices are bounded by 2. The fact that SINRgraphs with non-vanishing interference have bounded degrees originates from [DBT05, Theorem 1];however, in that paper, only the trivial assertion that SINR graphs with degrees bounded by 1 do notpercolate was proven, and this has not been improved before the present paper.Finally, in our third main result, Theorem 2.3, we state that in the case of the Poisson-SINR graphwith constant powers, indeed, the critical intensity for percolation in the presence of interference canbe represented via the critical intensity of an associated Poisson–Gilbert graph. This result extendsthe two-dimensional statement [DFM +
06, Theorem 1] to higher dimensions, whereas its proof is ratherdifferent from the one in [DFM + Setting and main results
For λ > , let X λ = { ( X i , P i ) } i ∈ I be an i.i.d. marked Cox point process (CPP) in R d × [0 , ∞ ) for d ≥ , with directing measure λ Λ ⊗ µ where Λ is stationary with E [Λ( Q )] = 1 and Q n = [ − n/ , n/ d for n > . Here, µ is a Borel probability measure on [0 , ∞ ) , the common distribution of the marks BENEDIKT JAHNEL AND ANDRÁS TÓBIÁS P = { P i } i ∈ I . We consider the SINR graph with vertex set given by the first component of X λ , whichwe denote by X λ = { X i } i ∈ I . Here, every pair X i , X j ∈ X λ of vertices with X i = X j , is connected byan edge if and only if P i ℓ ( | X i − X j | ) > τ (cid:0) N o + γ X k ∈ I \{ i,j } P k ℓ ( | X k − X j | ) (cid:1) and P j ℓ ( | X i − X j | ) > τ (cid:0) N o + γ X k ∈ I \{ i,j } P k ℓ ( | X k − X i | ) (cid:1) . (2.1)In (2.1), τ > is fixed and called the SINR threshold , the constant N o ≥ represents noise , r ℓ ( r ) ∈ [0 , ∞ ) is referred to as the path-loss function and γ ≥ is called the interference-cancellation factor .The random variables P are often called random powers and the term I ( X i , X j , X λ ) = X k ∈ I \{ i,j } P k ℓ ( | X k − X j | ) is referred to as interference . We will use the notation G γ ( X λ ) to indicate the SINR graph, suppressingthe dependencies on τ , N o and ℓ , but highlighting the dependence on γ , see also Figure 1 for anillustration. We refer to [DBT05, Section 1] for further interpretation of the modeling parameters. Figure 1.
A typical realization of a Cox-SINR graph (with blue vertices and blackedges) with directing measure given by the edge-length measure of a two-dimensionalPoisson–Voronoi tessellation (in red) in a box, with N o = 1 , γ = 0 . , τ = 0 . , constantpowers equal to , and a suitable path-loss function ℓ .The SINR graph has a nice interpretation in the study of device-to-device telecommunication systemswhere the devices X λ can communicate directly with each other if their mutual distance, representedby the path-loss function, and their individual powers, are sufficiently strong to overcome thermalnoise plus all the interference coming from the other devices. If this is the case, then the possibility tocommunicate is represented by an undirected edge.Our main interest lies in percolation properties of the SINR graph, as has been first studiedin [DBT05, DFM +
06, FM07]. A cluster in G γ ( X λ ) is a maximal connected component. We saythat G γ ( X λ ) percolates if G γ ( X λ ) contains an unbounded component. Clusters and percolation aredefined analogously in the case of any graph having a vertex set that is included in R d , d ≥ , and islocally finite. Here we focus on the following key quantities. First, the critical interference-cancellationfactor is defined as γ ( λ ) = sup (cid:8) γ > P ( G γ ( X λ ) percolates ) > (cid:9) . (2.2) INR PERCOLATION WITH RANDOM POWERS 5
In words, it represents the maximal amount of interference that can be added to the system and stillmaintain percolation. Second, the critical intensity is defined as λ ∗ c = inf { λ > γ ( λ ′ ) > , ∀ λ ′ > λ } , (2.3)which describes the smallest intensity such that for all larger intensities the addition of a small amountof interference does not destroy percolation.For the statement of our first main result, we assume certain decorrelation and connectivity propertiesfor the directing measure Λ of the underlying CPP. The precise definitions for Λ to be stabilizing , b -dependent or asymptotically essentially connected are technical and will be presented in Definitions 3.1and 3.2 in Section 3, where we will also mention a number of relevant examples of random measuressatisfying these definitions. We denote by P o a generic power random variable distributed accordingto µ and put P sup = ess sup µ . We call the pathloss function ℓ well-behaved if ℓ is continuous, constanton [0 , d o ] for some d o ≥ , strictly decreasing on [ d o , ∞ ) ∩ supp( ℓ ) , and satisfies R ∞ r d − ℓ ( r )d r < ∞ .Our first result establishes percolation for the SINR graph based on CPPs with random powers. Theorem 2.1.
Let d ≥ , N o , τ > , P sup = ∞ , Λ be stabilizing and ℓ be well-behaved. Then, λ ∗ c < ∞ holds if at least one of the following conditions is satisfied:(1) ℓ has unbounded support, Λ is b -dependent for some b > , and E [exp( α Λ( Q ))] < ∞ as wellas E [exp( αP o )] < ∞ holds for some α > , or(2) ℓ has bounded support, E [ P o ] < ∞ , and Λ is asymptotically essentially connected, or(3) ℓ has bounded support, E [ P o ] < ∞ , and sup supp( ℓ ) is larger than c , where c is a finite constantdepending on Λ , τ and N o .As discussed in the introduction, Theorem 2.1 extends similar results known for the case of a ho-mogeneous PPP in R with constant powers to the one of a CPP in R d , d ≥ with random andpossibly unbounded powers. Let us note that the complementary assertion that λ ∗ c > can be de-duced in certain cases based on recent results on Cox–Gilbert graphs with random radii, see [JTC20]and Section 4.1 for more details.Our second main result establishes a uniform upper bound on the critical interference-cancellationfactor. For this assertion we assume the basic non-degeneracy property that X λ is nonequidistant ,which is satisfied for a very large class of CPPs including many relevant examples for wireless telecom-munication systems, see Section 3. This means that for all i, j, k, l ∈ I , | X i − X j | = | X k − X l | > implies { i, j } = { k, l } and | X i | = | X j | implies i = j , almost surely. Clearly, this property implies thatthe point process X λ is simple, further, if nonequidistantness of X λ holds for some λ > , then it holdsfor all λ > . As for a (pathological) counterexample, note that for Λ being the sum of Dirac measuresat the points of the randomly shifted lattice Z d + U , where U is a uniform random variable in [0 , d ,the associated CPP is simple, stationary, but not nonequidistant. Theorem 2.2.
Let d ≥ , N o ≥ and τ, λ > , and assume that X λ is nonequidistant for all λ > .Then γ ( λ ) ≤ / (2 τ ) .Note that we do not require any stabilization or connectedness and also impose no direct restrictionson µ and on ℓ . The proof of Theorem 2.2 rests on showing absence of percolation in the SINR graphwith a maximal degree given by . The fact that SINR graphs with γ > have degrees less than / ( τ γ ) is already stated in [DBT05, Theorem 1]; an immediate consequence of this assertion is thatthere is no percolation in the case γ ≥ /τ when degrees are at most 1. These claims can easily be seento hold for any simple point process in any dimension, although in [DBT05] only the case of a two-dimensional homogeneous PPP was considered. Theorem 2.2 is the first improvement of this bound BENEDIKT JAHNEL AND ANDRÁS TÓBIÁS since then, applicable to stationary CPPs and thus in particular also covering the case of homogeneousPPPs in all dimensions.Finally, our third main result states that the critical intensity parameter for the SINR graph can berepresented as the critical threshold for percolation of an associated Gilbert graph in any dimension.For this we assume a simpler setting in which
Λ(d x ) equals the Lebesgue measure d x , i.e., the CPPis in fact a PPP, and the powers are non-random and given by p > . Note that for γ = 0 , the SINRgraph is in fact a Poisson–Gilbert graph (cf. [Gil61]) with connection radius given by r B = ℓ − ( τ N o /p ) , (2.4)which is a well-defined quantity if ℓ (0) > τ N o /p and the conditions of Theorem 2.1 on ℓ hold.Recall that the Gilbert graph based on a simple point process Y with connection radius r > hasvertex set Y and an edge between two different points of Y whenever the distance of the two points is lessthan r , and the name Poisson–Gilbert graph corresponds to the case when Y is a homogeneous PPP. Itis a standard result in continuum percolation that for the Poisson–Gilbert graph with connection radius < r < ∞ in d ≥ dimensions, there exists a unique critical intensity < λ c ( r ) < ∞ that separates asupercritical regime, where λ > λ c ( r ) , in which the Gilbert graph percolates with probability one anda subcritical regime, where λ < λ c ( r ) , in which the Gilbert graph does not percolate almost surely,cf. for example [MR96, Section 3]. Theorem 2.3.
Let d ≥ , N o , τ, p > , Λ(d x ) = d x and ℓ be well-behaved with ℓ (0) > τ N o /p , then λ ∗ c = λ c ( r B ) .Theorem 2.3 extends the result [DFM +
06, Theorem 1] to dimensions d ≥ using new techniques,see Section 4 for details.In the following section we present our main technical conditions together with examples for whichour main theorems are applicable.3. Stabilization, asymptotic essential connectedness and examples
The following definitions were recently introduced in [HJC19] in order to prove existence of a uniquenon-trivial critical intensity threshold for Cox–Gilbert graphs with fixed connectivity threshold. Letus recall that Q n = [ − n/ , n/ d for n > and d ∈ N , let Q n ( x ) = Q n + x denote the box with sidelength n , centered at x ∈ R d , and dist( x, A ) := inf {| x − y | : y ∈ A } for x ∈ R d and A ⊂ R d . We startwith the definition of stabilization that can be understood as a quantitative spatial mixing propertyof the directing measure of a CPP. Definition 3.1 (Stabilization) . The random measure Λ is called stabilizing if there exists a randomfield of stabilization radii R = { R x } x ∈ R d defined on the same probability space as Λ such that, writing R ( Q n ( x )) = sup y ∈ Q n ( x ) ∩ Q d R y , n ≥ , x ∈ R d , the following hold.(1) (Λ , R ) is jointly stationary,(2) lim n ↑∞ P ( R ( Q n ) < n ) = 1 , and(3) for all n ≥ , non-negative bounded measurable functions f , and finite ϕ ⊂ R d with dist( x, ϕ \{ x } ) > n for all x ∈ ϕ , the following random variables are independent: f (Λ Q n ( x ) ) { R ( Q n ( x )) < n } , x ∈ ϕ, where for a measurable set A ⊆ R d , Λ A denotes the restriction of Λ to A . INR PERCOLATION WITH RANDOM POWERS 7
A stronger form of stabilization is when Λ is b -dependent . That is, the restrictions Λ A and Λ B of Λ to the measurable sets A, B ⊂ R d are independent whenever dist( A, B ) > b for some b > . For b -dependence of subsets of Z d we will use the analogous definition but with dist replaced by the ℓ ∞ -distance. Next we give a definition of asymptotic essential connectedness, a suitable way of capturingconnectedness of the support of the directing measure of a CPP with high probability. Definition 3.2 (Asymptotic essential connectedness) . The stabilizing random measure Λ with stabi-lization radii R is asymptotically essentially connected if for all n ≥ , whenever R ( Q n ) < n/ , wehave that(1) supp(Λ Q n ) contains a connected component of diameter at least n/ ,(2) any two connected components of supp(Λ Q n ) of diameter at least n/ are contained in thesame connected component of supp(Λ Q n ) .The class of stabilizing random measures includes a number of interesting and relevant examples,for instance directing measures given via random tessellations based on PPPs. As already provenin [HJC19, Section 3.1], for example the edge-length measures of Poisson–Voronoi tessellations areasymptotically essentially connected (and hence also stabilizing), and it was also pointed out there thatthe same property for
Poisson–Delaunay tessellations can be proven very similarly. It is neverthelesseasy to see that these intensity measures are not b -dependent for any b > . However, let us note thatthe edge-length measures of Poisson line tessellations in R are not even stabilizing.Stabilizing random measures that are absolutely continuous with respect to the Lebesgue measureinclude the directing measure of some modulated PPPs or shot-noise fields with compactly supportedkernel. For the purpose of the present paper, a modulated PPP is defined with directing measure Λ(d x ) = λ { x ∈ Ξ } d x + λ ′ { x / ∈ Ξ } d x , for some Poisson–Boolean model Ξ with constant connectionradii, where the definition of a Poisson-Boolean model (with constant connection radii) will be presentedat the beginning of Section 5.3, and λ, λ ′ ≥ . As noted in [HJC19, Section 2.1], the intensity measurethat this definition yields is easily seen to be b -dependent for some b > , and if λ and λ ′ are positive,then Λ is asymptotically essentially connected. There exist examples, both for λ > and λ ′ = 0 as well as λ = 0 and λ ′ > , such that asymptotic essential connectedness fails, see [Tób20, Section2.5.1] for details. However, if Ξ is in the supercritical regime for percolation and λ > , then Λ isasymptotically essentially connected, which follows from [PP96, Theorems 2 and 5], as was observedin [HJC19, Section 2.1]. The general definition of a modulated PPP can be found in [CSK +
13, Section5.2.2]; here, the construction is similar to the case presented here, but Ξ can be a general randomclosed subset of R d , and hence the arising directing measure need not even be stabilizing, as explainedin [Tób20, Section 2.5.1]. Finally, without going into details, let us mention that the case when Ξ is aPoisson–Boolean model with random radii, it is possible that the corresponding directing measure isstabilizing but not b -dependent for any b > , see [JTC20, Example 3.4]. Shot-noise fields have directing measures of the form
Λ(d x ) = P i ∈ I κ ( Y i − x )d x , with ( Y i ) i ∈ I ahomogeneous PPP and κ : R d → [0 , ∞ ) compactly supported, cf. [HJC19, Example 2.2]. They arealways b -dependent for some b > but not asymptotically essentially connected in general, see [HJC19,Section 2.1], but in some relevant cases they are, see [Tób20, Section 2.5.1].4. Methods and discussion
In this section, we lay out the strategies for the proofs of our main results, and comment on limitationsand further extensions of the statements presented.4.1.
Strategy of proof and discussion for Theorem 2.1.
As mentioned in the introduction, thestatement of Theorem 2.1 is an extension of the results of [Löf19] to the case of stabilizing CPPs.
BENEDIKT JAHNEL AND ANDRÁS TÓBIÁS
For the proof, we combine the approach used for [Löf19, Theorem 4.5] for handling random radii andthe approach used for [Tób20, Theorem 2.4] for dealing with the spatial correlations of the directingmeasure Λ of the CPP. To begin with, an easy coupling argument, see [Tób19, Section 4.2.3.4], impliesthat as long as the powers are bounded, all positive results of [Tób20] about percolation in the Cox-SINR graph for asymptotically essentially connected Λ are applicable. More precisely, we have thefollowing proposition for the Cox-SINR graph with random bounded powers. Proposition 4.1. [Tób20, Theorem 2.4 and Proposition 2.7]
Let d ≥ , N o , τ > , P ( P o > > , Λ be stabilizing and ℓ be well-behaved. If P sup < ∞ and ℓ (0) > τ N o /P sup , then λ ∗ c < ∞ holds if at leastone of the following conditions is satisfied:(1) ℓ has unbounded support, Λ is b -dependent for some b > and E [exp( α Λ( Q ))] < ∞ holds forsome α > , and at least one of the following conditions hold: Λ is asymptotically essentiallyconnected, or P sup is sufficiently large, or(2) ℓ has bounded support, and Λ is asymptotically essentially connected, or(3) ℓ has bounded support, and sup supp( ℓ ) and P sup are both sufficiently large. Note that we have formulated the Condition (1) in Proposition 4.1 more generally than what wasstated in [Tób20], and the Condition (3) does not appear in [Tób20]. However, the proof from [Tób20]can also be adapted to these more general cases. Indeed, let us first explain how the case of a constantpower µ = δ p , p > , can be handled using the methods of [Tób20]. The cases where Λ is asymptoticallyessentially connected in Proposition 4.1 for constant powers are covered by [Tób20, Theorem 2.4,Part (2)]. Further, the methods of the proof of [Tób20, Proposition 2.7] apply to the case when Λ is stabilizing but not necessary asymptotically essentially connected. To see this, note that thearguments of that proof require the connection radii r B (see (2.4)) to be large enough. Now, if supp( ℓ ) is unbounded, then one can always make r B arbitrarily large via choosing the power value p sufficientlylarge, which corresponds to the case of large P sup = p in (1). Else, this is not always possible because sup p> ℓ − ( τ N /p ) equals the finite number sup supp( ℓ ) . However, once sup supp( ℓ ) is sufficiently large,one can make r B sufficiently large such that the proof of [Tób20, Proposition 2.7] becomes applicable.Hence, we see that Proposition 4.1 indeed follows from [Tób20] for fixed p > . Now, if µ is notconcentrated at one point, then one can always choose p ≥ p > such that µ ([ p , ∞ )) > and µ (( p , ∞ )) = 0 . Then, if p is sufficiently large, then the above arguments imply that there exists aninfinite connected component in the subgraph of the SINR graph spanned by all vertices X i where i ∈ I is such that P i > p , for all sufficiently large λ > and all sufficiently small γ > . Here, webound all power values corresponding to the interferences by p from above. See Section 5.1.1, Step 1for further details of a very similar argument. We conclude that λ ∗ c < ∞ holds under the assumptionof Proposition 4.1. Given Proposition 4.1, in the present paper it suffices to prove the case when P sup = ∞ . We prove Theorem 2.1 in Section 5.1.Let us comment on some further aspects of Theorem 2.1. First, as for Condition (2) in Theorem 2.1,an extension to the general stabilizing case is not possible in general. Indeed, even if P o has very heavytails, as soon as supp( ℓ ) is bounded, the radii of the associated Cox–Gilbert graph with random radiiare bounded. Then, it is not hard to exhibit examples of stabilizing directing measures Λ , such that λ ∗ c = ∞ , see the examples in [Tób20, Section 2.5.1].Second, if Λ is such that Λ( Q ) is almost surely bounded, then the exponential-moment condition E [exp( α Λ( Q ))] < ∞ (4.1)of condition (1) in Theorem 2.1 clearly holds for all α > . E.g., this is the case for the modulated PPPwith λ, λ ′ ≥ . Further, (4.1) holds for shot-noise fields for all α > , see e.g. [Tób20, Section 2.5.1].For Poisson–Voronoi and Poisson–Delaunay tessellations, the b -dependence assumption in (1) fails forall b > , and hence percolation in the SINR graph can only be concluded for compactly supported ℓ . INR PERCOLATION WITH RANDOM POWERS 9
On the other hand, it was verified in [JT20] that for these two kinds of tessellations in two dimensions, E [exp( α Λ( Q ))] < ∞ holds for all α > ; it is not known whether the same holds in higher dimensions.Third, the moment conditions on P o may look surprising at first. Indeed, why do we need to upperbound moments of P o in order to guarantee percolation in an SINR graph? This is indeed counterin-tuitive in view of the Gilbert graph since there larger radii would lead to better connectivity. However,in the SINR graph, as mentioned above, larger powers also increase interference and thus also mightdecrease connectivity. The classical approach used in [DFM +
06, BY13, Tób20] to establish percola-tion in SINR graphs is to show that the underlying Gilbert graph satisfies some strong connectivityproperties and at the same time the interferences can be uniformly bounded on large connected areas.We follow this approach as well, however, the random powers dictate several workarounds.Fourth, the Condition (1) in Theorem 2.1 is not necessarily optimal. However, we believe that ifpercolation with unbounded supp( ℓ ) and without exponential moments of P o is possible, then theproof for this statement must be rather different from ours. An interference-control argument may notbe possible at all, instead one should be able to show that the SINR values are sufficiently large formany transitions yielding satisfactory connectivity of the network for percolation. Let us mention asimilar problem. It was conjectured in [DBT05] that in the case with constant powers, in order to havepercolation in the SINR graph for large λ , ℓ has only to have integrable tails but it may explode at zero.However, the setting where lim r ↓ ℓ ( r ) = ∞ is such that the classical interference-control argument, asexhibited in [DFM +
06] and adapted to the case of random powers in Section 5.1.1, certainly cannotwork. Indeed, the interferences are almost-surely finite but they have infinite expectation, see [Dal71],hence there is no hope to apply a version of the exponential Markov inequality. Let us also note thatthe results of [Dal71] also imply that, if the tails of ℓ are not integrable, then SINR graphs with γ > have no edges.Finally, under the assumptions of Theorem 2.1 on ℓ , for γ = 0 , the SINR graph G ( X λ ) is aGilbert graph with i.i.d. random radii R i = ℓ − ( τ N o /P i ) . Let R o denote a generic random variablehaving the same distribution as R i . Now, if all other parameters are kept fixed, it is easy to seethat γ P ( G γ ( X λ ) percolates ) is decreasing. Hence, if almost surely there is no percolation in theSINR graph for γ = 0 , then the same holds for all γ > . Further, as already mentioned, percolationproperties of Gilbert graphs can equivalently be expressed in terms of the corresponding Booleanmodels. This way, the recent result [JTC20, Theorem 2.6, Part (2)] about existence of a subcriticalphase in Cox–Boolean models immediately implies the following assertion. If Λ is φ -stabilizing and R o is unbounded with E [ R do ] < ∞ , then λ ∗ c > . Here, the notion of φ -stabilization (cf. [JTC20, Definition2.5]) is very similar to our definition of stabilization, and many relevant stabilizing examples are also φ -stabilizing. This observation complements the result λ ∗ c < ∞ in Theorem 2.1. Moreover, it improvesthe assertion of [Tób19, Section 4.2.3.4] that λ ∗ c < ∞ holds for bounded P o (equivalently, bounded R o ) if Λ is stabilizing and ℓ satisfies the assumptions of Theorem 2.1 with ℓ (0) > τ N o / essinf µ or ℓ (0) ≤ τ N o /P sup .4.2. Strategy of proof and discussion for Theorem 2.2.
We call a maximal connected componentin a graph a cluster . As already pointed out in [DBT05, Theorem 1], for γ > , all degrees in G γ ( X λ ) ,where X λ is a PPP, are less than / ( τ γ ) for any choice of λ, τ > and N o ≥ . In other words, allvertices in G γ ( X λ ) have at most / ( τ γ ) neighbors. It is not hard to see that this property remainstrue if the PPP is replaced by a CPP, or even any simple point process, see [Tób19, Section A.3].Thanks to the degree bounds, any such Cox-SINR graph with random powers for which γ ≥ /τ hasno infinite cluster since it has degrees bounded by 1. For γ ∈ [1 / (2 τ ) , /τ ) , we have an a priori degreebound of 2, which implies that all maximal connected components of SINR graphs are finite cycles orpaths that are infinite in zero, one or two directions. This reminds of a one-dimensional percolationmodel, and thus the conjecture is that it contains no infinite clusters under general assumptions on the Figure 2.
A typical realization of a Cox-SINR graph (with blue vertices and blackedges) with directing measure given by the edge-length measure of a two-dimensionalPoisson–Voronoi tessellation (in red) in a box, with N o = τ = 1 , constant powers equalto , and a suitable path-loss function ℓ . The interference-cancellation factor is setto γ = 1 / (2 τ ) . We see only a few vertices having degree two, the largest connectedcomponent is of size three, and there are no cycles in the graph. As indicated byProposition 4.2 the graph is highly disconnected.directing measure of the CPP, see Figure 2 for an illustration. The following proposition shows thatthis is indeed true for the Cox-SINR graph with random powers. Proposition 4.2.
Let d ≥ , N o ≥ , τ > and γ ≥ / (2 τ ) , then for Λ nonequidistant, P ( G γ ( X λ ) percolates ) = 0 . The statement of Theorem 2.2 is an immediate consequence of Proposition 4.2, the proof of whichcan be found in Section 5.2. The proof employs a fine configuration-wise analysis of the SINR graph,which seems to be new in the literature. Moreover, we expect the proof to hold for SINR graphs basedon a large class of simple nonequidistant stationary point processes.Let us comment on a further aspect of Theorem 2.2. It can be observed that the proof of Proposi-tion 4.2 does not use the precise numerical relation γ ≥ / (2 τ ) , but rather just the fact that the SINRgraph has degrees at most 2. Hence, once Λ is nonequidistant, the result holds as soon as the SINRgraph has degrees bounded by 2. Note that for example if N o > , P o bounded and ℓ continuous on [0 , ∞ ) , then one can derive a stricter upper bound on the degrees (depending on several parameters)along the lines of the proof of [DBT05, Theorem 1].4.3. Strategy of proof for Theorem 2.3.
As mentioned previously, we have G ( X λ ) = g r B ( X λ ) forall λ > in the Poisson-SINR graph with fixed powers p , where r B is defined in (2.4). We use X λ insteadof X λ since the marks are non-random. Moreover, note that the increase of the interference-cancellationfactor γ can only lead to edges being removed from the graph and hence there is a monotonicity of G γ ( X λ ) with respect to γ . Additionally, there is a monotonicity of g r B ( X λ ) with respect to λ , whichtogether implies that λ ∗ c ≥ λ c ( r B ) . We have the following equivalence result from [DFM +
06] for thetwo-dimensional Poisson-SINR graphs.
Theorem 4.3. [DFM +
06] Let d = 2 , N o , τ, p > , Λ(d x ) = d x , and ℓ be well-behaved. Then λ ∗ c = λ c ( r B ) . INR PERCOLATION WITH RANDOM POWERS 11
In words, this result states that for any λ > such that the Poisson–Gilbert graph g r B ( X λ ) issupercritical, there exists γ > such that also the Poisson-SINR graph G γ ( X λ ) percolates. In anextended context of SINR graphs, it was shown that this percolation is preserved if the transmittersforming a PPP experience additional interference coming from a weakly α -sub-PPP, see [BY13, Section3.4].The proof of Theorem 4.3 employs Russo–Seymour–Welsh type arguments for the Poisson–Gilbertgraph in two dimensions, see [MR96, Section 4] and [DFM +
06, Section 3]. These arguments do not havea known analogue in the Poisson case for d ≥ , or in the general Cox case even for d = 2 . Note thatthe results of [Tób20] only imply that λ ∗ c < ∞ for d ≥ and Λ(d x ) = d x . However, [HJC19, Section2.1] includes some further observations about Gilbert graphs in d ≥ dimensions, originating fromresults of [PP96], that allow us to conclude the analogue of Theorem 4.3 for the higher-dimensionalPoisson case. The proof of Theorem 2.3 will be carried out in Section 5.3, where we will also recall theresults of [PP96] that are relevant for our analysis.5. Proofs
For the proofs it will be convenient to define the SINR of X i , X j ∈ X λ , X i = X j , via SINR( X i , X j , X λ ) = P i ℓ ( | X i − X j | ) N o + γ P k ∈ I \{ i,j } P k ℓ ( | X k − X j | ) . (5.1)5.1. Proof of Theorem 2.1.
Let us first carry out the proof under Condition (1) of the theorem inSection 5.1.1. The proof under the Conditions (2) and (3) of the theorem are presented in Sections 5.1.2and 5.1.3, respectively.5.1.1.
Proof of Theorem 2.1 Part (1) . For fixed λ and γ , in order to show that G γ ( X λ ) percolates, itsuffices to verify that a subgraph of it contains an infinite cluster. Our proof consists of four steps.First, for γ, λ > , we define a subgraph that is included in a Cox–Gilbert graph (with constantconnection radii). Second, we map this subgraph to a lattice percolation model and show that thisdiscrete model percolates for large λ for a suitable choice of auxiliary parameters. In particular,since Λ is only assumed stabilizing, the connection radius of the Gilbert graph must be large enoughso that the graph percolates for large λ . In this step, we are able to employ multiple argumentsof [DFM +
06, HJC19, Tób20]. Our interference-control assertion, Proposition 5.2, is presented here.Third, using the subgraph, we make a choice of γ > such that percolation in the discrete modelimplies percolation in the SINR graph G γ ( X λ ) , which is done analogously to [DFM + +
06, Tób20] for SINR graphswith constant powers and arguments used in [Löf19] for Poisson-SINR graphs with random powers.
STEP 1.
A subgraph of the SINR graph.
We first present a general construction of a subgraph of G γ ( X λ ) for γ, λ > . Let r o > d o . Sinceboth P o and supp( ℓ ) are unbounded, we have p ( r o ) := P (cid:0) ℓ − (cid:0) τ N o /P o (cid:1) ≥ r o (cid:1) = P (cid:16) P o ≥ τ N o /ℓ ( r o ) (cid:17) > . Let us define the independent thinning X λ, − := { X i ∈ X λ : P o ≥ τ N o /ℓ ( r o ) } of X λ with survival probability p ( r o ) . According to [Kin93, Colouring Theorem], X λ, − is a CPP withdirecting measure λp ( r o )Λ . Now, let us define a subgraph G − γ ( X λ ) of G γ ( X λ ) as follows. The vertex set is X λ, − , and two vertices X i , X j ∈ X λ, − , X i = X j , are connected by an edge if and only if SINR − ( X i , X j , X λ ) := (cid:0) τ N o /ℓ ( r o ) (cid:1) ℓ ( | X i − X j | ) N o + γ P k ∈ I \{ i,j } P k ℓ ( | X k − X j | ) > τ (5.2)and the analogously defined SINR − ( X j , X i , X λ ) also exceeds τ . Note that for X i , X j ∈ X λ, − , in thenumerator of SINR( X i , X j , X λ ) , for the power of X i we have P i ≥ τ N o /ℓ ( r o ) , whereas the denominatorsof (5.1) and (5.2) are equal, and the same holds with the roles of i and j interchanged. Hence, G − γ ( X λ ) is indeed a subgraph of G γ ( X λ ) for any γ ≥ . As for γ = 0 , G − ( X λ ) equals the Cox–Gilbert graph g r o ( X λ, − ) with connection radius r o and vertex set X λ, − . In words, in order to obtain G − γ ( X λ ) from G γ ( X λ ) , one first thins out vertices with small powers, in order to get rid of vertices with small valuesof the connection radius r i B , where r i B = ℓ − ( τ N o /P i ) . (5.3)Then, one bounds the powers of the remaining vertices by τ N o /ℓ ( r o ) from below. STEP 2.
Mapping the subgraph to a lattice-percolation problem and percolation on the lattice.
Now we are in a position to adapt to the setting of [Tób20, Section 3.2.2] and use strong connectivityof g r o ( X λ, − ) in case r o is sufficiently large and λ is chosen according to r o . Together with an interference-control argument presented below, this will allow us to verify Theorem 2.1 Part (1).First, let us recall the definition of rescalings of a Gilbert graph that were also used in [Tób20].For c > and a Gilbert graph G with connection radius r > , deterministic vertex set V ⊂ R d , d ≥ , and edge set E = { ( x, y ) ∈ V × V : x = y, | x − y | < r } , the graph cG is defined with vertexset cV = { cx : x ∈ V } and edge set cE = { ( cx, cy ) : ( x, y ) ∈ E } . It is easy to see that cG is aGilbert graph with vertex set cV and connection radius cr . For Gilbert graphs with random vertexsets (e.g., if the vertex set is given by a random simple point process), rescalings of the graph aredefined realizationwise. From [HJC19, Section 7.1, Proof of Theorem 2.9 (Convergence in BoundedDomains)] we know that in the coupled limit e r ↑ ∞ , e λ ↓ and e λ e r d = e ̺ > , we have that e r − g e r ( X e λ ) converges weakly to the graph g ( Y e ̺ ) , where Y ̺ is a homogeneous PPP with intensity ̺ . Let us notethat in [HJC19, Section 7.1] this convergence is formulated equivalently for the Boolean model and thecrucial point is that the convergence is guaranteed only in compact domains.Let ̺ c (1) be such that the Poisson–Gilbert graph g ( Y ̺ c (1) ) is critical. Then, due to the scaleinvariance of Poisson–Gilbert graphs [MR96, Section 2.2], for ̺ > ̺ c (1) , we can choose a smallerintensity ̺ ′ < ̺ such that g ( Y ̺ ′ ) is still supercritical. Now, for r > d o , we define r o ( r ) = r ( ̺/̺ ′ ) /d , λ ( r ) = ̺ ′ r − d ( p ( r o ( r ))) − and p ( r ) = τ N o /ℓ ( r o ( r )) . Noting that g r ( X λ ( r ) , − ) is a Cox–Gilbert graphwith connection radius r and stabilizing intensity p ( r o ( r )) λ ( r ) = ̺ ′ r − d , we have that r − g r ( X λ ( r ) , − ) converges to the supercritical graph g ( Y ̺ ′ ) on compact sets, as r tends to infinity.Further, recalling that R denotes the stabilization radii of Λ , we put R ( Q ) = sup x ∈ Q ∩ Q d R x for anymeasurable set Q ⊆ R d .Using these notions, we construct a renormalized percolation process on Z d as follows. For n ≥ and r > d o , the site z ∈ Z d is ( r, n ) -good if(1) R ( Q rn ( rnz )) < rn/ ,(2) X λ ( r ) , − ∩ Q rn ( rnz ) = ∅ , and(3) every X i , X j ∈ X λ ( r ) , − ∩ Q rn ( rnz ) are connected by a path in g r ( X λ ( r ) , − ) ∩ Q rn ( rnz ) .The site z ∈ Z d is ( r, n ) -bad if it is not ( r, n ) -good. Note that the process of ( r, n ) -good sites is 7-dependent thanks to the definition of stabilization. The following lemma has been verified in [Tób20,Section 3.2.2]. However, since in [Tób20] it was not formulated as a lemma and two different proofsare presented for d = 2 and d ≥ , we provide a self-contained proof here for the reader’s convenience. INR PERCOLATION WITH RANDOM POWERS 13
Lemma 5.1. [Tób20]
Assume that the general conditions of Theorem 2.1 plus the conditions in Part (1) hold. Then, for all sufficiently large λ > and for all n ≥ and r > d o with rn sufficiently large, thereexists q A = q A ( λ, rn ) < such that for any N ∈ N and pairwise distinct z , . . . , z N ∈ Z d , P ( z , . . . , z N are all ( r, n ) -bad ) ≤ q NA . (5.4) Further, for any ε > , one can choose λ and rn sufficiently large such that q A < ε .Proof. For z ∈ Z d , we write J n,r ( z ) for the event that z satisfies (2) and (3) in the definition of ( n, r ) -goodness. Then, for any n, r under consideration, the process of ( n, r ) -good sites is 7-dependent bythe definition of stabilization. Further, we write F n ( z ) for the event that in the definition of ( n, -goodness, the PPP Y ̺ ′ with intensity ̺ ′ = λ ( r ) r d satisfies (2) with X λ ( r ) , − replaced by Y ̺ ′ and (3) with g r ( X λ ( r ) , − ) replaced by g ( Y ̺ ′ ) everywhere. The probability of F n ( z ) is independent of the choice of z and tends to 1 as n → ∞ thanks to the arguments of [HJC19, Section 5.2], since the constant directingmeasure of the PPP Y ̺ ′ is certainly asymptotically essentially connected. Using a union bound andthe well-known scale invariance of Poisson–Gilbert graphs, namely that for e λ, e r > , e r − g e r ( X e λ ) equals g ( X e λ e r d ) in distribution, we conclude that for z ∈ Z d , P ( z is ( n, r ) -bad ) ≤ P (cid:16) R ( Q nr ( nrz )) ≥ nr/ (cid:17) + P ( F n ( z ) c ) + | P ( F n ( z ) c ) − P ( J n,r ( z ) c ) | , which can be made arbitrarily close to zero by first choosing n large and then r large according to n ,due to the weak convergence of r − g r ( X λ ) to g ( Y ̺ ′ ) on Q n ( nz ) as r → ∞ , λ ( r ) → , r d λ ( r ) = ̺ ′ .Hence, applying [LSS97, Theorem 0.0], for all sufficiently large n and large enough r chosen accordingto n , the 7-dependent process of ( n, r ) -good sites is stochastically dominated from below by a super-critical independent site percolation process. Moreover, the probability of a site of the independentsite percolation process being closed can be made arbitrarily close to 0 via further increasing nr . Thisimplies the lemma. (cid:3) We further proceed similarly to [DFM +
06, Tób20] by defining ‘shifted’ versions of the path-lossfunction ℓ . For a ≥ , define ℓ a ( r ) = ℓ (0) (cid:8) r < a √ d/ (cid:9) + ℓ (cid:0) r − a √ d/ (cid:1) (cid:8) r ≥ a √ d/ (cid:9) . (5.5)Note that ℓ = ℓ . Now, we define the shot-noise processes I a ( x ) = P i ∈ I P i ℓ a ( | x − X i | ) , I ( x ) = P i ∈ I P i ℓ ( | x − X i | ) , x ∈ R d , and note that I ( x ) = I ( x ) . By the triangle inequality, for a ≥ , I ( x ) ≤ I a ( z ) holds for any z ∈ R d and x ∈ Q a ( z ) . Now, the interference-control argument consists in verifying the following proposition.For z ∈ Z d , let us write B r,n,M ( z ) = { I rn ( rnz ) ≤ M } . Proposition 5.2.
Assume that the general conditions of Theorem 2.1 plus Part (1) hold. Then, forall λ > , for all n ≥ and r > d o with rn sufficiently large and for all M > sufficiently large, thereexists q B = q B ( λ, rn, N ) < such that for all N ∈ N and for all pairwise distinct z , . . . , z N ∈ Z d wehave P ( B r,n,M ( z ) c ∩ . . . ∩ B r,n,M ( z N ) c ) ≤ q NB . (5.6) Further, for any ε > and λ > , one can choose rn and M sufficiently large such that q B < ε . The proof of this proposition is postponed until Step 4. Once we have shown Proposition 5.2, we canderive the following corollary using a standard argument (see e.g. the proof of [DFM +
06, Proposition3] or the one of [Tób20, Proposition 3.1]). For z ∈ Z d let us define C r,n,M ( z ) = { z is ( r, n ) -good } ∩{ I rn ( rnz ) ≤ M } . Corollary 5.3.
Assume that the general conditions of Theorem 2.1 plus Part (1) hold. Then, forall sufficiently large λ > , for all r > d o and n ≥ with rn sufficiently large and for all M > sufficiently large, there exists q C = q C ( λ, rn, M ) < such that for all N ∈ N and for all pairwisedistinct z , . . . , z N ∈ Z d we have P ( C r,n,M ( z ) c ∩ . . . ∩ C r,n,M ( z N ) c ) ≤ q NC . Further, for any ε > , one can choose λ, rn, M sufficiently large such that q C < ε . STEP 3.
Percolation in the subgraph of the SINR graph.
Having Corollary 5.3 and employing a Peierls argument (cf. [Gri99, Section 1.4]), we conclude that for λ, rn, M sufficiently large, the process of ( r, n ) -good sites z ∈ Z d such that I rn ( rnz ) ≤ M percolates.Thanks to the exact same arguments as in [HJC19, Section 5.2, Proof of Theorem 2.6], this impliespercolation of the Cox–Gilbert graph G − ( X λ ( r ) ) = g r o ( r ) ( X λ ( r ) , − ) . From this point of the proof itis classical to derive that G − γ ( X λ ( r ) ) percolates for small γ > , see [DFM +
06, Section 3.3]. For theconvenience of the reader, let us give the details here. We define γ ′ = N o p ( r ) M (cid:16) ℓ ( r ) ℓ ( r o ( r )) − (cid:17) = ℓ ( r o ( r )) τ M (cid:16) ℓ ( r ) ℓ ( r o ( r )) − (cid:17) > , where the strict inequality holds because r o ( r ) > r > d o and ℓ has unbounded support. Then we have p ( r ) ℓ ( r ) N o + γ ′ p ( r ) M = τ. Now, let X i , X j ∈ X λ ( r ) , − be situated in Q rn ( rnz ) respectively Q rn ( rnz ′ ) for some sites z, z ′ ∈ Z d included in the same infinite cluster of the process of ( r, n ) -good sites z ∈ Z d satisfying I rn ( rnz ) ≤ M such that | X i − X j | < r . Then, for γ < γ ′ , we have SINR( X i , X j , X λ ) ≥ SINR − ( X i , X j , X λ ) > p ( r ) ℓ ( r ) N o + γ ′ p ( r ) M = τ. Thus, X i and X j are connected by an edge in G − γ ( X λ ) . Hence, G γ ( X λ ) also percolates. Thus, we canconclude Theorem 2.1 as soon as we have verified Proposition 5.2. STEP 4.
Proof of Proposition 5.2: the interference-control argument.
Similarly to [Tób20, Section 3.1.1], we split the interference into two parts. For x ∈ R d , n ≥ and r > , we put I in6 rn ( x ) = X X i ∈ X λ ∩ Q rn √ d ( x ) ℓ rn ( | X i − x | ) , I out6 rn ( x ) = X X i ∈ X λ \ Q rn √ d ( x ) ℓ rn ( | X i − x | ) . Then, for
M > , if I rn ( x ) > M , then I in6 rn ( x ) > M/ or I out6 rn ( x ) > M/ . Using a union boundand the fact that in Proposition 5.2, M can be chosen arbitrarily large, it suffices to conclude theproposition both with B r,n,M ( z i ) replaced by B in r,n,M ( z i ) and with B r,n,M ( z i ) replaced by B out r,n,M ( z i ) everywhere in (5.6) for all i ∈ { , . . . , N } , where for z ∈ Z d we write B in r,n,M ( z ) = { I in6 rn ( rnz ) ≤ M } and B out r,n,M ( z ) = { I out6 rn ( rnz ) ≤ M } . Indeed, having these assertions, we can combine them similarly toCorollary 5.3.We now verify Proposition 5.2 with B r,n,M ( · ) replaced by B in r,n,M ( · ) everywhere. For this assertion,instead of the assumption that P o and Λ( Q ) have some exponential moments, it suffices if they havea first moment (for Λ( Q ) this is automatic since E [Λ( Q )] = 1 by assumption). To be more precise,we prove the following lemma. INR PERCOLATION WITH RANDOM POWERS 15
Lemma 5.4.
Assume that the general conditions of Theorem 2.1 plus Part (1) hold. Further, let Λ bestabilizing and E [ P o ] < ∞ . Then, for all λ > , for all n ≥ and r > d o with rn sufficiently large andfor all M > sufficiently large, there exists q B = q B ( λ, rn, N ) < such that for all N ∈ N and for allpairwise distinct z , . . . , z N ∈ Z d we have P ( B in r,n,M ( z ) c ∩ . . . ∩ B in r,n,M ( z N ) c ) ≤ q NB . (5.7) Further, for any ε > and λ > , one can choose rn and M sufficiently large such that q B < ε .Proof. We use the following auxiliary discrete percolation process. A site z ∈ Z d is ( r, n ) -tame if(1) R ( Q rn √ d ( rnz )) < rn/ , and(2) I in6 rn ( rnz ) ≤ M .A site z ∈ Z d is ( r, n ) -wild if it is not ( r, n ) -tame. The process of ( r, n ) -tame sites is ⌈ √ d + 1 ⌉ -dependent according to the definition of stabilization. Thus, it follows from dependent-percolationtheory [LSS97, Theorem 0.0] that, in order to verify Lemma 5.4, it suffices to show that for all λ > , P ( o is ( r, n ) -wild ) can be made arbitrarily close to zero by choosing first rn sufficiently large and then M large enough accordingly. We have P ( o is ( r, n ) -wild ) ≤ P ( R ( Q rn √ d ( rnz )) ≥ rn/
2) + P ( I in6 rn ( rnz ) > M ) . The first term can be made arbitrarily small by choosing rn large enough, thanks to the definition ofstabilization. Moreover, by the definition of ℓ a , see (5.5), I in6 rn ( o ) = X X i ∈ X λ ∩ Q rn √ d ( o ) P i ℓ rn ( | X i | ) ≤ ℓ (0) X X i ∈ X λ ∩ Q rn √ d ( o ) P i . In particular, using that the point process X λ is independently marked with P i having marginaldistribution µ , and that Λ is stationary with E [Λ( Q )] = 1 , it follows that E [ I in6 rn ( o )] ≤ ℓ (0) λ E [ P o ] E [Λ( Q rn √ d )] = (12 rn √ d ) d ℓ (0) λ E [ P o ] . Thus, for any n ≥ and r > , P ( I in6 rn ( o ) > M ) can be made arbitrarily small by choosing M largeenough, given that E [ P o ] < ∞ . Thus, the statement of the lemma follows. (cid:3) It remains to verify Proposition 5.2 with B r,n,M ( · ) replaced by B out r,n,M ( · ) everywhere. More precisely,thanks to the exponential-moment and b -dependence assumption on Λ , the proof can be completedanalogously to the proof of [Tób20, Proposition 3.3] starting from [Tób20, Equation (3.15)], as soonas we have verified the following lemma. ([Tób20] also assumed that ℓ (0) ≤ , but since M can bemade arbitrarily large in Proposition 5.2, ℓ (0) ≤ can be assumed without loss of generality since for ℓ continuous, the function e ℓ = ℓ/ℓ (0) satisfies e ℓ (0) = 1 , and for a ≥ , we have ℓ a = ℓ (0) e ℓ a and hence I a ( x ) = ℓ (0) P i ∈ I P i e ℓ a ( | x − X i | ) .) Lemma 5.5.
Under the general assumptions of Theorem 2.1 plus Part (1) , there exists a constant c o = c o ( µ, ℓ ) > such that for all sufficiently small s > , for all λ > , n ≥ and r > d o with rn > sufficiently large and for all large enough M > , for all N ∈ N and pairwise distinct z , . . . , z N ∈ Z d we have P ( B out r,n,M ( z ) c ∩ . . . ∩ B out r,n,M ( z N ) c ) ≤ E h exp (cid:16) c o λs N X i =1 Z R d \ Q rn √ d ( rnz i ) ℓ rn ( | rnz i − x | )Λ(d x ) (cid:17)i . (5.8)Indeed, the right-hand side of (5.8) is the same as the one of [Tób20, Equation (3.15)], and theassumptions on Λ in the two proofs are also the same. Proof of Lemma 5.5.
We start with an estimate originating from [DFM +
06, Section 3.2]. By Markov’sinequality, for any s > , P ( B out r,n,M ( z ) c ∩ . . . ∩ B out r,n,M ( z N ) c ) = P ( I out6 rn ( rnz ) > M, . . . , I out6 rn ( rnz N ) > M ) ≤ P (cid:16) N X i =1 I out6 rn ( rnz i ) > N M (cid:17) ≤ e − sNM E h exp (cid:16) s N X i =1 X X k ∈ X λ \ Q rn √ d ( nz i ) P k ℓ rn ( | rnz i − X k | ) (cid:17)i . (5.9)The randomness of the power values P k prevents us from continuing the proof analogously to [DFM + X λ = { ( X i , P i ) } i ∈ I is a CPP in R d × [0 , ∞ ) with directing measure Λ ⊗ µ , where we recall that µ = P ◦ P − o is the distribution of P o . Hence, applying the Laplace functional of a CPP (cf. [Kin93,Sections 3.2, 6]) to the function f : R d × [0 , ∞ ) → [0 , ∞ ) , f ( x, p ) = s N X i =1 pℓ rn ( | x − rnz i | ) { x ∈ R d \ Q rn √ d ( rnz i ) } , we obtain E h exp (cid:16) s N X i =1 X X k ∈ X λ \ Q rn √ d ( rnz i ) P k ℓ rn ( | rnz i − X k | ) (cid:17)i (5.10) = E h exp (cid:16) λ Z R d \ Q rn √ d ( rnz i ) Z ∞ (cid:16) exp (cid:16) sp N X i =1 ℓ rn ( | rnz i − x | ) (cid:17) − (cid:17) µ (d p )Λ(d x ) (cid:17)i . Thanks to the exponential-moment assumption on P o from (1), the moment-generating function α E [exp( αP o )] = Z ∞ e αp µ (d p ) is infinitely differentiable at α = 0 with first derivative R ∞ pµ (d p ) = E [ P o ] < ∞ . Note that P Ni =1 ℓ rn ( | rnz i − x | ) is uniformly bounded in x ∈ R d , rn , N and pairwise distinct z , . . . , z N ,see [Tób20, Lemma 3.6]. Consequently, for any C > , the following holds for all sufficiently small s > (depending on C ), Z ∞ (cid:16) exp (cid:16) sp N X i =1 ℓ rn ( | rnz i − x | ) (cid:17) − (cid:17) µ (d p ) ≤ Cs E [ P o ] N X i =1 ℓ rn ( | rnz i − x | ) . (5.11)For such s , plugging (5.11) back into (5.10), starting from (5.9) we obtain P ( B out r,n,M ( z ) c ∩ . . . ∩ B out r,n,M ( z N ) c ) ≤ E h exp (cid:16) C E [ P o ] λs N X i =1 Z R d \ Q rn √ d ( rnz i ) ℓ rn ( | rnz i − x | )Λ(d x ) (cid:17)i , (5.12)which is (5.8) with c o = C E [ P o ] . With this we conclude the proof of the lemma. (cid:3) INR PERCOLATION WITH RANDOM POWERS 17
Proof of Theorem 2.1 Part (2) . Since Λ is asymptotically essentially connected, the criticalintensity λ c ( r ) for the Poisson–Gilbert graph with connectivity threshold r is finite, more precisely, λ c ( r ) < ∞ holds for any r > according to [HJC19, Theorem 2.4]. Note further that the connectionradii ( r i B ) i ∈ I , defined in (5.3), are bounded by d max = sup { x ≥ x ∈ supp( ℓ ) } . The proof ofTheorem 2.1 Part (2) can be obtained as an adaptation of the proof of Part (1) of the same theoremas follows.First, one defines the subgraph of the SINR graph analogously to Step 1 of the proof of Theorem 2.1Part (1). Next, one takes the Step 2 but for r ∈ ( d o , d max ) arbitrary and fixed instead of letting r ↑ ∞ , and for r o ( r ) > r such that r o ( r ) still lies in the interval ( d o , d max ) on which ℓ is strictlydecreasing. This way, choosing rn sufficiently large will be equivalent to choosing n large enough (forfixed r ). Further, one alters the choice of λ ( r ) : now, λ ( r ) has to be chosen so large that the processof ( r, n ) -good sites percolates for some n ≥ , which is possible for any fixed r ∈ ( d o , d max ) since Λ isasymptotically essentially connected, cf. [HJC19, Section 5.2]. Next, Step 3 is also applicable for allchoices of the parameters where the underlying discrete model percolates. Finally, let us explain howto complete the proof of Proposition 5.2 under the mere assumption that E [ P o ] < ∞ . Since supp( ℓ ) isbounded, for all sufficiently large n ≥ the following holds for all z ∈ Z d I rn ( rnz ) = X X i ∈ X λ ∩ Q rn +2 d max ( rnz ) P i ℓ rn ( | X i − rnz | ) ≤ X X i ∈ X λ ∩ Q rn √ d ( rnz ) P i ℓ rn ( | X i − rnz | ) = I in6 rn ( rnz ) . (5.13)Hence, it remains to control the inner part of the interference, which can be done analogously toLemma 5.4 once E [ P o ] < ∞ , given that Λ is stabilizing. Hence, we conclude Theorem 2.1 Part (2).5.1.3. Proof of Theorem 2.1 Part (3) . In the case when Λ is only stabilizing and P sup = ∞ , we observethat the proof of Theorem 2.1 Part (1) stays valid if supp( ℓ ) is bounded but the following assumptionholds: sup supp( ℓ ) is sufficiently large such that sup supp( ℓ ) > inf { r > ∃ n ≥ and λ > such that ( r, n ) -good sites percolate } , where the infimum is finite because Λ is stabilizing. Indeed, in this situation, Lemma 5.1 (as in [Tób20,Section 3.2.2]) holds as well. Further, (5.13) holds for all sufficiently large n for all z ∈ Z d , and thereforeone can complete the proof under the assumptions of Lemma 5.4, i.e., for Λ stabilizing and P o such that E [ P o ] < ∞ , without requiring b -dependence of Λ for any b > or existence of exponential moments of Λ( Q ) or P o .5.2. Proof of Proposition 4.2.
The strategy of the proof of the Proposition 4.2 is the following.We first show that up to P -null sets, clusters are either finite or infinite in both directions, i.e., theycontain no vertex of degree 1 in case they are infinite, see Lemma 5.6 below. Next, we assume fora contradiction that there exists an infinite cluster with positive probability. Then, we introduce aprocedure that removes points from the infinite cluster that is closest to the origin in a certain sense.Thanks to elementary properties of the SINR graph, in the resulting configuration, the infinite clusterstill remains infinite, but it contains a vertex of degree 1. Hence, the probability that the process takesvalues in the set of the resulting configurations is zero. What remains to show afterwards is that alsothe probability that the process takes place in the set of original configurations is zero, which leads tothe desired contradiction. At this point it will be useful to compare the resulting configuration withan independent thinning of the original configuration in a certain ball, and this is where we make useof the fact that the underlying point process is a stationary CPP.We assume throughout the proof that γ ≥ / (2 τ ) , so that degrees in G γ ( X λ ) are bounded by two,and that X λ is nonequidistant (for all λ > ). We can also assume that P ( P o > > in what follows, since otherwise the statement is trivially true. We start the proof with the following lemma, whichexcludes infinite paths that have an endpoint in case the degrees are bounded by two, in a substantiallymore general setting. Lemma 5.6.
Let g ( X ) be a random graph based on a stationary marked point process X = { ( X i , P i ) } i ∈ I with values in R d × Z , where the mark space ( Z, Z ) is an arbitrary measurable space, X = { X i } i ∈ I isthe vertex set, and such that the degree of all X i ∈ X , deg( X i ) , is bounded by , almost surely. Let X have a finite intensity and consider the point process of degree-one points in infinite clusters X = X i ∈ I δ X i { deg( X i ) = 1 , X i is part of an infinite cluster in g ( X ) } . Then, P ( X ( R d ) = 0) = 1 . We will apply this lemma to the SINR graph g ( X ) = G γ ( X λ ) with λ arbitrary and γ ≥ / (2 τ ) and Z = [0 , ∞ ) . The proof is based on a variant of the mass-transport principle (cf. [CDS20, Section 4.2]for instance). Proof of Lemma 5.6.
First, using the union bound and stationarity, it is enough to show that E [ X ( Q )] = 0 . Let us define the point process of points in infinite clusters that are at distanceequal to k ∈ N o from a point in X , X k = X i ∈ I δ X i { X i is part of an infinite cluster and has graph distance k from X } . Thanks to the degree bound, every infinite cluster has at most one point in X and E [ X k ( Q )] = E [ X ( Q )] , for all k ∈ N o , by stationarity. However, P k ≥ E [ X k ( Q )] ≤ E [ X ( Q )] < ∞ and thus E [ X ( Q )] = 0 . (cid:3) Let us denote by ( C i ) ≤ i 1) = 0 . (5.14)We view X λ as the canonical process X λ ( ω ) = ω on the set N of marked point configurations ω in R d × [0 , ∞ ) such that ω = { x i : ( x i , p i ) ∈ ω } is an infinite, locally-finite, nonequidistant pointconfiguration on R d . The set of such point configurations ω will be denoted by N . Note that N and N are equipped with the corresponding evaluation σ -fields.Now we introduce an ordering in R d × [0 , ∞ ) , which orders the points of the set according to thereceived signal power at a given point y ∈ R d (or equivalently, according to the received SINR values SINR( · , y, ω ) at y ). Definition 5.7. Let ( x, p ) , ( z, r ) ∈ R d × [0 , ∞ ) and y ∈ R d . We say that ( x, p ) transmits a strongersignal to y than ( z, r ) does if one of the following conditions is satisfied:(1) pℓ ( | x − y | ) > rℓ ( | z − y | ) , or(2) pℓ ( | x − y | ) = rℓ ( | z − y | ) and | x − y | < | z − y | .When talking about the marked CPP X λ , we will always assume that transmitters are associatedwith their own transmitted signal powers, and hence we will say “ X i transmits a stronger signal to X j than X l does” instead of “ ( X i , P i ) transmits a stronger signal to X j than ( X l , P l ) does”, for any i, j, l ∈ I such that i = j and l = j . Then, it is easy to see that for X λ such that X λ is nonequidistant,almost surely the following holds. For all i ∈ I , the relation “ X i transmits a stronger signal to X j than X l does” is a total ordering (i.e., irreflexive, antisymmetric and transitive, with any two elements INR PERCOLATION WITH RANDOM POWERS 19 being comparable) on the set { ( i, l ) ∈ I : i = j and l = j } , which we call the ordering of signal-weighted distance from receiver X i . This fact indeed relies on the tiebreaking mechanism (2): e.g., if ℓ is constant on some interval (which is possible under the assumption of Proposition 4.2 and even underthe stronger assumption of Theorem 2.1), then (1) does not define a total ordering on its own.For ω ∈ N and x o ∈ ω , we can consider the vector V ( x o , ω ) = ( V n ( x o , ω )) n ∈ N of the markedpoints of ω ordered increasingly according to signal-weighted distance from receiver x o . Then, wedefine V i ( x o , ω ) as the first component of V i ( x o , ω ) , which we call the i -th nearest neighbor of x o insignal-weighted order . In particular, V ( x o , ω ) = x o . Note that if the distribution µ is concentrated inone point p > , i.e., µ = δ p , then the i -th nearest neighbor of x o in signal-weighted order is just the i -th nearest neighbor of x o with respect to Euclidean distance.Now, if x o has degree two in G γ ( X λ ( ω )) , then x o must be connected by an edge to both V ( x o , ω ) and V ( x o , ω ) since the degree bound applies already for the edges towards x o . Moreover, both V ( x o , ω ) and V ( x o , ω ) must also have x o as one of their first two nearest neighbors in signal-weighted order,that is, x o ∈ (cid:8) V (V i ( x o , ω ) , ω ) , V (V i ( x o , ω ) , ω ) (cid:9) , for all i ∈ { , } . These signal-weighted nearest neighbor relations hold almost surely, in particularfor every nonequidistant configuration ω . The goal of using the configuration space N is to entirelyexclude configurations that offend the degree bound or the signal-weighted nearest neighbor relationsor are not nonequidistant.In the event { L ≥ } , let Z = ( Z, R ) denote the closest point to the origin that has degree two andis contained in an infinite cluster. Without loss of generality, we will assume that this cluster is alwaysequal to C . Now, Proposition 4.2 immediately follows once we have verified the following proposition. Proposition 5.8. Consider the event { L ≥ } and define the random variable I = inf { i ≥ i ( Z, X λ ) ∈ C } . Then, under the assumptions of Proposition 4.2, for any i ≥ , we have P ( { L ≥ } ∩ { I = i } ) = 0 . (5.15) Proof of Proposition 4.2. Using a union bound and noting that { L ≥ } ⊂ { I < ∞} , Proposition 5.8implies P ( L ≥ 1) = 0 , which is (5.14), and thus the proof of Proposition 4.2 is finished. (cid:3) Proof of Proposition 5.8. For ω ∈ { L ≥ } , by definition, we have that Z ( ω ) is connected by an edgeboth to V ( Z ( ω ) , ω ) and V ( Z ( ω ) , ω ) in G γ ( X λ ( ω )) . Further, thanks to the degree bound of two,in the event { L ≥ } , V ( Z ( ω ) , ω ) and V ( Z ( ω ) , ω ) have no further joint neighbor in G γ ( X λ ( ω )) since otherwise C ( ω ) has a loop and cannot be infinite by the degree bound. This way, for any i ≥ , there exists l ∈ { , } such that V i ( Z ( ω ) , ω ) and V l ( Z ( ω ) , ω ) are not connected by an edge in G γ ( X λ ( ω )) . Let us denote the corresponding V l ( Z ( ω ) , ω ) by M i ( ω ) , and define M i ( ω ) = V ( Z ( ω ) , ω ) if neither V ( Z ( ω ) , ω ) nor V ( Z ( ω ) , ω ) is connected to V i ( Z ( ω ) , ω ) by an edge. The element of { V ( Z ( ω ) , ω ) , V ( Z ( ω ) , ω ) } not being equal to M i ( ω ) is denoted by N i ( ω ) . We will write Q for thesignal power transmitted by M i ( ω ) .Let us fix i ≥ . Let ω ∈ { L ≥ } be such that I ( ω ) = i . Let us define a thinned configuration ω i = ω \ { ( M i ( ω ) , Q ) , V ( Z ( ω ) , ω ) , . . . , V i − ( Z ( ω ) , ω ) } . We claim for P -almost all ω ∈ { L ≥ } ∩ { I = i } also ω i ∈ { L ≥ } . For this, first note that theremoval of finitely many points and their associated edges from an infinite cluster does not changethe property of the cluster to be infinite. However, the removal of points can still change the edgestructure of the remaining points. In order to exclude this, we can use the following fundamentalproperty of the SINR graph. Assume that ω , ω ′ are elements of N such that ω ⊆ ω ′ . Then, for all x, y ∈ ω , if SINR( x, y, ω ′ ) > τ , then SINR( x, y, ω ) > τ , which is clear from (5.1). In words, if weremove some vertices from an SINR graph, then edges of the SINR graphs between the remainingpoints stay preserved.Then, our next claim is that for ω ∈ { L ≥ } ∩ { I = i } , we have that ω i is contained in B = { η : L ( η ) ≥ and C ( η ) contains a point of degree one } ⊂ { L ≥ } . The proof of this claim in the simplest case i = 3 is illustrated in Figure 3. For general i ≥ , recall Z V = M V = N V Figure 3. A visualization of the case I ( ω ) = 3 for some realization ω ∈ { L ≥ } . V = V ( Z ( ω ) , ω ) is contained in the infinite cluster C = C ( ω ) of the SINR graph G γ ( ω ) including Z = Z ( ω ) , and it is not a neighbor of M = M ( ω ) , which in thisexample equals V = V ( Z ( ω ) , ω ) , whereas V = V ( Z ( ω ) , ω ) = N = N ( ω ) . Hence,if V has degree two in C , then there are various possibilities respecting the degreebound of two to connect V to C so that it is not connected to M by an edge. V caneither be a direct neighbor of V (see dashed line) or a later point of the path from Z toinfinity starting with the edge from Z to V (dash-dotted lines) or a non-direct neighborof V on the path from Z to infinity starting with the edge from Z to V (dotted lines).Now, removing M from the realization, both edges adjacent to V are preserved. Alsoall edges from Z to infinity starting with the edge from Z to V are preserved, hence Z is still contained in an infinite cluster, but the edge from Z to V is removed. In theresulting new configuration, the second-nearest neighbor of Z in signal-weighted orderis V , and hence this is the only point of the configuration that could be connected to Z by an edge. But V still cannot have degree 3 or more, hence it cannot be connectedto Z , which implies that in the new configuration Z is in an infinite cluster containinga point of degree one.that Z cannot have degree higher than two in G γ ( X λ ( ω i )) , whereas it has degree at least one andits cluster C ( ω i ) is infinite in G γ ( X λ ( ω i )) . Note also that the edge between Z ( ω ) and N i ( ω ) stillexists in G γ ( X λ ( ω i )) . Further, if Z ( ω ) has degree two in G γ ( X λ ( ω i )) , then it is connected to thesecond-nearest neighbor of Z ( ω ) in signal-weighted order in ω i , which is V ( Z ( ω ) , ω i ) = V i ( Z ( ω ) , ω ) ,whereas V ( Z ( ω ) , ω i ) = N i ( ω ) . Now, since ω / ∈ B , ω ∈ { L ≥ } and V i ( Z ( ω ) , ω ) ∈ C ( ω ) , itfollows that V i ( Z ( ω ) , ω ) has degree equal to two in G γ ( X λ ( ω )) . Further, it is neither connected to M i ( ω ) by an edge nor to Z ( ω ) in this graph. Hence, both edges adjacent to V i ( Z ( ω ) , ω ) also exist in G γ ( X λ ( ω i )) . But since V i ( Z ( ω ) , ω ) has degree at most two in G γ ( X λ ( ω i )) , it follows that Z ( ω ) and V i ( Z ( ω ) , ω ) are not connected by an edge in this graph. Hence, ω i ∈ B , which implies the claim.Note that by Lemma 5.6, the set B is a P -null set, i.e., P ( { ω i : ω ∈ { L ≥ } ∩ { I = i }} ) = 0 . (5.16)This implies (5.15) and concludes the proof of Proposition 5.8 as soon as the following lemma is verified. Lemma 5.9. Under the assumptions of Proposition 4.2, for any i ≥ , P ( { L ≥ } ∩ { I = i } ) > implies P ( { ω i : ω ∈ { L ≥ } ∩ { I = i }} ) > . INR PERCOLATION WITH RANDOM POWERS 21 By Lemma 5.9, where we show that if the collection of thinned configurations is contained in a P -null set, also the non-thinned configurations form a P -null set, we see that (5.16) implies (5.15),which concludes the proof of Proposition 5.8. (cid:3) Proof of Lemma 5.9. Let us fix i ≥ and assume that P ( { L ≥ } ∩ { I = i } ) > . Then, by continuityof measures, there exists K > such that P (cid:0)(cid:8) ω ∈ { L ≥ } ∩ { I = i } : V j ( Z ( ω ) , ω ) ∈ B K ( o ) , ∀ j ∈ { , . . . , i } (cid:9)(cid:1) > , where B K ( o ) denotes the open Euclidean ball of radius K in R d . Hence, there exists n ≥ i such that P ( C i,K,n ) > , where C i,K,n = (cid:8) ω ∈ { L ≥ } ∩ { I = i } : (cid:0) ω ∩ B K ( o ) (cid:1) = n + 1 and V j ( Z ( ω ) , ω ) ∈ B K ( o ) , ∀ j ∈ { , . . . , i } (cid:9) . Conditional on the event C i,K,n , the marked point process ( X λ \ { Z } ) ∩ B K ( o ) has precisely n points X , . . . X n .Now, for some fixed q ∈ (0 , , we can represent X λ as X λ, ∪ X λ, as follows. For K > , let B K ( o ) denote the open ℓ -ball of radius K around o . Let X λ, be given as the union of X λ \ ( B K ( o ) × [0 , ∞ )) and the independent thinning of X λ ∩ ( B K ( o ) × [0 , ∞ )) with survival probability q , and let X λ, bethe complementary thinning. That is, conditional on X λ , X λ, ∩ ( B K ( o ) × [0 , ∞ )) contains each pointof X λ ∩ ( B K ( o ) × [0 , ∞ )) with probability q independent of the other points of this point process,and it contains no other points. Note further that X λ, and X λ, ∩ ( B K ( o ) × [0 , ∞ )) are independentthinnings of X λ ∩ ( B K ( o ) × [0 , ∞ )) with survival probability − q respectively q , moreover, X λ, = X λ \ X λ, , X λ, \ ( B K ( o ) × [0 , ∞ )) = ∅ , and X λ, \ ( B K ( o ) × [0 , ∞ )) = X λ \ ( B K ( o ) × [0 , ∞ )) . Inorder to provide a precise construction of the thinned processes, we choose a sequence ( J m ) m ∈ N ofi.i.d. Bernoulli random variables with parameter q that is independent of X λ , and given the realization ω = X λ ( ω ) = ( V i ( Z ( ω ) , ω )) i ∈ N , the realizations of X λ, ( ω ) and X λ, ( ω ) are defined as follows,depending also on ( J m ) m ∈ N : X λ, ( ω ) = X λ, ( ω , ( J m ) m ∈ N ) = { V m ( Z ( ω ) , ω ) : J m = 1 , V m ( Z ( ω ) , ω ) ∈ B K ( o ) }∪ { Z ( ω ) } ∪ { V m ( Z ( ω ) , ω ) : V m ( Z ( ω ) , ω ) ∈ R d \ B K ( o ) } and X λ, ( ω ) = X λ, ( ω , ( J m ) m ∈ N ) = { V m ( Z ( ω ) , ω ) : J m = 0 , V m ( Z ( ω ) , ω ) ∈ B K ( o ) } . It is clear that the projections X λ, , X λ, of X λ, respectively X λ, to the R d -coordinate are nonequidis-tant, further, X λ, can be represented as a random variable with values in N , defined on an enlargedprobability space (Ω ′ , F ′ , P ′ ) governing both the point process X λ and the sequence ( J m ) m ∈ N . Inparticular, P ′ ( X λ ∈ · ) = P ( X λ ∈ · ) .The next property of stationary and nonequidistant CPPs is crucial for completing the proof ofLemma 5.9. Lemma 5.10. Let Λ be stationary and nonequidistant. Then, for any K > , the law of X λ, isabsolutely continuous with respect to the one of X λ for any K > . To be more precise, the absolute continuity is meant in this lemma in the following way, with respectto the probability space (Ω ′ , F ′ , P ′ ) on which X λ, and X λ are jointly defined with P ′ ( X λ ∈ · ) = P ( X λ ∈· ) . Let G ∈ F ′ be any event such that P ′ ( X λ, ∈ G ) > , then we have P ′ ( X λ ∈ G ) > . Proof of Lemma 5.10. Let F be an element of the evaluation σ -algebra of N such that P ′ ( X λ, ∈ F ) > . We have to show that then also P ( X λ ∈ F ) > . Under the assumption that P ′ ( X λ, ∈ F ) > , bycontinuity of measures, we can find K, l ∈ N such that ε := P ′ ( X λ, ∈ F, X λ, ∩ ( B K ( o ) × [0 , ∞ ))) = l ) > . (5.17) In other words, we have < ε = P ′ ( X λ, ∈ G ) where G = { ω ∈ F : ω ∩ ( B K ( o ) × [0 , ∞ ))) = l } .Thus, P ( X λ ∈ F ) ≥ P ′ ( X λ ∈ G, X λ, = X ) ≥ P ′ ( X λ, ∈ G ) P ′ ( X λ, = X λ | X λ, ∈ G )= ε P ′ ( X λ, = X λ | X λ, ∈ G ) , and further, P ′ ( X λ, = X λ | X λ, ∈ G ) ≥ P ′ ( X λ, = X , X λ, ∈ G ) = P ′ ( X λ, = ∅ , X λ, ∈ G ) . (5.18)According to (5.17), we have < ε = P ′ ( X λ, ∈ G ) = ∞ X n =0 a n , where a n = E ′ (cid:2) P ′ ( X λ, ∈ G | Λ) { Λ( B K ( o )) ∈ [ n, n + 1) } (cid:3) , and thus there exists m ∈ N with a m > .Now, conditional on Λ , X λ, is an i.i.d. marked PPP, and hence a PPP on R d × [0 , ∞ ) , which alsoimplies that the complementary thinnings X λ, and X λ, are independent given Λ , see [Kin93, ColouringTheorem and Marking Theorem]. Hence, we obtain P ′ ( X λ, = ∅ , X λ, ∈ G ) = E ′ (cid:2) P ′ ( X λ, = ∅ | Λ) P ′ ( X λ, ∈ G | Λ) (cid:3) = ∞ X n =0 E ′ (cid:2) e − (1 − q )Λ( B K ( o )) P ′ ( X λ, ∈ G | Λ) { Λ( B K ( o )) ∈ [ n, n + 1) } (cid:3) ≥ ∞ X n =0 e − (1 − q )( n +1) a n ≥ e − (1 − q )( m +1) a m > , which verifies the lemma that the distribution of X λ, is absolutely continuous with respect to the oneof X λ . (cid:3) Given Lemma 5.10, we now finish the proof of Lemma 5.9. Thanks to the assumption that P ( C i,K,n ) > and using the definition of X λ, , P ′ (cid:0) X λ, ∈{ ω i : ω ∈ { L ≥ } ∩ { I = i }} (cid:1) ≥ P ′ (cid:0) X λ, ∈ { ω i : ω ∈ C i,K,n } (cid:1) ≥ P ′ (cid:0) X λ, ∈ { ω i : ω ∈ C i,K,n } , X λ ∈ C i,K,n (cid:1) = P ( C i,K,n ) P ′ (cid:0) X λ, ∈ { ω i : ω ∈ C i,K,n }| X λ ∈ C i,K,n (cid:1) = P ( C i,K,n ) q n − i +2 (1 − q ) i − > . (5.19)Finally, by Lemma 5.10, under P ′ the distribution of X λ, is absolutely continuous with respect to theone of X λ . Hence, it follows from (5.19) that P (cid:0) X λ ∈ { ω i : ω ∈ { L ≥ } ∩ { I = i }} (cid:1) > , which implies the lemma. (cid:3) Proof of Theorem 2.3. This proof is similar to the one of Theorem 2.1 Part (1) but simpler. Thenew proof ingredient that we use here is the strong connectivity of any supercritical Poisson–Booleanmodel [PP96, Theorems 2 and 5] in case d ≥ , which allows us to improve the result that λ ∗ c < ∞ to λ ∗ c = λ c ( r B ) . First we introduce an adequate discrete percolation model and then we control theinterferences.Throughout the proof X λ = { X i } i ∈ I denotes a homogeneous PPP with intensity λ in R d , and wewrite X λ instead of X λ since marks are non-random. Let us introduce the notion and elementary INR PERCOLATION WITH RANDOM POWERS 23 properties of Boolean models with (constant) radius r > . The Poisson–Boolean model B ( X λ , r ) (with constant connection radii r ) is defined as B ( X λ , r ) = [ i ∈ I B r ( X i ) = X λ ⊕ B r ( o ) . Connecting any two different points X i , X j ∈ X λ by an edge whenever | X i − X j | < r, (5.20)we obtain the Poisson–Gilbert graph g r ( X λ ) with connection radius r . Percolation in this Gilbertgraph is equivalent to the existence of an unbounded connected component in B ( X λ , r ) , which wealso refer to as percolation. This way, one can speak about subcritical, critical and supercriticalPoisson–Boolean models.Recall the definition of the radius r B from (2.4) and let us fix λ > λ c ( r B ) for the remainder ofthis section. Thanks to scale invariance of Poisson–Boolean models [MR96, Section 2.2] and the well-behavedness of ℓ , we can fix r ∈ ( d o , r B ) such that the Poisson–Boolean model B ( X λ , r/ associatedto g r ( X λ ) is still supercritical. The next lemma is an immediate consequence of the results in [PP96,Section 1]. Lemma 5.11 ([PP96]) . Let B ( X λ , r/ be a supercritical Poisson–Boolean model and let x ∈ R d .With probability tending to one as n ↑ ∞ , we have that(1) B ( X λ , r/ ∩ Q n ( x ) contains a connected component of diameter at least n/ ,(2) any two connected components of B ( X λ , r/ ∩ Q n ( x ) of diameter at least n/ each are containedin the same connected component of B ( X λ , r/ ∩ Q n ( x ) . Using Lemma 5.11, we construct a renormalized percolation process on Z d . For z ∈ Z d , let Ξ n ( z ) denote the union of all connected components of B ( X λ , r/ ∩ Q n ( z ) that are of diameter at least n/ .For n ≥ , we say that the site z ∈ Z d is n -good if(1) Ξ n ( nz ) = ∅ , and(2) for any z ′ ∈ Z d with | z − z ′ | ∞ ≤ , it holds that all pairs of connected components C of Ξ n ( nz ) and C ′ of Ξ n ( nz ′ ) are contained in the same connected component of B ( X λ , r/ ∩ Q n ( nz ) .The site z ∈ Z d is n -bad if z is not n -good. We have the following lemma. Lemma 5.12. Under the assumptions of Theorem 2.3, for all n ≥ sufficiently large, there exists q A = q A ( λ, n ) ∈ (0 , such that for any N ∈ N and pairwise distinct z , . . . , z N ∈ Z d we have P ( z , . . . , z N are all n -bad ) ≤ q NA . Further, for any ε > , for all large enough n one can choose q A such that q A < ε .Proof. For z ∈ Z d , { z is n -good } is measurable with respect to X λ ∩ ( Q n ( nz ) ⊕ B r/ ( o )) , which iscontained in X λ ∩ Q n ( nz ) for all n large enough, hence for all sufficiently large n the process of n -goodsites is 7-dependent thanks to the independence property of the PPP X λ . Hence, using a standardargument (using dependent percolation theory [LSS97], like in the proof of Lemma 5.4), it suffices toverify that lim sup n ↑∞ P ( o is n -bad ) = 0 . (5.21)The limit (5.21) can be verified along the lines of the proof of [HJC19, Theorem 2.6] using an adequateinterpretation of the Poisson–Boolean model. More precisely, in view of Definition 3.2, the assertion ofLemma 5.11 is equivalent to the statement [HJC19, Section 2.1] that the (for all sufficiently large b > ) b -dependent directing random measure Λ given as Λ(d x ) = λ { x ∈ B ( X λ , r/ } d x is asymptoticallyessentially connected, where λ > is such that E [Λ( Q )] = 1 . (cid:3) The other essential proof ingredient is the interference control. We recall the “shifted” path-lossfunctions ℓ a (5.5) and the shot-noise processes I a ( x ) , I ( x ) from Section 5.1, and also that by thetriangle inequality, for a ≥ , I ( x ) ≤ I a ( z ) holds for any z ∈ R d and x ∈ Q a ( z ) .For n ≥ and M > , we say that z ∈ Z d is ( n, M ) -tame if I n ( nz ) ≤ M and ( n, M ) -wild otherwise.Then we have the following assertion, which holds for all λ such that B ( X λ , r/ is supercritical. Lemma 5.13. [Tób20] Under the assumptions of Theorem 2.3, for fixed n ≥ , for all sufficientlylarge M > , there exists q B = q B ( λ, n, M ) ∈ (0 , such that for any N ∈ N and pairwise distinct z , . . . , z N ∈ Z d we have P ( z , . . . , z N are all ( n, M ) -wild ) ≤ q NB . Further, for ε > , for any n ≥ , for all sufficiently large M one can choose q B such that q B < ε .Proof. Clearly, the Lebesgue measure Λ is asymptotically essentially connected, b -dependent for any b > , and Λ( Q ) has all exponential moments. Hence the lemma can be proven very similarly to[Tób20, Proposition 3.3] under the condition (2b) in [Tób20, Theorem 2.4]. The only difference is thatin [Tób20], I n ( nz ) was considered instead of I n ( nz ) , but this makes no qualitative difference for theproof. Further, the additional condition in [Tób20, Theorem 2.4] that ℓ (0) ≤ can be assumed to holdwithout loss of generality, for the same reason as in the proof of Proposition 5.2 (see at the beginningof Step 4 in the proof of Theorem 2.1). (cid:3) Equipped with these results, we can now prove our main theorem. Proof of Theorem 2.3. For n ≥ and M > , we say that the site z ∈ Z d is ( n, M ) -nice if it isboth n -good and ( n, M ) -tame. We claim that for all sufficiently large n and accordingly chosen largeenough M , the process of ( n, M ) -nice sites percolates. Indeed, this follows by combining the estimatesof Lemmas 5.11 and 5.13 similarly to Corollary 5.3 and carrying out a Peierls argument.We claim that this assertion implies percolation in G γ ( X λ ) for small γ > . Indeed, let n, M be solarge that the process of ( n, M ) -nice sites percolates, and such that Q n ( o ) ⊕ B r/ ( o ) ⊆ Q n ( o ) . Usinga standard argument (cf. [DFM + 06] or Step 3 in the proof of Theorem 2.1 Part (1)), one can choose γ > sufficiently small such that for any ( n, M ) -tame site z , all connections in g r ( X λ ) ∩ Q n ( nz ) alsoexist in G γ ( X λ ) ∩ Q n ( nz ) .Now, analogously to [HJC19, Section 5.2], we can argue as follows. Let C be an infinite connectedcomponent of the process of sites that are ( n, M ) -nice. Let z, z ′ ∈ C and { z = z, z , . . . , z k − , z k = z ′ } a path in C connecting z and z ′ . Then, thanks to n -goodness, for any j = 0 , . . . , k and for any X j ∈ X λ such that B r/ ( X j ) ∩ Q n ( nz j ) ⊆ Ξ n ( nz j ) we have that X j and X j +1 are in the sameconnected component of B ( X λ , r/ ∩ Q n ( nz j ) . In other words X j and X j +1 are connected in thePoisson–Gilbert graph g r ( X λ ) via a path in Q n ( nz j ) , where the additional unit of n comes from thefact that centers of balls in the Boolean model might lie in a neighboring box. 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