aa r X i v : . [ m a t h . P R ] S e p Gibbs Random Graphs
Pablo A. Ferrari, Eugene A. Pechersky, Valentin V. Sisko and Anatoly A. Yambartsev October 16, 2018 Dobrushin laboratory of Institute for Information Transmission Problems of RussianAcademy of Sciences,19, Bolshoj Karetny, Moscow, Russia.E-mail: [email protected] Federal University of Fluminense, Institute of Matematics, Rua Mario Santos Braga, s/n,24020-140 Niter´oi, RJ, Brazil.E-mail: [email protected]ff.br Department of Statistics, Institute of Mathematics and Statistics, University of S˜ao Paulo,Rua do Mat˜ao 1010, CEP 05508–090, S˜ao Paulo SP, Brazil.E-mail: [email protected]
Abstract
Consider a discrete locally finite subset Γ of R d and the completegraph (Γ , E ), with vertices Γ and edges E . We consider Gibbs mea-sures on the set of sub-graphs with vertices Γ and edges E ′ ⊂ E . TheGibbs interaction acts between open edges having a vertex in com-mon. We study percolation properties of the Gibbs distribution of thegraph ensemble. The main results concern percolation properties ofthe open edges in two cases: (a) when the Γ is a sample from homo-geneous Poisson process and (b) for a fixed Γ with exponential decayof connectivity. Keywords: random graphs, Gibbs fields, percolation.AMS 2000 Subject Classifications: 60K35, 82C22, 05C80. Introduction
Let a sample Γ ⊂ R d of a point process be a locally finite set of R d . Weconsider an ensemble of graphs whose vertices are the points of Γ and whose edges are the set of unordered pairs of points in Γ. Each edge can be openor close; we study probability distributions on the set of configurations ofopen edges. The classical example is the Erd¨os-R´enyi’s random graph whereeach edge is open independently of the others with some probability. Manyworks have studied this model and correlates (see for example [1], [2]). Inthis paper we introduce interactions between edges and/or vertices and studythe associated Gibbs measures.Given a configuration of open edges, we say that two edges collide if bothof them are open and they have a vertex in common. We call monomers those vertices that are extreme of no open edge. A positive energy is paidby each collision and by each monomer. Furthermore to any open edge it isassigned a positive energy proportional to its length. This energy function isdescribed explicitly in (2.1) below. Roughly speaking, measures associated tothis energy function give more weight to configurations with few monomers,no collisions and short edges.In Theorem 1 we prove the existence of an infinite volume Gibbs measureassociated to the energy function. Call dimer an open edge not colliding withany other edge. In Theorem 2 we show that the ground states for a typicalconfiguration Γ is composed only by monomers and dimers. Theorem 3 givesconditions for the uniqueness of the ground state. Then we consider theproblem of percolation for two distributions of the point process. In the firstcase we consider that Γ is a sample of Poisson process and prove that if thedensity is small, then there is no percolation (Theorem 4). Second we ask Γto satisfy a ε -hard core condition (that is, the ball of radius ε around eachpoint of Γ has no other point of Γ), where ǫ is an arbitrary small positivenumber and prove that there is no percolation with probability 1 in this case(Theorem 5).To prove Theorem 5 we construct a process called cluster branching pro- ess and use a coupling of the paths of the process and the configurations ofthe Gibbs random graph.The open question is the non-percolation for Poisson vertices set havingarbitrary large rate λ . Of course, the large density of Poisson vertices mustbe compensated by a large temperature. Let Γ be sample of a point process and consider the complete graph G =(Γ , E ), where Γ is the set of vertices and E is the set of all unordered pairs[ γ , γ ] ⊂ Γ. The set E can be represented as E = S γ ∈ Γ E γ where E γ is theset of edges e ∈ E γ such that γ ∈ e . The length of edge e = [ γ , γ ] is definedby L ( e ) = | γ − γ | .Let Ω = { , } E be the set of configurations ω : E → { , } . The edge e ∈ E is called open with respect to ω if ω ( e ) = 1, and the edge is closed if ω ( e ) = 0. Often we shall use the term open/close with no mentions theconfiguration ω .We define the graph G ( ω ) = (Γ , E ( ω )) as the subgraph of G whose edgesare E ( ω ), the open edges of ω . The degree d ω ( γ ) of a vertex γ ∈ Γ with therespect of ω ∈ Ω is d ω ( γ ) = { e ∈ E γ : ω ( e ) = 1 } , the number of open edgescontaining γ . We shall use both the configuration ω and the graph G ( ω ) asthe synonyms.Our goal is to define a Gibbs distribution on the ensembleΩ = { ω : ω ∈ Ω , d ω ( γ ) < ∞ for all γ } the set of graphs whose vertices have finite degree. Introduce the followingformal Hamiltonian H ( ω ) = X e : ω ( e )=1 L ( e ) + X γ ∈ Γ φ ω ( γ ) , (2.1)3here φ ω ( γ ) is a “penalty” energy function defined by φ ω ( γ ) = h , if d ω ( γ ) = 0;0 , if d ω ( γ ) = 1; h (cid:0) d ω ( γ )2 (cid:1) if d ω ( γ ) > . (2.2)where h and h are fixed positive parameters. Notice that φ ω ( γ ) dependsonly on the degree d ω ( γ ). It defines the energy of a pair interaction betweenopen edges from E γ .An aspect which is not standard for Gibbs field constructions is thatthe potential function φ ω depends on infinite number of ‘sites’. The edgesplay here the role of sites in a lattice. To know that φ ω ( γ ) = 0 we have tocheck that ω ( e ) = 0 for infinite many e ∈ E γ . However the generalization ofthe usual Gibbs construction for this case is rather easy and does not causespecial considerations. Therefore further we do not concern this peculiarity.The description of the infinite volume Gibbs measure with Hamiltonian H requires the definition of finite volume Gibbs measures. Taking a finitevolume Λ ⊂ R d the set of points Γ Λ = Γ ∩ Λ is finite. The complete finitegraph G Λ = (Γ Λ , E Λ ) has edges E Λ connecting all pairs of points of Γ Λ . LetΩ Λ be the set of configurations ω : E Λ → { , } . The Gibbs state P Λ on Ω Λ with the ‘free’ boundary condition is defined by P Λ ( ω ) = exp {− βH Λ ( ω ) } Z Λ , (2.3)for ω ∈ Ω Λ , where the parameter β is the inverse temperature, Z Λ is thenormalizing constant and H Λ ( ω ) = X e ∈ E Λ ( ω ) L ( e ) + X γ ∈ Γ Λ φ ω ( γ ) , (2.4)where the set E Λ ( ω ) ⊆ E Λ is the set of all open edges in Λ.Since Ω is compact, there exists a Gibbs distribution on Ω which mayhave infinite-degree vertices. We show later finiteness of the degrees withprobability 1. This implies that the Hamiltonian 2.4 generates an infinitevolume Gibbs field concentrated on Ω.4 Main results
A point set Γ is weakly homogeneous if for any γ ∈ Γ and any β > T γ ( β ) = X e ∈ E γ e − βL ( e ) < ∞ , (3.1)where E γ is the set of all possible edges with common extreme γ . A pointset Γ is strongly homogeneous ifsup γ ∈ Γ T γ ( β ) < ∞ . (3.2)Since the possible unbounded contribution to the sum (3.1) comes fromaccumulation of vertices that are close to γ , if Γ consists of hard core ballcenters of a fixed radius, then Γ is strongly homogeneous. We show laterin Lemma 4 that for a Poisson process with law π λ , almost all Γ is weaklyhomogeneous but not strongly homogeneous. Theorem 1.
For any weakly homogeneous Γ and any < h < h the Gibbsrandom graph distributions associated to the Hamiltonian H defined in (2.1) are concentrated on Ω . Notice that the theorem concerns all possible Gibbs measures P associatedto the Hamiltonian H ; the problem of uniqueness is not discussed in thisarticle. A configuration b ω is a local perturbation of ω ∈ Ω if b ω = ω and there existsa finite volume Λ ⊂ R d such that b ω coincides with ω for edges not includedin Λ: b ω ( e ) = ω ( e ) for all e Λ. A configuration ω ∈ Ω is a ground state iffor any local perturbation b ω of ωH ( b ω ) − H ( ω ) ≥ . Theorem 2.
For any finite-local Γ and any < h < h there exists at leastone ground state. Furthermore if ω is a ground state of the Gibbs randomgraph distribution, then d ω ( γ ) ≤ for every γ ∈ Γ . Moreover, the length of edges in a ground state are less than h . Theorem 3.
Let π λ be the distribution of a homogeneous Poisson processwith rate λ > . There exists λ g such that if λ < λ g , then for π λ -almost all Γ the ground state of the Gibbs random graph with vertices Γ is unique. Let ω ∈ Ω. The set E ( ω ) is split into a set of maximal connected componentswhich we call clusters . For a locally finite Γ, we say that the associated toΓ Gibbs random graph measure P on Ω percolates if there exists an infinitecluster in E ( ω ) with P probability 1. Otherwise we say that the Gibbsrandom graph measure P on Ω does not percolate.In the next theorem we establish non percolation of P when Γ is a Poissonprocess with small intensity λ . Theorem 4.
Let π λ be the distribution of a Poisson process with rate λ > and Γ chosen with π λ . Then in the region F = (cid:26) ( λ, T ) : λ ≤ h + J ( T ) (cid:27) , (3.3) where J ( T ) = Z ∞ h e − x/T e − x/T + e − h /T d x, the Gibbs measure P associated to Γ does not percolate, π λ -almost surely. Theorem 5.
Let Γ be strongly homogeneous. Then there exists a criticaltemperature T c (Γ) such that for all T < T c (Γ) the Gibbs measure P associatedto Γ does not percolate. Let γ be a point of Γ, and Σ γ be a set of all configurations defined on E γ and having a finite degree at γ . That is, any σ ∈ Σ γ is the restriction of aconfiguration ω ∈ Ω to E γ , σ = ω E γ ; in this case we call σ a star centered at γ , or simply a star. Clearly σ ⊂ ω . Let d σ := d ω ( γ ).Let Λ ⊂ R d be a finite volume and γ ∈ Λ; let E Λ γ be the set of edgescontained in Λ having γ as its end, and Σ Λ γ be the set of the configurationson E Λ γ which are restrictions of the configurations from Ω Λ . Lemma 1.
Let the point γ ∈ Λ , consider σ ∈ Σ Λ γ and Ω Λ ( σ ) = { ω ∈ Ω Λ : σ ⊂ ω } , and assume that the number of open edges | σ | in σ is greater orequal than 2 then P Λ (Ω Λ ( σ )) ≤ e − β P e : σ ( e )=1 L ( e ) − βh ( dσ ) e βh ( d σ +1) (4.1) Proof.
Let V ( σ ) be the set of all vertices belonging to open edges of thestar-configuration σ V ( σ ) = { γ ′ : [ γ, γ ′ ] ∈ E Λ γ ( σ ) } . To any configuration ω ∈ Ω Λ ( σ ) we associate a configuration ˜ ω without thestar σ ; in ˜ ω the point γ is isolated:˜ ω ( e ) = ( ω ( e ) if e / ∈ σ, , if e ∈ σ. (4.2)7he transformation ω ˜ ω (taking out the star σ from ω ) changes thepenalty weights only at the vertices in { γ } ∪ V ( σ ). Namely, for any point γ ′ ,the penalty weight in configuration ˜ ω is φ ˜ ω ( γ ′ ) = φ ω ( γ ′ ) if γ ′ / ∈ V ( σ ) ∪ { γ } ,h (cid:0) d ω ( γ ′ ) − (cid:1) + h δ ( d ω ( γ ′ ) −
1) if γ ′ ∈ V ( σ ) ,h if γ ′ = γ, (4.3)where δ ( · ) is Kronecker symbol. Consider the possible changes of the energycaused by the star σ removal from configuration ω. The difference betweenenergies is H (˜ ω ) − H ( ω ) = − X e ∈ σ L ( e ) + X γ ′ ∈ V ( σ ) ∪{ γ } ( φ ˜ ω ( γ ′ ) − φ ω ( γ ′ )) (4.4)and φ ˜ ω ( γ ′ ) − φ ω ( γ ′ ) = (4.5) h − h (cid:0) d σ (cid:1) if γ ′ = γ, − h ( d ω ( γ ′ ) −
1) if γ ′ ∈ V ( σ ) and d ω ( γ ′ ) ≥ ,h if γ ′ ∈ V ( σ ) and d ω ( γ ′ ) = 1 . We used here the fact that for any γ ′ ∈ V ( σ ) its degree in configurations ω and ˜ ω satisfy equality d ˜ ω ( γ ′ ) = d ω ( γ ′ ) −
1, and we used that (cid:0) k (cid:1) − (cid:0) k − (cid:1) = k − k ≥ . Let us denote∆ φ ω ( γ ′ ) := φ ˜ ω ( γ ′ ) − φ ω ( γ ′ ) . σ is P Λ (Ω Λ ( σ )) = X ω ∈ Ω Λ ( σ ) P Λ ( ω ) = 1 Z Λ X ω ∈ Ω Λ ( σ ) e − βH ( ω ) = 1 Z Λ X ω ∈ Ω Λ ( σ ) e − β ( H ( ω ) − H (˜ ω )) e − βH (˜ ω ) = 1 Z Λ X ω ∈ Ω Λ ( σ ) e − β P e ∈ σ L ( e )+ β P γ ′∈ V ( σ ) ∪{ γ } ∆ φ ω ( γ ′ ) e − βH (˜ ω ) (4.6)Here and further instead of the sum P e : σ ( e )=1 we write simply P e ∈ σ . In the last expression of (4.6) the factor e − βH (˜ ω ) does not depend on σ and the factor e − β P e ∈ σ L ( e ) does not depend on ω . However the factor e β P γ ′∈ V ( σ ) ∪{ γ } ∆ φ ω ( γ ′ ) depends on both σ and ω . To find an upper bounddepending only on σ we represent the energy of the difference as X γ ′ ∈ V ( σ ) ∪{ γ } ∆ φ ω ( γ ′ ) = h − h (cid:18) d σ (cid:19) + X γ ′ ∈ V ( σ ) d ω ( γ ′ )=1 h − X γ ′ ∈ V ( σ ) d ω ( γ ′ ) > h ( d ω ( γ ′ ) − ≤ h + h d σ − h (cid:18) d σ (cid:19) . The above inequality we obtain if we assume that all vertices in V ( σ ) haveits degrees equal to 1.Thus we obtain the following estimate for P Λ (Ω Λ ( σ )). Let Ω( γ ) denotethe set of configurations where γ is isolate point, then it follows from (4.6)that P Λ (Ω Λ ( σ )) ≤ e − β P e ∈ σ L ( e )+ β ( h − h ( dσ ) )+ βh d σ Z Λ X ˜ ω ∈ Ω( γ ) e − βH (˜ ω ) . (4.7)Noting that 1 Z Λ X ˜ ω ∈ Ω( γ ) e − βH (˜ ω ) = P Λ (Ω( γ )) <
19e obtain the estimation (4.1) of the lemma.
Remark . The estimate (4.1) does not depend on Λ when V ( σ ) ⊂ Λ.We can generalize the lemma for the case when there is an “environment”.Let B and F be some nonempty sets of edges of E Λ γ without intersection B ∩ F = ∅ . Let B be the set of open edges and F be the set of closed edges.Introduce a configuration of the “environment” µ on B ∪ Fµ ( e ) = ( , if e ∈ B , if e ∈ F Consider the following sets of edges G Λ γ, B ∪ F = E Λ γ \ ( B ∪ F ) . And let Σ Λ γ,B ∪ F be the set of all configurations on G Λ γ, B ∪ F . If σ ∈ Σ Λ γ,B ∪ F then the degree ofthe point γ is equal to the number of open edges on σ (we denote it by d σ )plus the number of the open edges | B | in the “environment” µ (denote it by d µ ). As before denote Ω Λ ( σ ) and Ω Λ ( µ ) the sets of all configurations whichinclude the star-configuration σ and the configuration µ correspondingly. Lemma 2.
Let σ ∈ Σ Λ γ ,B ∪ F then P Λ (cid:0) Ω Λ ( σ ) (cid:12)(cid:12) Ω Λ ( µ ) (cid:1) ≤ e − β P e : σ ( e )=1 L ( e ) − βh ( dσ +12 ) e βh d σ . (4.8)The proof of Lemma 2 is similar to the proof of Lemma 1. The differ-ence in the right hand side between (4.8) and (4.1) can be explained in thefollowing way. The set B is nonempty, thus the number of interacted pairsis at least (cid:0) d σ (cid:1) + d σ = (cid:0) d σ +12 (cid:1) . That provides the energy h (cid:0) d σ +12 (cid:1) . Removingthe star σ we can obtain at most d σ isolated points, but not d σ + 1 as in the(4.1), because now the point γ cannot become isolated point. Let Γ be weakly homogeneous. 10 emma 3.
The following inequality E Λ d σ ( γ ) ≤ ∞ X k =0 e − βh ( k ) + βh ( k +1) ( T γ ( β )) k ( k − holds for the mean value of the vertex degrees. Where E Λ is the expectationwith respect to the probability P Λ and T γ ( β ) is defined in (3.1).Proof. The assertion of the lemma follows from the inequalities E Λ d σ ( γ ) = X σ d σ ( γ ) P (Ω Λ ( σ )) = ∞ X k =0 k X σ : | E γ ( σ ) | = k P (Ω Λ ( σ )) ≤ ∞ X k =0 ke − βh ( k ) + βh ( k +1) X σ : | E γ ( σ ) | = k e − β P e ∈ Eγ ( σ ) L ( e ) ≤ ∞ X k =0 ke − βh ( k ) + βh ( k +1) ( T γ ( β )) k k ! . Remark . We note that, when Λ contains the star, then the estimationdoes not depend on Λ . Thus it gives an uniform over Λ upper estimation,which holds when Λ ր R d The theorem 1 follows now from the finiteness of E Λ d σ ( γ ) (see (4.9) andRemark 4.2)Any sample of Poisson process is weakly homogeneous. It shows the next Lemma 4.
Almost all samples Γ from Poisson distribution π λ are weaklyhomogeneous.Proof. Let γ ∈ Γ. Consider a sequence of rectangles U n = [ − l n , l n ] d , n ≥ γ with size length l n . We chose l n = ( n + 1) /d . Then any ring11 n = U n − U n − , n ≥
0, except U − = ∅ , has its volume equal to 1. If γ ∈ W n then for e = h γ, γ ′ i the inequality L ( e ) ≥ l n − holds. Let ξ n be anumber of points from Γ located in W n . The variables ξ n are independentrandom variables having Poisson distribution with the parameter λ (sincethe volume off W n is equal to 1). Therefore the following series converges T γ = X e ∈ E γ e − L ( e ) ≤ ∞ X n =0 ξ n e − l n − = ∞ X n =0 ξ n e − n d < ∞ a.s.The convergence with probability 1 follows from the convergence of the se-ries of the expectations and the variances of the random variables ξ n e − n d (Theorem of ”two series”, [5]). Proof of Theorem ω be a ground state and therebe a vertex γ ∈ Γ ∩ Λ such that d ω ( γ ) ≥
2. Let e = [ γ , γ ] be the incidentto the vertex γ in graph ω , ω ( e ) = 1. Let ˜ ω be the new configuration suchthat ˜ ω is the same as ω with the exception that the edge e is now removed:˜ ω ( e ) = 0. Then we have H Λ ( ω ) − H Λ (˜ ω ) = ( L ( e ) + ( d ˜ ω ( γ ) + d ˜ ω ( γ )) h if d ˜ ω ( γ ) ≥ ,L ( e ) + d ˜ ω ( γ ) h − h if d ˜ ω ( γ ) = 0 . Since 0 < h < h , we have H Λ ( ω ) − H Λ (˜ ω ) > . There is no edges in a ground with its length L greater than 2 h , sincethe energy of two monomers is 2 h < L. V n ) of increasing cubes covering R d = S n V n . We build a ground state ofthe model by a sequence of reconstructions of an initial configuration. Itis reasonable to take the initial configuration satisfying the property provedabove. For example, we can take the configuration ω with no edges, that isthe configuration of all monomers. Let ω n be a configuration in V n havingthe minimal energy over all configurations in V n . There exists a sequence ( ω ′ i )of configurations which is a subsequence of ( ω n ), that is ω ′ i = ω n i , such thatthere exists a limit lim i →∞ ω ′ i ( e ) for every e ∈ E . Moreover, the sequence( ω ′ i ) can be chosen such that ω ′ j ( e ) ≡ const for all j ≥ i when e ∈ E V i . Theconfiguration ω ′ = S i ω ′ i is one of the ground states. Indeed, let b ω be a localperturbation of ω ′ . There exists V i such that { e : b ω ( e ) = ω ′ ( e ) } ⊆ E V i . Forany i > i let b ω i be the configuration equal to the restriction of b ω on E V i .The configuration b ω i is the perturbation of ω ′ i therefore H V i ( b ω i ) − H V i ( ω ′ i ) ≥ . Moreover, the fact, that for any i there is no edges with length greater that2 h , means that there exists i ≥ i such that H ( b ω ) − H ( ω ′ ) = H V i ( b ω i ) − H V i ( ω ′ i ) ≥ . That proves that the any local perturbation of ω ′ increase the energy. Thus ω ′ is really the ground state. Proof of Theorem h . Another observation is that there exists a critical intensity λ c suchthat there is no boolean percolation with radius h for all λ < λ c (see [3],Theorem 3.3).Thus, for λ < λ c any process configuration Γ is an union of finite clustersΓ = ∪ ∞ i =1 Γ i , | Γ i | < ∞ , and for any i = j and for any γ ∈ Γ i , γ ′ ∈ Γ j thedistance | γ − γ ′ | > h . There are open edges only inside of the clusters Γ i .Since Γ i are finite there exists a unique configuration of open edges in everyΓ i minimizing the energy. 13 .4 Proof of Theorem 4 and 5 on non-percolation Proof of Theorem ν on Ω which does notpercolate and stochastically dominates the Gibbs measure P . We can applythis method for small rates λ of Poisson measure π λ and low temperature ofthe distribution of Gibbs random graph.On the set Ω of the configurations we define the following Bernoulli mea-sure ν ν ( ω ( e ) = 1) = e − βL ( e ) e − βL ( e ) + e − βh (4.10)independently for any e ∈ E . This measure forms the random - connectedmodel (see [3], ch. 6), which is driven by Poisson process with the rate λ andconnected function g ( x ) = e − βx e − βx + e − βh , (4.11)the probability of two points to be connected on the distance x . The proofof Theorem 4 is a direct application of Holley’s inequality (see [4], Theorem4.8). It is shown in the next two lemmas. Lemma 5.
The following inequality P ( ω ( e ) = 1 | ω e ) ≤ ν ( ω ( e ) = 1) (4.12) holds for any ω e , where P ( ω ( e ) = 1 | ω e ) is the Gibbs conditional probabilityof ( ω ( e ) = 1) given a configuration ω e out of the edge e .Proof of Lemma 5. Let e = [ γ , γ ]. Then the conditional probability in(4.12) depends on a configuration on ( E γ ∪ E γ ) \ { e } : P ( ω ( e ) = 1 | ω e ) = P ( ω ( e ) = 1 | ω γ ∪ ω γ ) , where ω γ and ω γ are configurations on E γ \ { e } and E γ \ { e } respectively,and ∪ means the conjugation of the configurations.Consider three cases: 14. ω γ = ω γ ≡ ω γ , ω γ ≡ . This case has the symmetrical version ω γ ≡ , ω γ ω γ , ω γ P ( ω ( e ) = 1 | ω γ ∪ ω γ ≡
0) = e − βL ( e ) e − βL ( e ) + e − βh = g ( L ( e )) = ν ( ω ( e ) = 1)which means that for the case 1 Holly’s inequality holds.Case 2. Let E γ ( ω γ ) ( E γ ( ω γ ) ) be the set of the open edges of con-figurations ω γ ( ω γ ) . Let m := | E γ ( ω γ ) | . Since the edge e is open then itinteracts with m open edges from E γ ( ω γ ) . Then P ( ω ( e ) = 1 | ω γ ∪ ω γ ) = e − βL ( e ) − βmh e − βL ( e ) − βmh + e − βh < e − βL ( e ) e − βL ( e ) + e − βh < e − βL ( e ) e − βL ( e ) + e − βh = ν ( ω ( e ) = 1) . Case 3. P ( ω ( e ) = 1 | ω γ ∪ ω γ ) = e − βL ( e ) − βmh e − βL ( e ) − βmh + 1 < e − βL ( e ) e − βL ( e ) + 1 < e − βL ( e ) e − βL ( e ) + e − βh = ν ( ω ( e ) = 1)where m = | E γ ( ω γ ) | + | E γ ( ω γ ) | . In the next lemma we find the condition for the non-percolation of therandom - connected model, which dominate the Gibbs distribution.
Lemma 6.
In the region (3.3) there is no percolation in the random - con-nected model with Poisson rate λ and the connection function (4.11). roof. The assertion of the lemma is a consequence of the Theorem 6.1 of [3],which claims that a random - connected model with the connection function(4.11) does not percolate if λ Z ∞ g ( x )d x < . (4.13)Note that for any β > h > g ( x ) in (4.13) is finite.We represent the integral in (4.13) as Z g ( x )d x = Z h e − βx e − βx + e − βh d x + Z ∞ h e − βx e − βx + e − βh d x =: J ( T ) + J ( T ) , where T = 1 /β. The first integral J ( T ) on the right side of the above equalityis increasing and tends to 2 h as β → ∞ . The second integral tends to 0 as β → ∞ . Choosing λ such that λ < h + J ( T ) ≤ R g ( x ) dx we obtain the claim of the lemma. Proof of Theorem
The Cluster Branching Process
The proof of Theorem 5 is based on the construction of a non-homogeneous cluster branching process of the edges.An informal description of the cluster branching process is the following.Let B be some connected set of open edges which forms a cluster and let V
16e the set of vertices in the cluster. Consider the pair (
V, B ) as a connectedgraph. The graph distance ρ between two vertices is the number of edges ina shortest path connecting them. Fix a vertex γ in V . For any n ∈ N asphere with radius n and center γ is V ( n ) = { γ ∈ V : ρ ( γ, γ ) = n } , where V (0) = { γ } . The sequence { V ( i ) : i = 1 , , . . . } is a partition of V = ∪ ∞ i =0 V ( i ) . Then B = ∪ ∞ i =1 B ( i ) , where B ( n ) = { e = [ w, v ] ∈ B : w ∈ V ( n − and v ∈ V ( n − ∪ V ( n ) } . We interpret the set B ( n ) as n -th offspring generation of the ancestor set V ( n − . The set B ( n ) is a set of ’plant branches’ growing from a set of ’buds’ V ( n − . We think B ( n ) as the state of a branching cluster process at “time” n . This construction leads to an ambiguity since the edge [ w, v ] ∈ B ( n ) canbe the offspring of two ancestors v and w if v, w ∈ V ( n − . This problemcan be solved by introducing an order along which the embranchment iscontrolled. The order of the branching induces a dependence of the offsprings.Another peculiarity of the branching cluster process is interactions of theoffsprings having different ancestors. These properties differ the branchingcluster process from the standard branching processes.The formal definition of the branching cluster process can be made in thefollowing way. Construction of Cluster Branching Process . Recall that E γ is the setof all edges incident with the point γ ∈ Γ. As before we denote Σ γ = { , } E γ and Σ γ,D = { , } E γ \ D the set of all configurations on E γ and E γ \ D correspondingly, where D is some set of the edges.The path of the cluster branching process is a sequence of triples ( B ( n ) , V ( n ) , S n ).The distribution of the cluster branching process is denoted by P . The precisedefinition is the following. Let γ ∈ Γ be the starting point of a branchingprocess path. 17 nitial stage. B (0) := ∅ , V (0) := { γ } and S := ∅ . First stage.
Let us choose some set of edges B (1) ⊆ E γ which are theoffsprings of γ . With help of B (1) we construct the next objects V (1) := { γ : [ γ, γ ] ∈ B (1) } , S := E γ .In order to define the offspring probability P ( B (1) ) of the ancestor γ we introduce the star configuration σ γ ( e ) = ( , if e ∈ B (1) , , if e ∈ S \ B (1) and µ ( e ) = σ γ ( e ) . The path of the one step embranchment is B = B (1) .Then P ( B (1) ) := P (Ω( σ γ )) , where Ω( σ γ ) is the set of all configurations of Ω such that its projec-tion on E γ coincide with the star-configuration σ γ . It follows fromTheorem 1 that the number of the offsprings from one point is finite.
Second stage.
Having B (1) , V (1) , S we construct the next generation.Namely, we shall define the objects B (2) , V (2) , S . We shall do it suc-cessively according to an order in V (1) . The order is arbitrary. We needit to avoid the ambiguity in the definition of ancestors of an offspring e = [ w, v ] when w, v ∈ V (1) . Let k = | V (1) | . Suppose that the points in V (1) are enumerated in some way, V (1) = { γ (1)1 , ..., γ (1) k } . We constructsuccessively B (2) i , V (2) i , S ,i , i = 1 , . . . , k . Let us begin with the firstpoint γ (1)1 . Let B (2)1 be a subset of E γ (1)1 \ S which is a offspring set of γ (1)1 . Then 18 (2)1 := { γ : [ γ, γ (1)1 ] ∈ B (2)1 } ; S , := S ∪ E γ (1)1 .Since the set B (2)1 is from E γ (1)1 \ S the initial point γ can not belongto V (2)1 . However the points from V (1) may belong to V (2)1 .In order to define the offspring probability we introduce two configura-tions: σ γ (1)1 ( e ) = ( , if e ∈ B (2)1 , , if e ∈ E γ (1)1 \ B (2)1 and µ , ( e ) = ( , if e ∈ B , , , if e ∈ S , \ B , where the path B , = B (1) ∪ B (2)1 . We have described two steps of theprocess: branching from γ and from γ (1)1 .Further the upper index denotes the number of a stage and the lowerindex if it single denotes the number of a step in the stage. Doublelower indices contain both the step and the stage.The conditional probability of the offsprings B (2)1 of the ancestor γ (1)1 given the environment B is P ( B (2)1 | B ) := P (Ω( σ γ (1)1 ) | Ω( µ )) . (4.14)Assume we have constructed B (2) i , V (2) i , S ,i and also we have B ,i , µ ,i , i =1 , ..., m , where m < k . Doing the next branching of the point γ (1) m +1 choose some set B (2) m +1 from the set E γ (1) m +1 \ S ,m which means the off-spring set of γ (1) m +1 . Then V (2) m +1 = { γ : [ γ, γ (1) m +1 ] ∈ B (2) m +1 } ;19 ,m +1 := S ,m ∪ E γ (1) m +1 .Now we obtain the path B ,m +1 = B ,m ∪ B (2) m +1 . To define the offspring probability introduce the configuration σ γ (2) m +1 ( e ) = ( , if e ∈ B (2) m +1 , , if e ∈ E γ (1) m +1 \ S ,m +1 and configuration µ ,m +1 := µ ,m ∨ σ γ (1) m +1 . We use the sign ∨ to notate the concatenation of two configurationsdefined on non-intersected sets.The conditional probability of offsprings B (2) m +1 of the ancestor γ (1) m +1 given B ,m is P ( B (2) m +1 | B ,m ) := P (Ω( σ γ (1) m +1 ) | Ω( µ ,m )) (4.15)Having done the construction for i = 1 , . . . , k we obtain B (2) := ∪ k i =1 B (2) i ; V (2) := ∪ k i =1 V (2) i \ V (1) = { γ (2)1 , . . . , γ (2) k } ; B := B ,k and µ := µ ,k ; S := S ,k . Remark that the set ∪ k i =1 V (2) i can include points from V (1) . The pointsfrom the set (cid:16) ∪ k i =1 V (2) i (cid:17) ∩ V (1) can not have offsprings. Therefore theyare excluded from the next branching generation.( n + 1) th stage. Assume we have constructed B ( n ) , V ( n ) = { γ ( n )1 , ..., γ ( n ) k n } , S n and B n , µ n . B ( n +1) = ∪ k n i =1 B ( n +1) i is constructed in thesame way as in the second stage with objects V ( n +1) i , S n +1 ,i , B n +1 ,i , µ n +1 ,i and σ γ ( n +1) m . The offspring probabilities are defined in the same way P ( B ( n +1) m +1 | B n +1 ,m ) := P (Ω( σ γ ( n ) m +1 ) | Ω( µ n +1 ,m )) . (4.16)It completes the construction of the cluster branching process.We show next that the cluster branching processes posses the main featureof the usual branching processes, namely, if the expectation of the offspringnumber of one ancestor is less than 1 then the processes extinct. Lemma 7.
Assume that there exists ε > such that for n > either E ( | B ( n ) i | | B n,i − ) ≤ − ε when i > or E ( | B ( n ) i | | S n − ) ≤ − ε when i = 1 then E ( | B | ) < ∞ , where B = ∪ ∞ n =1 B ( n ) roof follows from the following equalities E (cid:2) | B ( n ) | (cid:3) = E (cid:2) ∞ X k =1 I {| B ( n − | = k } | B ( n ) | (cid:3) = ∞ X k =1 E h I {| B ( n − | = k } k X i =1 | B ( n ) i | i = ∞ X k =1 k X i =1 E h E h I {| B ( n − | = k } | B ( n ) i | (cid:12)(cid:12)(cid:12) B n,i − ii = ∞ X k =1 k X i =1 E h I {| B ( n − | = k } E h | B ( n ) i | (cid:12)(cid:12)(cid:12) B n,i − ii . We used the measurability of the event ( | B ( n − | = k ) with respect to the σ -algebra generated by B n,i − . We adopt the above that B n, = B n − .Next we obtain E (cid:2) | B ( n ) | (cid:3) ≤ (1 − ε ) ∞ X k =1 k E h I {| B ( n − | = k } i = (1 − ǫ ) E [ | B ( n − | ]Thus E (cid:2) | B | (cid:3) ≤ E [ | B (1) | ] /ǫ. The definition of the cluster branching process is done in a way such thatany maximal component of any configuration ω can be obtained as a clusterprocess path. It means the following.Let γ ∈ Γ, and ω be some configuration from Ω . Let C γ ( ω ) be themaximal connected component of open edges of ω containing γ . We con-struct a cluster branching process along ω where γ is the initial point of thecluster path. The only freedom in the cluster process path deriving is in the22hoice of the offsprings. Doing the coupling with chosen configuration ω wedefine σ γ ( n ) i as a projection of ω on the set of edges E γ ( n ) i \ S n,i − . Then theprobability to have a finite connected component C γ ( ω ) can be obtained asthe probabilities (4.15) of the branching cluster process path made along ω .As a consequence the following equality holds: for any point γ ∈ Γ P ( C γ is finite) = P ( B ( n ) not survives) (4.17)The following lemma finish the prove of the theorem. Lemma 8.
Let Γ be strongly homogeneous (see (3.2)). Then for any small ǫ > there exists β = β ( ǫ ) such that for all β > β E (cid:2) | B ( n ) i | (cid:12)(cid:12) B n,i − (cid:3) < − ǫ. (4.18) uniformly over i, n > . Here B n, = B n − .Proof. Let γ ( n − i be the branching point of which offsprings are B ( n ) i . Letthe previous path be B n,i − . It follows from (4.8) that: P (cid:16) | B ( n ) i | = m (cid:12)(cid:12)(cid:12) B n,i − (cid:17) ≤ e − β P e ∈ B ( n ) i L ( e ) − β ( m +12 ) e βh m . (4.19)Thus, E (cid:16) | B ( n ) i | (cid:12)(cid:12)(cid:12) B n,i − (cid:17) = ∞ X m =1 m X B ( n ) i : | B ( n ) i | = m P (cid:16) | B ( n ) i | = m (cid:12)(cid:12)(cid:12) B n,i − (cid:17) ≤ ∞ X m =1 m exp n − βh (cid:18) m + 12 (cid:19) + βh m o ×× X B ( n ) i : | B ( n ) i | = m exp n − β X e ∈ B ( n ) i L ( e ) o < ∞ X m =1 m exp n − βh (cid:18) m + 12 (cid:19) + βh m o ( T γ ( n +1) i ( β )) m m ! , (4.20)23here T γ ( β ) = X e ∈ E γ e − βL ( e ) . Since T γ ( β ) are uniformly bounded over γ ∈ Γ the choice of large enough β leads to (4.18).
1. Since the ground state of Gibbs Random Graph do not percolate thetheorems about the non-percolation show a kind of ”stability” of theground states.2. Condition of the existence of an infinite cluster is an open problem.
The work of E.P. was partly supported by CNPq grants 300576/92-7 and662177/96-7, (PRONEX) and FAPESP grant 99/11962-9, RFBR grants 07-01-92216 and 08-01-00105.The work of V.S. was partly supported by FAPESP grant 99/11962-9 andCNPq grant 306029/2003-0.The work of A.Ya. was partly supported by E26-170.008-2008 (PRONEX),”Edital Universal 2006” grant 471925/2006-3 and 306092/2007-7 (CNPq).
References [1] Bela Bollobas. Random graphs.
Cambridge studies in advanced mathe-matics.
Second Edition. Cambrige University Press, 2001.[2] Rick Durrett. Random graph Dynamics. Cambrige University Press, Oc-tober 2006. 243] Ronald Meester and Rahul Roy. Continuum Percolation.