Survival probabilities of high-dimensional stochastic SIS and SIR models with random edge weights
SSurvival probabilities of high-dimensionalstochastic SIS and SIR models with randomedge weights
Xiaofeng Xue ∗ Beijing Jiaotong University
Abstract:
In this paper, we are concerned with the stochastic SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered) models on high-dimensional latticeswith random edge weights, where a susceptible vertex is infected by an infectious neighborat rate proportional to the weight on the edge connecting them. All the edge weights areassumed to be i.i.d.. Our main result gives mean field limits for survival probabilities ofthe two models as the dimension grows to infinity, which extends the main conclusion givenin [13] for classic stochastic SIS model.
Keywords:
SIS model, SIR model, edge weight, survival probability, mean field limit.
In this paper, we are concerned with the stochastic SIS (susceptible-infected-susceptible)and SIR (susceptible-infected-recovered) models on high-dimensional lattices Z d . For lateruse, we introduce some notations. We use O to denote the origin of Z d . For each x ∈ Z d ,we denote by (cid:107) x (cid:107) the l norm of x , i.e., (cid:107) x (cid:107) = d (cid:88) i =1 | x i | for x = ( x , . . . , x d ). For 1 ≤ i ≤ d , we use e i to denote the i th basic unit-vector of Z d , i.e., e i = (0 , . . . , , i th , . . . , . For x, y ∈ Z d , we write x ∼ y when and only when (cid:107) x − y (cid:107) = 1. We use E d to denote (cid:110) { x, y } : x ∼ y (cid:111) , which is identified with the set of edges on Z d . For any set A , we denoteby | A | the cardinality of A .Let ρ be a random variable that P (0 ≤ ρ ≤ Θ) = 1 for some Θ ∈ (0 , + ∞ ) and P ( ρ > >
0, then we assume that { ρ ( e ) } e ∈ E d are i.i.d. copies of ρ . For e = { x, y } ∈ E d , we write ρ ( e ) as ρ ( x, y ). Note that ρ ( x, y ) = ρ ( y, x ).When { ρ ( e ) } e ∈ E d are given, the stochastic SIS model with edge weights { ρ ( e ) } e ∈ E d is acontinuous-time Markov process { C t } t ≥ with state space X = { C : C ⊆ Z d } ∗ E-mail : [email protected]
Address : School of Science, Beijing Jiaotong University, Beijing 100044,China. a r X i v : . [ m a t h . P R ] J un nd transition rate function given by C t → C t \ { x } at rate 1 if x ∈ C t ,C t ∪ { x } at rate λ d (cid:80) y : y ∼ x ρ ( x, y )1 { y ∈ C t } if x (cid:54)∈ C t , (1.1)where λ is a positive constant called the infection rate and 1 A is the indicator function ofthe event A .The stochastic SIR model with edge weights { ρ ( e ) } e ∈ E d is a continuous-time Markovprocess { ( S t , I t ) } t ≥ with state space X = { ( S, I ) : S ⊆ Z d , I ⊆ Z d , S ∩ I = ∅} and transition rate function given by( S t , I t ) → (1.2) ( S t , I t \ { x } ) at rate 1 if x ∈ I t , ( S t \ { x } , I t ∪ { x } ) at rate λ d (cid:80) y : y ∼ x ρ ( x, y )1 { y ∈ I t } if x ∈ S t . Both the SIS model and the SIR model describe the spread of epidemics on a graph.For the SIS model, each vertex is in one of two states, ‘susceptible’ or ‘infectious’. C t is theset of infectious vertices at moment t . An infectious vertex waits for an exponential timewith rate one to become susceptible while a susceptible vertex is infected by an infectiousneighbor at rate proportional to the weight on the edge connecting them. For the SIR model,each vertex is in one of three states, ‘susceptible’, ‘infectious’ or ‘recovered’. S t is the set ofsusceptible vertices and I t is the set of infectious vertices at the moment t . A susceptiblevertex is infected in the same way as that of the SIS model while an infectious vertex waitsfor an exponential time with rate one to become recovered. A recovered vertex can neverinfect neighbors or be infected again.The SIS model is also named as the contact process. The classic contact process isintroduced by Harris in [5], where ρ = 1. For a detailed survey of the classic contactprocess, see Chapter 6 of [8] and Part one of [9]. The contact process with i.i.d edge weightsis introduced by Chen and Yao in [15], where the complete convergence theorem of theprocess is proved. When P ( ρ = 1) = p = 1 − P ( ρ = 0), the model reduces to the contactprocess on clusters of bond percolation, which is also introduced by Chen and Yao in [2]to prove a similar complete convergence theorem. It is also interesting to put the randomweights on vertices instead of edges, where a susceptible vertex x with weight ρ ( x ) is infectedby an infectious neighbor y with weight ρ ( y ) at rate proportional to ρ ( x ) ρ ( y ). This model isintroduced by Peterson on the complete graph in [10], where a phase transition phenomenonconsistent with a mean-field analysis is shown. Xue studies the contact process with randomvertex weights on the oriented lattice in [11], where a limit theorem of the critical infectionrate is given. When the vertex weight takes 1 with probability p and takes 0 otherwise,the process reduces to that on clusters of site percolation, which is a special case of themodel introduced in [1] with N = 1. In [1], Bertacchi, Lanchier and Zucca study the contactprocess on C ∞ × K N , where C ∞ is unique infinite open cluster of the site percolation on Z d while K N is the complete graph with N vertices. Criteria to judge whether the processsurvives is given in [1].The initial motivation of the study in this paper is to extend the main result in [13],which gives the mean field limit for survival probability of high-dimensional classic contactprocess, to the case where the contact process is with random edge weights. We find out2hat the SIR model is a useful auxiliary tool for us to accomplish our objective and similarconclusion holds for the SIR model simultaneously according to our proof. We are inspireda lot by the technique introduced in [14], which gives asymptotic behavior of the criticalvalue of the high-dimensional SIR model on clusters of bond percolation. In this section we give our main results. First we introduce some notations and defini-tions. We assume that the edge weights { ρ ( e ) } e ∈ E d are defined under the probability space(Ω d , F d , µ d ). The expectation operator with respect to µ d is denoted by E µ d . For ω ∈ Ω d ,we write ρ ( e ) as ρ ( e, ω ) when we emphasize that the weight on e is with respect to therandom environment ω . For λ > ω ∈ Ω d , we denote by P λ,ω the probability measureof the SIS and SIR models on Z d with infection rate λ and edge weights { ρ ( e, ω ) } e ∈ E d . P λ,ω is called the quenched measure. We define P λ,d ( · ) = E µ d (cid:16) P λ,ω ( · ) (cid:17) = (cid:90) P λ,ω ( · ) µ d ( dω ) .P λ,d is called the annealed measure. When we do not need to distinguish the dimension d , we omit the subscript d in the above notations. For A ⊆ Z d , we write C t as C At when C = A . If A = { x } for x ∈ Z d , then we write C { x } t as C xt for simplicity. For any x ∈ Z d ,we write ( S t , I t ) as ( S xt , I xt ) when ( S , I ) = ( Z d \ { x } , { x } ).The following theorem is our main result, which gives mean field limits of the survivalprobabilities of the SIS and SIR models as the dimension d grows to infinity. Theorem 2.1.
Let O be the origin of Z d as we have defined in Section 1, then lim d → + ∞ P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = lim d → + ∞ P λ,d (cid:0) I Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = λEρ − λEρ for any λ ≥ Eρ , where Eρ is the expectation of ρ . Theorem 2.1 shows that for high-dimensional SIS and SIR models with random edgeweights, assuming that O is the unique infectious vertex at t = 0 while other vertices aresusceptible, then the probability that infectious vertices will never die out approximatelyequals ( λEρ − /λEρ . This result can be intuitively explained according to a mean-fieldanalysis. When the dimension d is large, it is not likely that infectious vertices will cluster,then | C t | decreases by one at rate | C t | and increases by one at rate approximate to (cid:88) x ∈ C t (cid:88) y : y ∼ x λ d ρ ( x, y ) ≈ λ | C t | Eρ according to the law of large numbers. Then, the embedded chain of | C t | is similar witha biased random walk on Z that increases by one with probability λEρλEρ +1 or decreases byone with probability λEρ +1 . Such a biased random walk starting at 1 does not visit zero atprobability ( λEρ − /λEρ .For the classic SIS model with ρ ≡
1, Theorem 2.1 shows thatlim d → + ∞ P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = λ − λ for λ ≥
1. This result is first given in [13] as far as we know.3imilar result with that in Theorem 2.1 for the bond percolation model is obtained in [7].In [7], Kesten studies the high-dimensional Fortuin-Kasteleyn cluster model, containing thebond percolation model as a special case. It is shown in [7] that the probability that O belongs to the infinite open cluster converges to the solution to the equation x = 1 − e − λx as the dimension d grows to infinity for the bond percolation model on Z d where an edge isopen with probability λ d with λ > P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) and P λ,d (cid:0) I Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) are increasing with λ , hence it is reasonable to define λ c ( d ) = sup (cid:8) λ : P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = 0 (cid:9) and β c ( d ) = sup (cid:8) λ : P λ,d (cid:0) I Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = 0 (cid:9) .λ c ( d ) is called the critical value of the contact process, since it is the maximum of infectionrates with which the infectious vertices die out with probability one. For similar reason, β c ( d )is called the critical value of the SIR model. The following conclusion about estimations of λ c ( d ) and β c ( d ) is an application of Theorem 2.1. Theorem 2.2. lim d → + ∞ β c ( d ) = 1 Eρ while lim sup d → + ∞ λ c ( d ) ≤ Eρ .
When ρ ≡
1, Theorem 2.2 shows that lim sup d → + ∞ λ c ( d ) ≤
1. A stronger conclusionthat lim d → + ∞ λ c ( d ) = 1 for the classic contact process is proved by Holley and Liggettin [6]. In [4], Griffeath gives another proof of this result and obtains a better upper boundof λ c ( d ). When P ( ρ = 1) = p = 1 − P ( ρ = 0), Theorem 2.2 shows that lim d → + ∞ β c ( d ) = p while lim sup d → + ∞ λ c ( d ) ≤ p for SIR and SIS models on clusters of bond percolation model.These two results are proved in [14] and [12] respectively.We believe that lim d → + ∞ λ c ( d ) = Eρ but have not found a proof yet. Since P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) is increasing with λ , a direct corollary of Theorem 2.1 is thatlim d → + ∞ P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = 0for any λ < Eρ . If this conclusion can be strengthened to that P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = 0for λ < Eρ and sufficiently large d , then we can claim that lim inf d → + ∞ λ c ( d ) ≥ Eρ andhence lim d → + ∞ λ c ( d ) = Eρ . We will work on this problem as a further study.According to the basic coupling of Markov process (see Section 3.1 of [8]), it is easy tocheck that P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) ≥ P λ,d (cid:0) I Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) . Therefore, to prove Theorem 2.1, we only need to show thatlim sup d → + ∞ P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) ≤ λEρ − λEρ (2.1)4nd lim inf d → + ∞ P λ,d (cid:0) I Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) ≥ λEρ − λEρ (2.2)for λ > Eρ .The proof of Equation (2.1) is given in Section 3. The core idea of the proof is as follows.For given large inter M and small positive constant (cid:15) , with high probability that every vertex x in the set (cid:8) u : (cid:107) u (cid:107) ≤ M (cid:9) satisfies that (cid:88) y : y ∼ x ρ ( x, y ) ≤ d ( Eρ + (cid:15) ) . Before the first moment when C t contains a vertex with l -norm larger than M , the embed-ded chain of | C t | is dominated from above by a biased random walk which increases by onewith probability λ ( Eρ + (cid:15) ) / λ ( Eρ + (cid:15) ) or decrease by one with probability 1 / λ ( Eρ + (cid:15) ).Such a biased random walk starting at 1 hits zero at least once with probability 1 /λ ( Eρ + (cid:15) ).The proof of Equation (2.2) is given in Section 4. The core idea of the proof is as follows.We divide Z d into two disjoint parts Γ and Γ . We first show that there exist d / verticesin Γ which are infected through paths on Γ with probability about ( λEρ − /λEρ . In thisstep we dominate the embedded chain of | I t | from below by another biased random walk.Then we show that these d / vertices infect at least d / vertices in Γ by edges connectingΓ and Γ with high probability. At last, we show that with d / initial infectious verticesin Γ , the SIR model confined to Γ survives with high probability. The approach in thisstep is inspired by the technique introduced in [14]. Since Γ and Γ are disjoint, the eventconcerned with in the third step is independent of the two events concerned with in the firstand second steps and hence the survival probability is at least the product of the probabilityof the third event and the probability that both the first and the second events occur. Tomake the above explanation rigorous, we introduce the definition of so-called infectious pathat the beginning of Section 4 that a vertex x has ever been infected when and only whenthere exists an infectious path from O to x .The proof of Theorem 2.2 is given in Section 5, which is an application of Theorem 2.1and the definition of infectious path introduced in Section 4. (2.1) In this section we give the proof of Equation (2.1). Throughout this section we assume that λ ≥ /Eρ . First we introduce some notations and definitions. For r >
0, we define B ( d, r ) = (cid:8) x ∈ Z d : (cid:107) x (cid:107) ≤ r (cid:9) as the set of vertices with l norm at most r . For M > (cid:15) >
0, we define A ( d, M, (cid:15) ) = (cid:8) ω ∈ Ω d : (cid:88) y : y ∼ x ρ ( x, y, ω ) ≤ d ( Eρ + (cid:15) ) , ∀ x ∈ B ( d, M ) (cid:9) as the set of random environments where every vertex x with l norm at most M satisfiesthat (cid:88) y : y ∼ x ρ ( x, y ) ≤ d ( Eρ + (cid:15) ) . According to the classic theory about large deviation principle, there exists J ( (cid:15) ) > µ d (cid:0) ω : 12 d (cid:88) y : y ∼ x ρ ( x, y, ω ) ≥ Eρ + (cid:15) (cid:1) ≤ e − dJ ( (cid:15) ) d ≥ x ∈ Z d , since { ρ ( x, y ) : y ∼ x } are 2 d independent copies of ρ . For each x ∈ B ( d, M ), there is an path from O to x with length at most M . For a path on Z d , eachstep has 2 d choices, therefore | B ( d, M ) | ≤ M (cid:88) l =0 (2 d ) l ≤ ( M + 1)(2 d ) M . As a result, µ d (cid:0) A ( d, M, (cid:15) ) (cid:1) ≥ − ( M + 1)(2 d ) M e − dJ ( (cid:15) ) (3.1)according to the Chebyshev’s inequality.We define { V n } n ≥ as the biased random walk on Z that P ( V n +1 − V n = 1) = λ ( Eρ + (cid:15) )1 + λ ( Eρ + (cid:15) ) = 1 − P ( V n +1 − V n = − n ≥ V = 1. For K ≥
0, we define τ K = inf { n ≥ V n = K } as the first moment when K is visited. According to classic theory about biased randomwalk, lim K → + ∞ τ K K = 1 λ ( Eρ + (cid:15) )1+ λ ( Eρ + (cid:15) ) − λ ( Eρ + (cid:15) ) = 1 + λ ( Eρ + (cid:15) ) λ ( Eρ + (cid:15) ) − . (3.2)Now we give the proof of Equation (2.1). Proof of Equation (2.1) . For given
M > (cid:15) > ω ∈ A ( d, M, (cid:15) ), C t with edge weights { ρ ( e, ω ) } e ∈ E d decreases by one at rate | C t | or increases by one at rate at most λ d (cid:88) x ∈ C t (cid:88) y : y ∼ x ρ ( x, y, ω ) ≤ λ d d ( Eρ + (cid:15) ) | C t | = λ ( Eρ + (cid:15) ) | C t | for t < inf { s : C s (cid:51) x for some x with (cid:107) x (cid:107) > M } . As a result, before the momentinf { s : C s (cid:51) x for some x with (cid:107) x (cid:107) > M } , the embedded chain of | C t | is dominated fromabove by { V n } n ≥ . Since all the infections occur between nearest neighbors, the state of { C t } t ≥ must jump at least M times to make C t contain a vertex with l norm lager than M . According to the above analysis, for given K > (cid:8) C t = ∅ for some t > (cid:9) ⊇ { τ < τ K , τ K < M } in the sense of coupling, where τ K is the first time K is visited by { V n } n ≥ as we havedefined. Therefore, for ω ∈ A ( d, M, (cid:15) ), P λ,ω (cid:0) C t = ∅ for some t > (cid:1) ≥ P ( τ < τ K ) − P ( τ K ≥ M ) (3.3)= λ ( Eρ + (cid:15) ) − ( λ ( Eρ + (cid:15) ) ) K − ( λ ( Eρ + (cid:15) ) ) K − P ( τ K ≥ M )according to classic theorem of biased random walk. Then, according to Equation (3.1), P λ,d (cid:0) C t = ∅ for some t > (cid:1) ≥ E µ d (cid:104) P λ,ω (cid:0) C t = ∅ for some t > (cid:1) A ( d,M,(cid:15) ) (cid:105) (3.4) ≥ (cid:104) λ ( Eρ + (cid:15) ) − ( λ ( Eρ + (cid:15) ) ) K − ( λ ( Eρ + (cid:15) ) ) K − P ( τ K ≥ M ) (cid:105)(cid:16) − ( M + 1)(2 d ) M e − dJ ( (cid:15) ) (cid:17) .
6e choose K = (cid:98) d / (cid:99) and M = 2 K λ ( Eρ + (cid:15) ) λ ( Eρ + (cid:15) ) − , then lim d → + ∞ P ( τ K ≥ M ) = 0 by Equation(3.2), lim d → + ∞ ( M + 1)(2 d ) M e − dJ ( (cid:15) ) = 0 andlim d → + ∞ λ ( Eρ + (cid:15) ) − ( λ ( Eρ + (cid:15) ) ) K − ( λ ( Eρ + (cid:15) ) ) K = 1 λ ( Eρ + (cid:15) ) . As a result, by Equation (3.4),lim inf d → + ∞ P λ,d (cid:0) C t = ∅ for some t > (cid:1) ≥ λ ( Eρ + (cid:15) ) . Since (cid:15) is arbitrary, we havelim inf d → + ∞ P λ,d (cid:0) C t = ∅ for some t > (cid:1) ≥ λEρ . (3.5)Equation (2.1) follows from Equation (3.5) directly. (2.2) The aim of this section is to prove Equation (2.2). Throughout this section we assume that λ > Eρ , since the case where λ = Eρ becomes trivial after the case where λ > Eρ is proved.This section is divided into four parts. In Subsection 4.1 we give the proof of Equation (2.2)based on Lemmas 4.3 and 4.4. The proof of Lemma 4.4 is given in Subsection 4.4 while theproof of Lemma 4.3 is given in Subsection 4.3. The proof of Lemma 4.3 utilizes Lemma 4.2.We give the proof of Lemma 4.2 in Subsection 4.2. (2.2) In this subsection we give the proof of Equation (2.2). First we introduce the definition ofthe infectious path. Let H d = (cid:8) ( x, y ) ∈ Z d × Z d : x ∼ y (cid:9) be the set of ordered pairs of neighbors on Z d , then we define X = [0 , + ∞ ) Z d × [0 , + ∞ ) H d . Therefore, an element in X can be written as ( Y, U ), where Y : Z d → [0 , + ∞ ) and U : H d → [0 , + ∞ ). Let F be the smallest sigma-field that { Y ( x ) } x ∈ Z d and { U ( x, y ) } ( x,y ) ∈ H d are measurable with respect to.For any ω ∈ Ω d , let ν ω be a probability measure on ( X , F ) that Y ( x ) is an exponentialtime with rate one for each x ∈ Z d and U ( y, z ) is an exponential time with rate λ d ρ ( y, z, ω )for each ( y, z ) ∈ H d while all these exponential times are independent under ν ω .For a self-avoiding path (cid:126)l = ( l , l , . . . , l n ) on Z d with length n , we say (cid:126)l is an infectiouspath (with respect to ( Y, U )) when and only when U ( l i , l i +1 ) < Y ( l i ) for 0 ≤ i ≤ n −
1. Wehave the following important lemma.
Lemma 4.1.
Let X be defined as in Section 1 and B be the smallest sigma-field containingall the finite cylinder sets included in X [0 , + ∞ )2 , then there exists a measurable mapping { ( (cid:98) S t , (cid:98) I t ) } t ≥ : ( X , F ) → (cid:16) X [0 , + ∞ )2 , B (cid:17) hat { ( (cid:98) S t , (cid:98) I t ) } t ≥ under the measure ν ω is a version of { ( S Ot , I Ot ) } t ≥ under the probabilitymeasure P λ,ω ( · ) for each ω ∈ Ω d and (cid:110) ( Y, U ) : x ∈ (cid:98) I t for some t > (cid:111) = (cid:110) ( Y, U ) : there exists an infectious path with respect to ( Y, U ) from O to x (cid:111) for any x (cid:54) = O . We omit the proof of Lemma 4.1 here since it is a little tedious while this lemma canbe explained intuitively and clearly. The intuitive explanation of Lemma 4.1 is as follows. Y ( x ) is the time x waits for to become recovered after x is infected, i.e., x becomes recoveredat moment t + Y ( x ) if x is infected at moment t . U ( x, y ) is the time x waits for to infectneighbor y after x is infected. The infection really occurs when U ( x, y ) < Y ( x ) and y isnot infected by other vertices before the moment t + U ( x, y ), where t is the moment when x is infected. As a result, for any x (cid:54) = O , if x has ever been infected, then there exists aself-avoiding path (cid:126)l = ( O, l , . . . , l n − , x ) that l i has ever infected l i +1 for 0 ≤ i ≤ n − U ( l i , l i +1 ) < Y ( l i ) for 0 ≤ i ≤ n −
1, i.e., (cid:126)l is an infectious path. On the other hand,if (cid:126)l = (
O, l , . . . , l n − , x ) is an infectious path, then we claim that l i has ever been infectedfor all 0 ≤ i ≤ n . This claim holds for i = 0 trivially since O ∈ I . Assuming that l i hasever been infected for some i < n , then there are two possible cases. The first case is that l i +1 ∈ I s for some s < inf { t : l i ∈ I t } + U ( l i , l i +1 ), then our claim holds for i + 1 trivially.The second case is that l i +1 (cid:54)∈ I s for any s < inf { t : l i ∈ I t } + U ( l i , l i +1 ), theninf { t : l i +1 ∈ I t } = inf { t : l i ∈ I t } + U ( l i , l i +1 )as we have introduced. As a result, our claim holds for i + 1 and then holds for all 0 ≤ i ≤ n according to the principle of mathematical induction. In conclusion, (cid:8) x has ever been infected (cid:9) = (cid:8) there is an infectious path from O to x (cid:9) . For simplicity, from now on we identify { ( (cid:98) S t , (cid:98) I t ) } t ≥ given by Lemma 4.1 with { ( S Ot , I Ot ) } t ≥ and identify ν ω with P λ,ω . This identification is permitted by Lemma 4.1. As a result, P λ,d can be considered as a probability measure on ( X , F ) and P λ,d ( · ) = E µ d (cid:16) P λ,ω ( · ) (cid:17) = E µ d (cid:16) ν ω ( · ) (cid:17) . For later use, we introduce some definitions. We defineΓ = (cid:110) x = ( x , . . . , x d ) ∈ Z d : d (cid:88) i = d −(cid:98) d log d (cid:99) +1 | x i | = 0 (cid:111) , Γ = (cid:110) x = ( x , . . . , x d ) ∈ Z d : x i ≥ d − (cid:98) d log d (cid:99) + 1 ≤ i ≤ d and d (cid:88) i = d −(cid:98) d log d (cid:99) +1 x i > (cid:111) and Γ = (cid:110) x = ( x , . . . , x d ) ∈ Γ : d (cid:88) i = d −(cid:98) d log d (cid:99) +1 x i = 1 (cid:111) , (cid:98) u (cid:99) = n when n is an integer and n ≤ u < n + 1.We say an infectious path is on a subgraph A of Z d when all the vertices on this pathbelong to A . We define D = (cid:110) x ∈ Γ : there is an infectious path on Γ from O to x (cid:111) . Note that D is a mapping from X to the power set of Γ . The following lemma is importantfor us to prove Equation (2.2). Lemma 4.2. lim inf d → + ∞ P λ,d (cid:16) | D | ≥ (cid:98) d / (cid:99) (cid:17) ≥ λEρ − λEρ . The proof of Lemma 4.2 will be given in Subsection 4.2.For any x ∈ Γ and any B ⊆ Γ , we define D ( x ) = (cid:110) y ∈ Γ : y ∼ x and U ( x, y ) < Y ( x ) (cid:111) and D ( B ) = (cid:83) w ∈ B D ( w ). The following lemma about D ( D ) is important for us to proveEquation (2.2). Lemma 4.3. lim inf d → + ∞ P λ,d (cid:0) | D ( D ) | > d / (cid:1) ≥ λEρ − λEρ . The proof of Lemma 4.3 is given in Subsection 4.3, where Lemma 4.2 will be utilized.For integer n ≥ B ⊆ Γ , we define D ( n, B ) = (cid:110) x ∈ Γ : (cid:107) x (cid:107) ≥ n and there exists an infectious pathon Γ from some vertex in B to (cid:107) x (cid:107) (cid:111) . Note that D ( n, B ) is a mapping from X to the power set of Γ for given n and B . Thefollowing lemma is crucial for us to prove Equation (2.2), where we use { A n i.o. } to denote (cid:84) n ≥ (cid:83) k ≥ n A k for a series of events { A n } n ≥ . Lemma 4.4.
For each d ≥ , let ∆( d ) = inf (cid:110) P λ,d (cid:0) D ( n, B ) (cid:54) = ∅ i.o. (cid:1) : B ⊆ Γ and | B | = (cid:98) d / (cid:99) (cid:111) , then lim d → + ∞ ∆( d ) = 1 . The proof of Lemma 4.4 is given in Subsection 4.4. The strategy of the proof is inspiredby the approach introduced in [14].At the end of this subsection we show how to utilize Lemmas 4.3 and 4.4 to proveEquation (2.2).
Proof of Equation (2.2) . According to the definitions of D , D ( B ) and D ( n, B ), for each n ≥ x ∈ D ( n, D ( D )), there exist y ∈ Γ and z ∈ Γ that the following threeconditions holds.(1) There is an infectious path on Γ from z to x .(2) y ∼ z and U ( y, z ) < Y ( y ).(3) There is an infectious path on Γ from O to y .9s a result, there is an infectious path from O to x , as it is shown in Figure 1.Figure 1: Infectious pathTherefore, by Lemma 4.1, D (cid:0) n, D ( D ) (cid:1) ⊆ (cid:91) t ≥ I t and hence (cid:110) D (cid:0) n, D ( D ) (cid:1) (cid:54) = ∅ i.o. (cid:111) ⊆ (cid:110) | (cid:91) t ≥ I t | = + ∞ (cid:111) . If there are infinite many vertices have ever been infected, then they can not all becomerecovered before a uniform moment
T < + ∞ , since each infected vertex waits for an inde-pendent copy of the exponential time with rate 1 to become recovered. As a result, (cid:110) | (cid:91) t ≥ I t | = + ∞ (cid:111) ⊆ (cid:8) I t (cid:54) = ∅ , ∀ t ≥ (cid:9) and hence P λ,d (cid:0) I t (cid:54) = ∅ , ∀ t ≥ (cid:1) ≥ P λ,d (cid:16) D (cid:0) n, D ( D ) (cid:1) (cid:54) = ∅ i.o. (cid:17) . (4.1)Conditioned on D ( D ) = B for some B ⊆ Γ , D ( n, B ) depends on { Y ( x ) : x ∈ Γ } (cid:83) { U ( y, z ) : y, z ∈ Γ } , which is independent of the random set D ( D ), since D ( D )depends on { Y ( x ) : x ∈ Γ } (cid:83) { U ( y, z ) : y ∈ Γ , z ∈ Γ (cid:83) Γ } . Therefore, P λ,d (cid:16) D (cid:0) n, D ( D ) (cid:1) (cid:54) = ∅ i.o. (cid:12)(cid:12)(cid:12) D ( D ) (cid:17) = P λ,d (cid:16) D (cid:0) n, B (cid:1) (cid:54) = ∅ i.o. (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B = D ( D ) (4.2) ≥ inf (cid:110) P λ,d (cid:16) D (cid:0) n, A (cid:1) (cid:54) = ∅ i.o. (cid:17) : A ⊆ Γ and | A | = k (cid:111)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = | D ( D ) | . It is obviously thatinf (cid:110) P λ,d (cid:16) D (cid:0) n, A (cid:1) (cid:54) = ∅ i.o. (cid:17) : A ⊆ Γ and | A | = k (cid:111)
10s increasing with k . As a result, by Equation (4.2), P λ,d (cid:16) D (cid:0) n, D ( D ) (cid:1) (cid:54) = ∅ i.o. (cid:17) = P λ,d (cid:16) D (cid:0) n, D ( D ) (cid:1) (cid:54) = ∅ i.o. (cid:12)(cid:12)(cid:12) | D ( D ) | > (cid:98) d / (cid:99) (cid:17) (4.3) × P λ,d (cid:0) | D ( D ) | > (cid:98) d / (cid:99) (cid:1) ≥ E λ,d (cid:32) inf (cid:110) P λ,d (cid:16) D (cid:0) n, A (cid:1) (cid:54) = ∅ i.o. (cid:17) : A ⊆ Γ and | A | = k (cid:111)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = | D ( D ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | D ( D ) | > (cid:98) d / (cid:99) (cid:33) × P λ,d (cid:0) | D ( D ) | > (cid:98) d / (cid:99) (cid:1) ≥ inf (cid:110) P λ,d (cid:16) D (cid:0) n, A (cid:1) (cid:54) = ∅ i.o. (cid:17) : A ⊆ Γ and | A | = (cid:98) d / (cid:99) (cid:111) × P λ,d (cid:0) | D ( D ) | > (cid:98) d / (cid:99) (cid:1) = ∆( d ) P λ,d (cid:0) | D ( D ) | > (cid:98) d / (cid:99) (cid:1) . By Equation (4.3), Lemmas 4.3 and 4.4,lim inf d → + ∞ P λ,d (cid:16) D (cid:0) n, D ( D ) (cid:1) (cid:54) = ∅ i.o. (cid:17) ≥ λEρ − λEρ . (4.4)Equation (2.2) follows from Equations (4.1) and (4.4) directly. In this subsection we give the proof of Lemma 4.2, which is similar with that of Equation(2.1). First we introduce some notations. For given
M > (cid:15) >
0, we define F ( d, M, (cid:15) ) = (cid:110) ω ∈ Ω : (cid:88) y : y ∼ x ρ ( x, y, ω ) > d ( Eρ − (cid:15) ) for all x ∈ B ( d, M ) (cid:111) . According to a similar analysis with that of Equation (3.1), there exist J ( (cid:15) ) > µ d (cid:16) F ( d, M, (cid:15) ) (cid:17) ≥ − ( M + 1)(2 d ) M e − dJ ( (cid:15) ) . (4.5)We choose (cid:15) sufficiently small such that λ ( Eρ − (cid:15) ) >
1. Then we assume that we deal with d sufficiently large such that Eρ − (cid:15) − (cid:0) M ( d ) (cid:1) Θ2 d − Θ2 d (cid:98) d log d (cid:99) ≥ Eρ − (cid:15) (4.6)where Θ is defined as in Section 1 while M ( d ) = 2 (cid:98) d / (cid:99) λ ( Eρ − (cid:15) )+1 λ ( Eρ − (cid:15) ) − . We define { W n } n ≥ as biased random walk on Z that W = 1 and P ( W n +1 − W n = 1) = λ ( Eρ − (cid:15) ) λ ( Eρ − (cid:15) ) + 1 = 1 − P ( W n +1 − W n = − . For each integer K ≥
0, we define φ K = inf { n ≥ W n = K } as the first moment when K is visited.Now we give the proof of Lemma 4.2. 11 roof of Lemma 4.2. We denote by { ( (cid:101) S t , (cid:101) I t ) : t ≥ } the SIR model with random edgeweights confined to the graph Γ with ( (cid:101) S , (cid:101) I ) = (Γ \ { O } , { O } ). Let (cid:101) I = ∪ t ≥ (cid:101) I t , thenaccording to a similar analysis with that leads to Lemma 4.1, D = (cid:101) I. (4.7)Let ψ = inf { t : | (cid:101) I t | = (cid:98) d / (cid:99)} , then by Equation (4.7), P λ,d (cid:0) | D | ≥ (cid:98) d / (cid:99) (cid:1) ≥ P λ,d (cid:0) ψ < + ∞ (cid:1) . (4.8)When the number of jumps of the state of (cid:101) I t is no more than M ( d ), there are at most1 + M ( d ) vertices that have ever been infected and all the infectious vertices belong to B ( d, M ( d )). As a result, for ω ∈ F ( d, M ( d ) , (cid:15) ) and { ( (cid:101) S t , (cid:101) I t ) : t ≥ } with random edgeweights with respect to ω , | (cid:101) I t | decreases by one at rate | (cid:101) I t | while increases by one at rate λ d (cid:88) x ∈ (cid:101) I t (cid:88) y : y ∼ x ρ ( x, y ) − λ d (cid:88) x ∈ (cid:101) I t (cid:88) y : y ∼ x ρ ( x, y )1 { y (cid:54)∈ Γ } − λ d (cid:88) x ∈ (cid:101) I t (cid:88) y : y ∼ x ρ ( x, y )1 { y ∈ (cid:101) I s for some s ≥ t } ≥ λ | (cid:101) I t | ( Eρ − (cid:15) ) − λ Θ2 d | (cid:101) I t |(cid:98) d log d (cid:99) − λ d | (cid:101) I t | ( M ( d ) + 1)Θ= λ | (cid:101) I t | (cid:16) Eρ − (cid:15) − (cid:0) M ( d ) (cid:1) Θ2 d − Θ2 d (cid:98) d log d (cid:99) (cid:17) ≥ λ | (cid:101) I t | ( Eρ − (cid:15) ) (This step utilizes Equation (4.6).)before the moment when the state of {| (cid:101) I t |} t ≥ jumps for the M ( d )th time. As a result,the embedded chain of {| (cid:101) I t |} t ≥ is dominated from below by { W n } n ≥ for 0 ≤ n < M ( d ).Therefore, for ω ∈ F ( d, M ( d ) , (cid:15) ) and { ( (cid:101) S t , (cid:101) I t ) } t ≥ with random edge weights with respectto ω , (cid:110) ψ < + ∞ (cid:111) ⊇ (cid:110) φ (cid:98) d / (cid:99) < φ , φ (cid:98) d / (cid:99) < M ( d ) (cid:111) in the sense of coupling. Therefore, for ω ∈ F ( d, M ( d ) , (cid:15) ), P λ,ω (cid:0) ψ < ∞ (cid:1) ≥ P (cid:0) φ (cid:98) d / (cid:99) < φ , φ (cid:98) d / (cid:99) < M ( d ) (cid:1) ≥ P (cid:0) φ (cid:98) d / (cid:99) < φ (cid:1) − P (cid:0) φ (cid:98) d / (cid:99) ≥ M ( d ) (cid:1) (4.9)= 1 − λ ( Eρ − (cid:15) ) − ( λ ( Eρ − (cid:15) ) ) (cid:98) d / (cid:99) − P (cid:0) φ (cid:98) d / (cid:99) ≥ M ( d ) (cid:1) according to the classic theory of biased random walk. By Equations (4.5) and (4.9), P λ,d (cid:0) ψ < ∞ (cid:1) ≥ E µ d (cid:104) P λ,ω (cid:0) ψ < ∞ (cid:1) F ( d,M ( d ) ,(cid:15) ) (cid:105) ≥ (cid:16) − λ ( Eρ − (cid:15) ) − ( λ ( Eρ − (cid:15) ) ) (cid:98) d / (cid:99) − P (cid:0) φ (cid:98) d / (cid:99) ≥ M ( d ) (cid:1)(cid:17) (4.10) × (cid:16) − ( M ( d ) + 1)(2 d ) M ( d ) e − dJ ( (cid:15) ) (cid:17) . d → + ∞ φ (cid:98) d / (cid:99) M ( d ) = 12and hence lim d → + ∞ P (cid:0) φ (cid:98) d / (cid:99) ≥ M ( d ) (cid:1) = 0 . (4.11)Then by Equations (4.10) and (4.11),lim inf d → + ∞ P λ,d (cid:0) ψ < ∞ (cid:1) ≥ − λ ( Eρ − (cid:15) )since lim d → + ∞ − ( M ( d ) + 1)(2 d ) M ( d ) e − dJ ( (cid:15) ) = 1 . Since (cid:15) is arbitrary, we havelim inf d → + ∞ P λ,d (cid:0) ψ < ∞ (cid:1) ≥ − λEρ = λEρ − λEρ . (4.12)Lemma 4.2 follows from Equations (4.8) and (4.12) directly. In this subsection we give the proof of Lemma 4.3. First we introduce some notationsand definitions. Let { Ψ( x ) } x ∈ Z d be i.i.d. exponential times with rate λ Θ and independentwith { Y ( x ) } x ∈ Z d and { U ( x, y ) } ( x,y ) ∈ H d under the measure P λ,ω for any ω ∈ Ω, where Θ isdefined as in Section 1. Note that to make the above definition rigorous we can expand X to (cid:101) X = X × [0 , + ∞ ) Z d and identify P λ,ω with the measure ν ω × π , where π is the probabilitymeasure of i.i.d exponential times with rate λ Θ. This is classic approach in measure theoryso we omit the details.For any A ⊆ Γ , we denote by q ( A ) the random event that Y ( x ) < Ψ( x ) for any x ∈ A .For any s > A ⊆ Γ , it is easy to check that E λ,d (cid:16) e − s | D ( A ) | (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) depends only on s and the cardinality of A . Hence we can reasonably define h ( d, s, K ) = E λ,d (cid:16) e − s | D ( A ) | (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) for s > A ⊆ Γ with | A | = K . The following lemma is crucial for us to prove Lemma4.3. Lemma 4.5.
For any s > , lim d → + ∞ h ( d, − log dd / s, (cid:98) d / (cid:99) ) = exp {− λsEρ λ Θ + 1) } . E λ,d (cid:16) D ( A ) (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) ≈ λd / Eρ d ( λ Θ + 1)for A ⊆ Γ with | A | = (cid:98) d / (cid:99) and large d . Then it is natural to check wether the distributionof log dd / | D ( A ) | conditioned on q ( A ) converges weakly to the Dirac measure on λEρ λ Θ+1) . Oneapproach to do so is the Laplace transform, i.e., the calculation of h ( d, − log dd / s, (cid:98) d / (cid:99) ).We give the proof of Lemma 4.5 at the end of this subsection. Now we show how toutilize Lemmas 4.2 and 4.5 to prove Lemma 4.3. Proof of Lemma 4.3.
By Lemma 4.2,lim inf d → + ∞ P λ,d (cid:0) | D ( D ) | > d / (cid:1) ≥ lim inf d → + ∞ P λ,d (cid:16) | D ( D ) | > d / (cid:12)(cid:12)(cid:12) | D | ≥ (cid:98) d / (cid:99) (cid:17) P λ,d (cid:16) | D | ≥ (cid:98) d / (cid:99) (cid:17) (4.13) ≥ λEρ − λEρ lim inf d → + ∞ P λ,d (cid:16) | D ( D ) | > d / (cid:12)(cid:12)(cid:12) | D | ≥ (cid:98) d / (cid:99) (cid:17) . We claim that P λ,d (cid:16) | D ( D ) | > d / (cid:12)(cid:12)(cid:12) | D | ≥ (cid:98) d / (cid:99) (cid:17) ≥ P λ,d (cid:16) | D ( A ) | > d / (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) , (4.14)where | A | = (cid:98) d / (cid:99) . Note that P λ,d (cid:16) | D ( A ) | > d / (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) depends only on | A | , not on thechoice of A . The explanation of Equation (4.14) is as follows. Conditioned on | D | ≥ (cid:98) d / (cid:99) ,there exists a subset B of D that | B | = (cid:98) d / (cid:99) . If | D ( B ) | > d / , then | D ( D ) | > d / .The event | D | ≥ (cid:98) d / (cid:99) relies on the values of { Y ( x ) } x ∈ Γ and { U ( x, y ) } x ∼ y,x,y ∈ Γ . Sothe event | D | ≥ (cid:98) d / (cid:99) is correlated with the event | D ( B ) | > d / and we do not ensure(though we guess) that they are positive correlated . However, the worst condition withrespect to Y ( · ) and U ( · , · ) on Γ for the probability that | D ( B ) | > d / occurs is that Y ( x ) < inf { U ( x, y ) : y ∼ x, y ∈ Γ } for any x ∈ B . Hence the probability that | D ( B ) | > d / occurs decreases if we replacethe condition | D | ≥ (cid:98) d / (cid:99) by that Y ( x ) < inf { U ( x, y ) : y ∼ x, y ∈ Γ } for each x ∈ B .inf { U ( x, y ) : y ∼ x, y ∈ Γ } is an exponential time with rate (cid:88) y : y ∼ x,y ∈ Γ λρ ( x, y )2 d ≤ λ Θ2 d (2 d − (cid:98) d log d (cid:99) ) ≤ λ Θ .λ Θ is the rate of the exponential time Ψ( x ). As a result, the probability that | D ( B ) | > d / occurs will further decrease if we replace the condition Y ( x ) < inf { U ( x, y ) : y ∼ x, y ∈ Γ } by Y ( x ) < Ψ( x ) for every x ∈ B , which leads to Equation (4.14).For A ⊆ Γ with | A | = (cid:98) d / (cid:99) and any s >
0, by Chebyshev’s inequality, P λ,d (cid:16) | D ( A ) | < d / (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) = P λ,d (cid:16) e − s log dd / | D ( A ) | > e − s d / d / log d (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) ≤ exp (cid:8) s d / d / log d (cid:9) h ( d, − log dd / s, (cid:98) d / (cid:99) ) . (4.15)14hen by Lemma 4.5,lim sup d → + ∞ P λ,d (cid:16) | D ( A ) | < d / (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) ≤ exp {− λsEρ λ Θ + 1) } . (4.16)Since s is arbitrary, let s → + ∞ , we havelim sup d → + ∞ P λ,d (cid:16) | D ( A ) | < d / (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) = 0and hence lim d → + ∞ P λ,d (cid:16) | D ( A ) | > d / (cid:12)(cid:12)(cid:12) q ( A ) (cid:17) = 1 . (4.17)Lemma 4.3 follows from Equations (4.13), (4.14) and (4.17) directly.At the end of this subsection we give the proof of Lemma 4.5. Proof of Lemma 4.5.
According to assumptions of our model, it is easy to check that h ( d, − log dd / s, (cid:98) d / (cid:99) ) = (cid:32) E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) < Ψ( O ) (cid:17)(cid:33) (cid:98) d / (cid:99) . (4.18)and E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) < Ψ( O ) (cid:17) (4.19)= E λ,d (cid:32) E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y ( O ) < Ψ( O ) (cid:33) . Note that in Equation (4.18) we utilize the fact that | D ( B ) | = (cid:80) x ∈ B | D ( x ) | since D ( x ) ∩ D ( y ) = ∅ for x (cid:54) = y .According to assumptions of the model, it is easy to check that E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) (cid:17) = (cid:34) E λ,d (cid:16) e − log dd / s Λ (cid:12)(cid:12)(cid:12) Y ( O ) (cid:17)(cid:35) (cid:98) d log d (cid:99) , (4.20)where Λ = (cid:40) U ( O, e d ) < Y ( O ) , U ( O, e d ) ≥ Y ( O )and e d = (0 , . . . , ,
1) as we have defined in Section 1.By direct calculation, E λ,d (cid:16) e − log dd / s Λ (cid:12)(cid:12)(cid:12) Y ( O ) (cid:17) = E (cid:16) e − log dd / s (1 − e − λρt d ) + e − λρt d (cid:17)(cid:12)(cid:12)(cid:12) t = Y ( O ) (4.21)= (cid:16) − (cid:0) − e − log dd / s (cid:1) E (cid:0) − e − λρt d (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) t = Y ( O ) , where ρ is as defined in Section 1 while E is the expectation operator with respect to ρ . ByLagrange Mean Value Theorem and the fact that e a = 1 + a + o ( a ), it is not difficult tocheck thatlim d → + ∞ (cid:98) d / (cid:99) (cid:40)(cid:16) − (cid:0) − e − log dd / s (cid:1) E (cid:0) − e − λρt d (cid:1)(cid:17) (cid:98) d log d (cid:99) − (cid:41) = − sλtEρ . (4.22)15y Equations (4.20), (4.21) and (4.22),lim d → + ∞ (cid:98) d / (cid:99) (cid:40) E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) (cid:17) − (cid:41) = − sλY ( O ) Eρ . (4.23)By Equation (4.23) and Dominated Convergence Theorem,lim d → + ∞ (cid:98) d / (cid:99) (cid:40) E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) < Ψ( O ) (cid:17) − (cid:41) (4.24)= − sλEρ E λ,d (cid:16) Y ( O ) (cid:12)(cid:12)(cid:12) Y ( O ) < Ψ( O ) (cid:17) = − λsEρ λ Θ + 1) . According to classic conclusion about calculus, if a d → c d → + ∞ while a d c d → b as d → + ∞ , then lim d → + ∞ (1 + a d ) c d = e b . Therefore, by Equation (4.24),lim d → + ∞ (cid:32) E λ,d (cid:16) e − log dd / s | D ( O ) | (cid:12)(cid:12)(cid:12) Y ( O ) < Ψ( O ) (cid:17)(cid:33) (cid:98) d / (cid:99) = exp {− λsEρ λ Θ + 1) } . (4.25)Lemma 4.5 follows from Equations (4.18) and (4.25) directly. In this subsection we give the proof of Lemma 4.4. First we introduce some definitions andnotations. For each n ≥
1, we use Ξ n to denote the set of self-avoiding paths on Z d withlength n . For each n ≥ x ∈ Γ and B ⊆ Γ , we define L n ( x ) = (cid:110) (cid:126)x =( x , . . . , x n ) ∈ Ξ n : x = x,x i − x i − ∈ (cid:8) ± e j : 1 ≤ j ≤ d − (cid:98) d log d (cid:99) (cid:9) for each i that (cid:98) log d (cid:99) (cid:45) i while x i − x i − ∈ (cid:8) e j : d − (cid:98) d log d (cid:99) + 1 ≤ j ≤ d (cid:9) for each i that (cid:98) log d (cid:99) | i (cid:111) and L n ( B ) = (cid:83) w ∈ B L n ( w ), where we use a | b to denote that b is divisible by a and { e j : 1 ≤ j ≤ d } are defined as in Section 1.For any (cid:126)x, (cid:126)y ∈ L n ( B ), we define σ ( (cid:126)x, (cid:126)y ) = (cid:8) ≤ i ≤ n : there exists j that 0 ≤ j ≤ n and y i = x j (cid:9) and ζ ( (cid:126)x, (cid:126)y ) = (cid:8) ≤ i ≤ n − j that 0 ≤ j ≤ n − y i = x j while y i +1 = x j +1 } . Let { α n } n ≥ be a self-avoiding random walk on Γ that α ∈ Γ and (cid:101) P (cid:16) α n − α n − = e j (cid:12)(cid:12)(cid:12) α l , ≤ l ≤ n − (cid:17) = 1 (cid:98) d log d (cid:99) d − (cid:98) d log d (cid:99) + 1 ≤ j ≤ d and n that (cid:98) log d (cid:99) | n while (cid:101) P (cid:16) α n − α n − = y (cid:12)(cid:12)(cid:12) α l , ≤ l ≤ n − (cid:17) = 1 | R ( (cid:126)α, n ) | for each n that (cid:98) log d (cid:99) (cid:45) n and y ∈ R ( (cid:126)α, n ), where R ( (cid:126)α, n ) = (cid:110) u : u − α n − ∈ (cid:8) ± e j : 1 ≤ j ≤ d − (cid:98) d log d (cid:99) (cid:9) and u (cid:54) = α l for all 0 ≤ l ≤ n − (cid:111) while (cid:101) P is the probability measure of { α n } n ≥ . We use (cid:126)α n to denote the path ( α , . . . , α n ),then it is easy to check that (cid:126)α n ∈ L n ( α ) for each n ≥
1. Note that | R ( (cid:126)α, n ) | ≥ d − (cid:98) d log d (cid:99) ) − (cid:98) log d (cid:99) > d . This is because (cid:80) dk = d −(cid:98) d log d (cid:99) +1 y ( k ) = (cid:80) dk = d −(cid:98) d log d (cid:99) +1 α n ( k ) for y ∈ R ( (cid:126)α, n ) (where y ( k ) is the k th coordinate of y ) while d (cid:88) k = d −(cid:98) d log d (cid:99) +1 α n ( k ) > d (cid:88) k = d −(cid:98) d log d (cid:99) +1 α l ( k )for each l that n − l > log d .It is easy to check that d (cid:88) k = d −(cid:98) log d (cid:99) +1 α n ( k ) = 1 + (cid:4) n (cid:98) log d (cid:99) (cid:5) (4.27)for each n ≥ α n and Γ . This property will be utilizedrepeatedly in the proof of Lemma 4.4.Let { q n } n ≥ be an independent copy of { α n } n ≥ and (cid:126)q n = ( q , . . . , q n ), then we define σ ( n ) = σ ( (cid:126)α n , (cid:126)q n ) = (cid:8) ≤ i ≤ n : there exists j that 0 ≤ j ≤ n and q i = α j (cid:9) and ζ ( n ) = ζ ( (cid:126)α n , (cid:126)q n ) = (cid:8) ≤ i ≤ n − j that 0 ≤ j ≤ n − q i = α j while q i +1 = α j +1 } . Furthermore, we define σ = (cid:91) n ≥ σ ( n ) = (cid:8) i ≥ j ≥ q i = α j (cid:9) and ζ = (cid:91) n ≥ ζ ( n ) = (cid:8) i ≥ j ≥ q i = α j and q i +1 = α j +1 (cid:9) . For any x, y ∈ Γ , we denote by (cid:101) P x,y the probability measure of { α n , q n } n ≥ with α = x and q = y . The expectation operator with respect to (cid:101) P x,y is denoted by (cid:101) E x,y . The followlemma is crucial for us to prove Lemma 4.4. 17 emma 4.6. For any B ⊆ Γ , P λ,d (cid:16) D ( n, B ) (cid:54) = ∅ i.o. (cid:17) ≥ | B | (cid:80) x ∈ B (cid:80) y ∈ B (cid:101) E x,y (cid:104) M | σ \ ζ | ( d + λ Θ λEρ ) | ζ | (cid:105) , where M = ( Eρ ) . The proof of Lemma 4.6 is given at the end of this subsection. Now we show how toutilize Lemma 4.6 to prove Lemma 4.4.
Proof of Lemma 4.4.
We define κ = inf { i ≥ j ≥ q i = α j } . If κ =+ ∞ , then | σ | = | ζ | = 0. As a result, (cid:101) E x,y (cid:104) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:105) = (cid:101) P x,y ( κ = + ∞ ) (4.28)+ (cid:101) E x,y (cid:32) (cid:101) E x,y (cid:16) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:12)(cid:12)(cid:12) κ < + ∞ (cid:17) { κ< + ∞} (cid:33) . We claim that there exists M > d that (cid:101) P x,y ( κ < + ∞ ) ≤ M (log d ) d (4.29)for any x, y ∈ Γ , x (cid:54) = y . The proof of Equation (4.29) will be given later.Reference [14] gives a detailed calculation of the upper bound of the function f ( C , C ) = (cid:101) E x,y (cid:16) C | σ \ ζ | C | ζ | (cid:17) for x = y = O . For the general case where ( x, y ) (cid:54) = ( O, O ), the calculationis still valid after modifying some details. According to a similar analysis with that leadsto Lemma 3.4 of [14], for any C , C >
0, there exists M > d, C , C , x, y that (cid:101) E x,y (cid:16) C | σ \ ζ | C | ζ | (cid:12)(cid:12)(cid:12) κ < + ∞ (cid:17) ≤ C + C C (cid:98) d log d (cid:99) + ∞ (cid:88) k =1 (cid:16) C d − (cid:98) d log d (cid:99) ) − (cid:98) log d (cid:99) + C (cid:98) d log d (cid:99)(cid:98) log d (cid:99) + M (log d ) C d (cid:17) k . (4.30)Let C = M and C = d + λ Θ λEρ for λ > Eρ , thenlim d → + ∞ C d − (cid:98) d log d (cid:99) ) − (cid:98) log d (cid:99) + C (cid:98) d log d (cid:99)(cid:98) log d (cid:99) + M (log d ) C d = 1 λEρ < . We choose c ∈ ( λEρ , d , dλEρ d − (cid:98) d log d (cid:99) ) − (cid:98) log d (cid:99) + dλEρ (cid:98) d log d (cid:99)(cid:98) log d (cid:99) + M (log d ) M d ≤ c and (cid:101) E x,y (cid:16) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:12)(cid:12)(cid:12) κ < + ∞ (cid:17) ≤ M (cid:2) d (1 − c ) λEρ (cid:3) (4.31)by Equation (4.30). 18y Equations (4.28), (4.29) and (4.31), for sufficiently large d and any B ⊆ Γ , (cid:101) E x,y (cid:104) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:105) ≤ M (cid:2) d (1 − c ) λEρ (cid:3) M (log d ) d (4.32)for any x, y ∈ B, x (cid:54) = y .By Equations (4.31) and (4.32), for sufficiently large d and any B ⊆ Γ with | B | = (cid:98) d / (cid:99) ,1 | B | (cid:88) x ∈ B (cid:88) y ∈ B (cid:101) E x,y (cid:104) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:105) = 1 | B | (cid:32) (cid:88) x ∈ B (cid:101) E x,x (cid:104) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:105) + (cid:88) x ∈ B (cid:88) y : y (cid:54) = x (cid:101) E x,y (cid:104) M | σ \ ζ | ( 2 d + λ Θ λEρ ) | ζ | (cid:105)(cid:33) ≤ | B | M (cid:2) d (1 − c ) λEρ (cid:3) | B | + | B | ( | B | − (cid:0) M (cid:2) d (1 − c ) λEρ (cid:3) M (log d ) d (cid:1) | B | = M (cid:2) d (1 − c ) λEρ (cid:3) (cid:98) d / (cid:99) + ( (cid:98) d / (cid:99) − (cid:0) M (cid:2) d (1 − c ) λEρ (cid:3) M (log d ) d (cid:1) (cid:98) d / (cid:99) and hence P λ,d (cid:16) D ( n, B ) (cid:54) = ∅ i.o. (cid:17) ≥ M (cid:2) d (1 − c ) λEρ (cid:3) (cid:98) d / (cid:99) + ( (cid:98) d / (cid:99)− (cid:0) M (cid:2) d (1 − c ) λEρ (cid:3) M d )2 d (cid:1) (cid:98) d / (cid:99) by Lemma 4.6. Then according to the definition of ∆( d ),∆( d ) ≥ M (cid:2) d (1 − c ) λEρ (cid:3) (cid:98) d / (cid:99) + ( (cid:98) d / (cid:99)− (cid:0) M (cid:2) d (1 − c ) λEρ (cid:3) M d )2 d (cid:1) (cid:98) d / (cid:99) and hence lim d → + ∞ ∆( d ) = 1 . To finish this proof, we only need to show that Equation (4.29) holds. According to thedefinition of κ , { κ < (cid:98) log d (cid:99)} = {∃ i < (cid:98) log d (cid:99) , j ≥ α j = q i } while {(cid:98) log d (cid:99) ≤ κ < + ∞} ⊆ {∃ i ≥ (cid:98) log d (cid:99) , j ≥ α j = q i } . Therefore, for x (cid:54) = y , (cid:101) P x,y ( κ < + ∞ ) ≤ (cid:101) P x,y ( ∃ i < (cid:98) log d (cid:99) , j ≥ α j = q i ) (4.33)+ (cid:101) P x,y ( ∃ i ≥ (cid:98) log d (cid:99) , j ≥ α j = q i ) . For x, y ∈ Γ , x (cid:54) = y , (cid:101) P x,y ( ∃ i < (cid:98) log d (cid:99) , j ≥ α j = q i ) ≤ + ∞ (cid:88) j =1 (cid:101) P x,y ( α j = x ) + (cid:98) log d (cid:99)− (cid:88) i =1 (cid:88) j ≥ (cid:101) P x,y ( q i = α j ) . and Equation (4.27), (cid:80) dk = d −(cid:98) d log d (cid:99) +1 x ( k ) = 1 while d (cid:88) k = d −(cid:98) d log d (cid:99) +1 α j ( k ) ≥ j ≥ (cid:98) log d (cid:99) . Hence, by Equation (4.26), + ∞ (cid:88) j =1 (cid:101) P x,y ( α j = x ) = (cid:98) log d (cid:99)− (cid:88) j =1 (cid:101) P x,y ( α j = x ) ≤ (cid:98) log d (cid:99)− (cid:88) j =1 sup u ∈ Z d (cid:101) P x,y ( α j = u ) (4.34)= (cid:98) log d (cid:99)− (cid:88) j =1 (cid:101) E x,y (cid:0) | R ( (cid:126)α, n ) | (cid:1) ≤ (cid:98) log d (cid:99) − d − (cid:98) d log d (cid:99) ) − (cid:98) log d (cid:99) . According to Equation (4.27), (cid:80) dk = d −(cid:98) d log d (cid:99) +1 α j ( k ) (cid:54) = (cid:80) dk = d −(cid:98) d log d (cid:99) +1 q i ( k ) when | i − j | > (cid:98) log d (cid:99) , therefore by Equation (4.26), (cid:98) log d (cid:99)− (cid:88) i =1 (cid:88) j ≥ (cid:101) P x,y ( q i = α j )= (cid:98) log d (cid:99)− (cid:88) i =1 (cid:88) j : | j − i |≤(cid:98) log d (cid:99) (cid:101) P x,y ( q i = α j ) ≤ (cid:98) log d (cid:99)− (cid:88) i =1 (cid:88) j : | j − i |≤(cid:98) log d (cid:99) sup u ∈ Z d (cid:101) P x,y ( q i = u ) (4.35)= (cid:98) log d (cid:99)− (cid:88) i =1 (cid:88) j : | j − i |≤(cid:98) log d (cid:99) (cid:101) E x,y (cid:0) | R ( (cid:126)q, n ) | (cid:1) ≤ ( (cid:98) log d (cid:99) − (cid:98) log d (cid:99) + 1)2( d − (cid:98) d log d (cid:99) ) − (cid:98) log d (cid:99) . By Equations (4.34) and (4.35), (cid:101) P x,y ( ∃ i < (cid:98) log d (cid:99) , j ≥ α j = q i ) ≤ d ) d (4.36)for sufficiently large d .For w = ( w , . . . , w d ) ∈ Z d , we use ϑ ( w ) to denote( w d −(cid:98) d log d (cid:99) +1 , . . . , w d ) ∈ Z (cid:98) d log d (cid:99) . Then, by Equation (4.27), {∃ i ≥ (cid:98) log d (cid:99) , j ≥ α j = q i } ⊆ (cid:8) ϑ ( α k (cid:98) log d (cid:99) ) = ϑ ( q k (cid:98) log d (cid:99) ) for some k ≥ (cid:9) . (4.37)According to the definitions of α n and q n , { ϑ ( α k (cid:98) log d (cid:99) ) } k ≥ and { ϑ ( q k (cid:98) log d (cid:99) ) } k ≥ are twoindependent oriented random walks on Z (cid:98) d log d (cid:99) . Then, according to the lemma given in [3]about the first collision time of two independent oriented random walks on the lattice, thereexists M which does not depend on d, x, y that (cid:101) P x,y (cid:0) ϑ ( α k (cid:98) log d (cid:99) ) = ϑ ( q k (cid:98) log d (cid:99) ) for some k ≥ (cid:1) ≤ M (cid:98) d log d (cid:99) . (4.38)By Equations (4.37) and (4.38), (cid:101) P x,y (cid:0) ∃ i ≥ (cid:98) log d (cid:99) , j ≥ α j = q i (cid:1) ≤ M (cid:98) d log d (cid:99) . (4.39)20y equations (4.33), (4.36) and (4.39), (cid:101) P x,y ( κ < + ∞ ) ≤ d ) d + M (cid:98) d log d (cid:99) (4.40)for sufficiently large d , where M does not depend on d, x, y . Equation (4.29) follows fromEquation (4.40) directly.At the end of this subsection, we give the proof of Lemma 4.6. The proof utilizes thefollowing Proposition given in [14]. Proposition 4.7. If A , A , . . . , A n are n arbitrary random events defined under the sameprobability space such that P ( A i ) > for ≤ i ≤ n and q , q , . . . , q n are n positive constantssuch that (cid:80) nj =1 q j = 1 , then P ( + ∞ (cid:91) j =1 A j ) ≥ n (cid:80) i =1 n (cid:80) j =1 q i q j P ( A i (cid:84) A j ) P ( A i ) P ( A j ) . This proposition is Lemma 3.3 of [14] and a detailed proof is given there.
Proof of Lemma 4.6.
For any B ⊆ Γ and (cid:126)l ∈ L n ( B ), we denote by G (cid:126)l the event that (cid:126)l isan infectious path. According to the definition of D ( n, B ), it is easy to check that (cid:8) D ( n, B ) (cid:54) = ∅ i.o. (cid:9) ⊇ (cid:92) n =1 (cid:91) (cid:126)l ∈ L n ( B ) G (cid:126)l . As a result, P λ,d (cid:0) D ( n, B ) (cid:54) = ∅ i.o. (cid:1) ≥ lim n → + ∞ P λ,d (cid:0) (cid:91) (cid:126)l ∈ L n ( B ) G (cid:126)l (cid:1) , (4.41)since (cid:83) (cid:126)l ∈ L n ( B ) G (cid:126)l ⊆ (cid:83) (cid:126)l ∈ L m ( B ) G (cid:126)l for n > m .For each (cid:126)l = ( l , . . . , l n ) ∈ L n ( B ), we define g (cid:126)l = (cid:101) P l (cid:0) (cid:126)α n = (cid:126)l (cid:1) | B | , where (cid:126)α n = ( α , . . . , α n ) , then it easy to check that (cid:80) (cid:126)l ∈ B g (cid:126)l = 1. Then, by Proposition 4.7, P λ,d (cid:0) (cid:91) (cid:126)l ∈ L n ( B ) G (cid:126)l (cid:1) ≥ (cid:80) (cid:126)x ∈ L n ( B ) (cid:80) (cid:126)y ∈ L n ( B ) g (cid:126)x g (cid:126)y P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) = 1 | B | (cid:80) x ∈ B (cid:80) y ∈ B (cid:80) (cid:126)x ∈ L n ( x ) (cid:80) (cid:126)y ∈ L n ( y ) (cid:101) P x (cid:0) (cid:126)α n = (cid:126)x (cid:1) (cid:101) P y (cid:0) (cid:126)q n = (cid:126)y (cid:1) P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) . (4.42)Now we deal with the factor P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) . According to our assumption of the model,the denominator P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) = n − (cid:89) l =0 P λ,d (cid:0) U ( x l , x l +1 ) < Y ( x l ) (cid:1) n − (cid:89) l =1 P λ,d (cid:0) U ( y l , y l +1 ) < Y ( y l ) (cid:1) , (cid:126)x and (cid:126)y are self-avoiding. According to our assumption of the model, for any y i (cid:54)∈ (cid:126)x and x j (cid:54)∈ (cid:126)y , the numerator P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) has factors P λ,d (cid:0) U ( x j , x j +1 ) < Y ( x j ) (cid:1) and P λ,d (cid:0) U ( y i , y i +1 ) < Y ( y j ) (cid:1) , which can be cancelled with the same factors in the denominator.For each l ∈ ζ ( (cid:126)x, (cid:126)y ), there exists 0 ≤ k ≤ n − u, v ∈ Z d that y l = x k = u and y l +1 = x k +1 = v . Then, the numerator P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) has the factor P λ,d (cid:0) U ( u, v ) < Y ( u ) (cid:1) while the denominator P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) has the factor (cid:104) P λ,d (cid:0) U ( u, v ) < Y ( u ) (cid:1)(cid:105) . Hence, P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) has the factor P λ,d (cid:0) U ( u, v ) < Y ( u ) (cid:1)(cid:104) P λ,d (cid:0) U ( u, v ) < Y ( u ) (cid:1)(cid:105) = 2 dλE (cid:0) ρ λρ d (cid:1) for each l ∈ ζ ( (cid:126)x, (cid:126)y ).For each m ∈ σ ( (cid:126)x, (cid:126)y ) \ ζ ( (cid:126)x, (cid:126)y ), there exists 0 ≤ r ≤ n − u , v , v ∈ Z d , v (cid:54) = v that y m = x r = u , y m +1 = v and x r +1 = v . Then, the denominator P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y )has the factor P λ,d (cid:0) U ( u , v ) < Y ( u ) (cid:1) P λ,d (cid:0) U ( u , v ) < Y ( u ) (cid:1) while the numerator has afactor at most P λ,d (cid:0) (cid:101) U ( u , v ) < Y ( u ) , (cid:101) U ( u , v ) < Y ( u ) (cid:1) , where (cid:101) U ( u , v ) , (cid:101) U ( u , v ) areindependent exponential times with rate λ Θ2 d and are independent with Y ( u ). Note that herewe replace U ( u , v ) and U ( u , v ) by (cid:101) U ( u , v ) and (cid:101) U ( u , v ) in case some other event in thenumerator depends on the exponential time U ( v , u ) or U ( v , u ), which are independentwith U ( u , v ) and U ( u , v ) under the quenched measure but positively correlated underthe annealed measure. As a result, for each m ∈ σ ( (cid:126)x, (cid:126)y ), P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) has the factor atmost P λ,d (cid:0) (cid:101) U ( u , v ) < Y ( u ) , (cid:101) U ( u , v ) < Y ( u ) (cid:1) P λ,d (cid:0) U ( u , v ) < Y ( u ) (cid:1) P λ,d (cid:0) U ( u , v ) < Y ( u ) (cid:1) = 2Θ (1 + λ Θ2 d )(1 + λ Θ d ) (cid:2) E ( ρ λρ d ) (cid:3) . Inclusion, P λ,d ( G (cid:126)x (cid:84) G (cid:126)y ) P λ,d ( G (cid:126)x ) P λ,d ( G (cid:126)y ) ≤ (cid:104) dλE (cid:0) ρ λρ d (cid:1) (cid:105) | ζ ( (cid:126)x,(cid:126)y ) | (cid:104) (1 + λ Θ2 d )(1 + λ Θ d ) (cid:2) E ( ρ λρ d ) (cid:3) (cid:105) | σ ( (cid:126)x,(cid:126)y ) \ ζ ( (cid:126)x,(cid:126)y ) | (4.43) ≤ (cid:104) d + λ Θ λEρ (cid:105) | ζ ( (cid:126)x,(cid:126)y ) | (cid:104) ( Eρ ) (cid:105) | σ ( (cid:126)x,(cid:126)y ) \ ζ ( (cid:126)x,(cid:126)y ) | for sufficiently large d .By Equations (4.42), (4.43) and the definitions of σ ( n ) , ζ ( n ), P λ,d (cid:0) (cid:91) (cid:126)l ∈ L n ( B ) G (cid:126)l (cid:1) ≥ | B | (cid:80) x ∈ B (cid:80) y ∈ B (cid:80) (cid:126)x ∈ L n ( x ) (cid:80) (cid:126)y ∈ L n ( y ) (cid:101) P x (cid:0) (cid:126)α n = (cid:126)x (cid:1) (cid:101) P y (cid:0) (cid:126)q n = (cid:126)y (cid:1)(cid:104) d + λ Θ λEρ (cid:105) | ζ ( (cid:126)x,(cid:126)y ) | (cid:104) ( Eρ ) (cid:105) | σ ( (cid:126)x,(cid:126)y ) \ ζ ( (cid:126)x,(cid:126)y ) | = 1 | B | (cid:80) x ∈ B (cid:80) y ∈ B (cid:101) E x,y (cid:16)(cid:0) d + λ Θ λEρ (cid:1) | ζ ( n ) | (cid:0) ( Eρ ) (cid:1) | σ ( n ) \ ζ ( n ) | (cid:17) . σ and ζ ,lim d → + ∞ P λ,d (cid:0) (cid:91) (cid:126)l ∈ L n ( B ) G (cid:126)l (cid:1) ≥ | B | (cid:80) x ∈ B (cid:80) y ∈ B (cid:101) E x,y (cid:16)(cid:0) d + λ Θ λEρ (cid:1) | ζ | (cid:0) ( Eρ ) (cid:1) | σ \ ζ | (cid:17) . (4.44)Lemma 4.6 follows directly from Equations (4.41) and (4.44). In this section, we give the proof of Theorem 2.2.
Proof of Theorem 2.2.
For given λ > Eρ , according to Theorem 2.1, P λ,d (cid:0) C Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) > d . Therefore, λ c ( d ) ≤ λ for sufficiently large d and hencelim sup d → + ∞ λ c ( d ) ≤ λ. Let λ → Eρ , we have lim sup d → + ∞ λ c ( d ) ≤ Eρ . (5.1)According to a similar analysis with that leads to Equation (5.1),lim sup d → + ∞ β c ( d ) ≤ Eρ . (5.2)Now we let λ = γEρ for given γ <
1. For a self-avoiding path (cid:126)l = (
O, l , . . . , l n )on Z d starting at O with length n , P λ,d (cid:0) (cid:126)l is an infectious path (cid:1) = E µ d (cid:104) P λ,ω (cid:0) (cid:126)l is an infectious path (cid:1)(cid:105) = E µ d (cid:104) n − (cid:89) i =0 P λ,ω (cid:0) U ( l i , l i +1 ) < Y ( l i ) (cid:1)(cid:105) = E µ d (cid:104) n − (cid:89) i =0 γρ ( l i ,l i +1 ,ω )2 dEργρ ( l i ,l i +1 ,ω )2 dEρ + 1 (cid:105) = (cid:104) γ dEρ E (cid:0) ρ γρ dEρ (cid:1)(cid:105) n ≤ γ n (2 d ) n . For a self-avoiding path on Z d , each step has at most 2 d choices. Therefore, the number ofself-avoiding paths with length n staring at O is at most (2 d ) n . As a result, P λ,n (cid:0) there exists an infectious path with length n starting at O (cid:1) ≤ (2 d ) n γ n (2 d ) n = γ n . (cid:80) + ∞ n =0 γ n < + ∞ for γ <
1, according to the Borel-Cantelli’s lemma, P λ,n (cid:0) there exist arbitrary long infectious paths starting at O (cid:1) = 0 . (5.3)By Lemma 4.1, { I Ot (cid:54) = ∅ , ∀ t ≥ }⊆ { there exist arbitrary long infectious paths starting at O } , since infectious vertices never die out when and only when there are infinite many verticeshave ever been infected. Then by Equation (5.3), P λ,n (cid:0) I Ot (cid:54) = ∅ , ∀ t ≥ (cid:1) = 0for λ = γEρ with γ < β c ( d ) ≥ γEρ for any γ <
1. Let γ →
1, we have β c ( d ) ≥ Eρ and hencelim d → + ∞ β c ( d ) = 1 Eρ (5.4)combined with Equation (5.2). Theorem 2.2 follows from Equations (5.1) and (5.4) directly. Acknowledgments.
The author is grateful to the financial support from the NationalNatural Science Foundation of China with grant number 11501542 and the financial supportfrom Beijing Jiaotong University with grant number KSRC16006536.
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