Critical value for the contact process with random edge weights on regular tree
aa r X i v : . [ m a t h . P R ] A ug Critical value for the contact process withrandom edge weights on regular tree
Xiaofeng Xue ∗ University of Chinese Academy of Sciences
Abstract:
In this paper we are concerned with contact processes with random edge weights onrooted regular trees. We assign i.i.d weights on each edge on the tree and assume thatan infected vertex infects its healthy neighbor at rate proportional to the weight on theedge connecting them. Under the annealed measure, we define the critical value λ c as themaximum of the infection rate with which the process will die out and define λ e as themaximum of the infection rate with which the process dies out at exponential rate. Weshow that these two critical values satisfy an identical limit theorem and give an preciselower bound of λ e . We also study the critical value under the quenched measure. We showthat this critical value equals that under the annealed measure or infinity according to adichotomy criterion. The contact process on a Galton-Watson tree with binomial offspringdistribution is a special case of our model. Keywords: contact process, regular tree, edge weight, critical value.
In this paper, we are concerned with contact processes with random edge weights on regulartrees. For each integer N ≥
1, we denote by T N the rooted regular tree where the root O has degree N and other vertices have degree N + 1. That is to say, each vertex produces N children and the root O has no ancestor while each other vertex has a father. The followingpicture describes a local area of T . O ❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟ ❏❏❏❏✡✡✡✡ ❉❉❉❉▲▲▲▲☞☞☞☞ ☎☎☎☎❉❉❉❉▲▲▲▲☞☞☞☞ ☎☎☎☎❉❉❉❉▲▲▲▲☞☞☞☞ ☎☎☎☎❉❉❉❉▲▲▲▲☞☞☞☞ ☎☎☎☎ ∗ E-mail : [email protected]
Address : School of Mathematical Sciences, University of ChineseAcademy of Sciences, Beijing 100049, China. x, y ∈ T N , we denote by x ∼ y when there is an edge connecting them.We denote by E N the set of edges on T N .Let ρ be a non-negative random variable such that P ( ρ ≤ M ) = 1 for some M ∈ (0 , + ∞ )and P ( ρ > > { ρ ( e ) } e ∈ E N are i. i. d. random variables such that for each e ∈ E N , ρ ( e ) and ρ have the same probability distribution. For e ∈ E N with endpoints x, y ∈ T N ,we write ρ ( e ) as ρ ( x, y ). When { ρ ( e ) } e ∈ E N is given, the contact process with edge weights { ρ ( e ) } e ∈ E N is a spin system with state space { , } T N and flip rates function given by c ( x, η ) = ( η ( x ) = 1 ,λ P y : y ∼ x ρ ( x, y ) η ( y ) if η ( x ) = 0 (1.1)for any ( x, η ) ∈ T N × { , } T N , where λ is a positive parameter called the infection rate.The assumption P ( ρ ≤ M ) = 1 ensures the existence of our process according to thebasis theory constructed in [5] and [8].Intuitively, the process describes the spread of an infection disease. Vertices in state 1are infected individuals while vertices in state 0 are healthy. An infected individual waitsfor an exponential time with rate 1 to recover. A healthy vertex x is infected by its infectedneighbor y at a rate proportional to ρ ( x, y ). That is to say, the larger ρ ( x, y ) is, the fasterthe disease spreads from y to x .When ρ ≡
1, our model degenerates to the classic contact process, which is introducedin [6] by Harris. In [13], Pemantle first considers contact processes on trees. The two books[9] and [11] written by Liggett give a detailed introduction for the study of classic contactprocesses on lattices and trees.When P ( ρ = 1) = 1 − P ( ρ = 0) = p ∈ (0 , B ( N, p ) and also can beseen as contact process on open clusters of bond percolation on tree. In [14], Pemantle andStacey study contact processes and branching random walks on Galton-Watson trees. Theyshow that on some Galton-Watson trees the branching random walk has one phase transitionwhile the contact process has two. Contact processes on clusters of bond percolation onlattices are studied by Chen and Yao in [3]. They show that the complete convergencetheorem holds.In this paper, we are concerned with contact processes with random edge weights. It isalso interesting to consider the process with random vertex weights. In detail, each vertex x is assigned a weight ρ ( x ). Infected vertex x infects healthy neighbor y at rate proportionalto ρ ( x ) ρ ( y ). This model concludes contact process on clusters of site percolation as a specialcase. In [1], Bertacchi, Lanchier and Zucca study contact processes on C ∞ × K N , where C ∞ is the infinite open cluster of site percolation and K N is a complete graph with N vertices.Criterions to judge whether the process will survive are given in [1]. Contact processes withrandom vertex weights on complete graphs are introduced in [15] by Peterson. In [15], it isshown that the critical value of the model is inversely proportional to the second moment ofthe vertex weight. Xue extends this result to the case where the graph is oriented lattice in[17]. In [16], Xue studies contact processes with random vertex weights on general regulargraphs and obtains a lower bound of the critical value of the model.2 Main results
In this section we give main results of this paper. First we introduce some notations anddefinitions. For each N ≥
1, we assume that { ρ ( e ) } e ∈ E N are defined on the probability space (cid:0) Ω N , F N , µ N (cid:1) . We write (Ω N , F N , µ N ) briefly as (Ω , F , µ ) when there is no misunderstand-ing. For any ω ∈ Ω, we denote by P ωλ the probability measure of the contact process on T N with edge weights { ρ ( e, ω ) } e ∈ E N and infection rate λ . P ωλ is called the quenched measure.The expectation operator with respect to P ωλ is denoted by E ωλ . We define P Nλ ( · ) = Z P ωλ ( · ) µ N ( dω ) , which is called the annealed measure. The expectation operator with respect to P Nλ isdenoted by E Nλ . When there is no misunderstanding, we write P Nλ and E Nλ briefly as P λ and E λ .For any t ≥
0, we denote by η t the configuration of the contact process at moment t .The value of vertex x at moment t is denoted by η t ( x ). For any t >
0, let C t = { x ∈ T N : η t ( x ) = 1 } be the set of infected vertices at t . We write C t as C Ot when C = { O } .Since ∅ is an absorbed state of the process { C t } t ≥ and the contact process is an attractivespin system (see section 3.2 of [9]), for any λ > λ , P λ ( ∀ t ≥ , C Ot = ∅ ) ≥ P λ ( ∀ t ≥ , C Ot = ∅ ) . (2.1)By (2.1), it is reasonable to define the following critical value. For each N ≥
1, we define λ c ( N ) = sup { λ : P Nλ ( ∀ t ≥ , C Ot = ∅ ) = 0 } . (2.2)We write λ c ( N ) as λ c when there is no misunderstanding.Supposing that only O is infected at t = 0, then when λ < λ c , with probability onethere will be no infected vertices eventually, which means that the disease dies out. When λ > λ c , with positive probability there will be always some vertices in the infected state,which means that the disease survives. The case of λ = λ c is difficult. In [2], Bezuidenhoutand Grimmett show that the critical classic contact process on lattice dies out. We guesssame conclusion holds for our model but have not find a way to prove it yet.When λ < λ c , lim t → + ∞ P λ ( C Ot = ∅ ) = 0 . It is natural to ask whether P λ ( C Ot = ∅ ) converges to 0 at an exponential rate. So it isnatural to define the following critical value. For any N ≥
1, we define λ e ( N ) = sup { λ : lim sup t → + ∞ t log P Nλ ( C Ot = ∅ ) < } . (2.3)It is obviously that λ e ≤ λ c . Does λ e equal λ c ? Section 6.3 of [9] shows that the answer ispositive for classic contact process on Z . We have no idea whether λ e = λ c for our model.Now we give our main results. Our first result is a criterion to judge whether λ c ∈ (0 , + ∞ ). 3 heorem 2.1. If P ( ρ >
0) = 1 , then for each N ≥ , < λ c ( N ) < + ∞ . If P ( ρ > < ,then < λ c ( N ) < + ∞ for N > /P ( ρ > and λ c ( N ) = + ∞ for N ≤ /P ( ρ > . We can not judge whether λ c < + ∞ for the case where N = 1 and P ( ρ >
0) = 1. Weguess in this case there is no common conclusion. More information about the distributionof ρ is needed. For example, if there exists ǫ > P ( ρ > ǫ ) = 1, then it is easy tosee that λ c ∈ (0 , + ∞ ) since classic contact process on Z has finite critical value (see Section6.1 of [9] and [10]).To describe λ c and λ e more accurately, we obtain a limit theorem of λ c , λ e and a preciselower bound of λ e . Theorem 2.2.
For ρ satisfies that P ( ρ > > and P (0 ≤ ρ ≤ M ) = 1 for some M ∈ (0 , + ∞ ) , lim N → + ∞ N λ c ( N ) = lim N → + ∞ N λ e ( N ) = 1 Eρ . (2.4)
Furthermore, λ e ( N ) ≥ (cid:0) N Eρ + M Eρ (cid:1) − . (2.5)Theorem 2.2 show that λ c , λ e ≈ / ( N Eρ ), which is inversely proportional to the degreeof the root and the mean of the edge weight.Let us see some examples. When ρ ≡
1, Theorem 2.2 shows thatlim N → + ∞ N λ c ( N ) = 1and λ c ( N ) ≥ / ( N + 1), which is the estimation of critical value for classic contact processon regular tree given in [13].When P ( ρ = 1) = 1 − P ( ρ = 0) = p ∈ (0 , B ( N, p ) that lim N → + ∞ N pλ c ( N ) = 1and λ c ( N ) ≥ N p + 1 /p .
These two estimations do not occur in former references.The critical value λ c is defined under the annealed measure. It is natural to consider thecritical value of the process with fixed edge weights { ρ ( e, ω ) } e ∈ E N for some ω ∈ Ω. Hence,for any ω ∈ Ω N , we define b λ c ( ω, N ) = sup { λ : P ωλ ( ∀ t, C Ot = ∅ ) = 0 } . (2.6)For ω ∈ Ω N , if there is a cut-off Π of T N separating O from infinity such that ρ ( e, ω ) = 0for each e ∈ Π, then it is easy to see that b λ c ( ω, N ) = + ∞ . We can show that except thiscase, b λ c ( ω, N ) = λ c ( N ), which means the critical values under the annealed measure andquenched measure are equal. To introduce our result rigorously, we introduce some notationsand definitions. 4or any ω ∈ Ω N , we define L ( ω ) = { e ∈ E N : ρ ( e, ω ) > } . (2.7)For each x ∈ T N , there is an unique path p ( O, x ) from O to x which does not backtrack.We write O → ω x when and only when each edge of p ( O, x ) belongs to L ( ω ). We define D ( ω ) = { x ∈ T N : O → ω x } (2.8)and A N = { ω : | D ( ω ) | = + ∞} . (2.9)It is obviously that D ( ω ) forms a Galton-Watson tree with offspring distribution B ( N, q )and 1 − µ N ( A N ) is the extinction probability of the tree, where q = P ( ρ > P ( B ( N, q ) = k ) = (cid:18) Nk (cid:19) q k (1 − q ) N − k for 1 ≤ k ≤ N .Now we can give our result of the critical value under the quenched measure. Theorem 2.3. If P ( ρ >
0) = 1 , then for each N ≥ , there exists K N ∈ F N such that µ N ( K N ) = 1 and b λ c ( ω, N ) = λ c ( N ) ∈ (0 , + ∞ ) for any ω ∈ K N , where λ c ( N ) is the same as that in (2.2) .If P ( ρ > < , then when N ≤ /P ( ρ > , b λ c ( ω, N ) = + ∞ for any ω ∈ Ω N . When N > /P ( ρ > , then µ N ( A N ) > and there exists K N ⊆ A N such that µ N ( A N \ K N ) = 0 and b λ c ( ω, N ) = λ c ( N ) ∈ (0 , + ∞ ) for any ω ∈ K N . For any ω A N , b λ c ( ω, N ) = + ∞ . In conclusion, theorem 2.3 shows that b λ c ( ω, N ) ∈ { λ c ( N ) , + ∞} with probability one.Furthermore, { ω : b λ c ( ω, N ) = λ c ( N ) } = A N and { ω : b λ c ( ω, N ) = + ∞} = Ω N \ A N in the sense of ignoring a set with probability 0.The proofs of our main results are divided into three sections. In Section 3, we willgive an upper bound of λ c , which shows that lim sup N → + ∞ N λ c ( N ) ≤ /Eρ . The coreidea is to compare the contact process with a SIR epidemic model. This section also gives5ost part of the proof of Theorem 2.1 except showing that λ c >
0. In Section 4, we willprove that λ e ( N ) ≥ (cid:0) N Eρ + M Eρ (cid:1) − and hence λ c >
0, which accomplishes the proof ofTheorem 2.2 and Theorem 2.1. The main approach is to compare the contact process withthe binary contact path process introduced in [4] by Griffeath. In technique, we need toestimate the number of paths (may backtrack) from O with given length on the tree. Werelate this problem to simple random walk on regular tree. In Section 5, we give the proof ofTheorem 2.3. Our approach is inspired by the classic method of proving extinction criterionfor Galton-Watson trees. λ c In this section we will prove that lim sup N → + ∞ N λ c ( N ) ≤ /Eρ . The following lemma givesan upper bound of λ c ( N ), which is crucial for our proof. Lemma 3.1. If λ satisfies that N E (cid:2) λρ λρ (cid:3) > , then λ c ( N ) ≤ λ. We give the proof of Lemma 3.1 at the end of this section. First we utilize Lemma 3.1to prove that lim sup N → + ∞ N λ c ( N ) ≤ /Eρ . Proof of lim sup N → + ∞ N λ c ( N ) ≤ /Eρ . For γ >
1, let λ = γNEρ , then N E (cid:2) λρ λρ (cid:3) = γEρ E (cid:2) ρ γρNEρ (cid:3) . According to Domination Convergence Theorem,lim N → + ∞ γEρ E (cid:2) ρ γρNEρ (cid:3) = γEρ Eρ = γ > . Therefore, for sufficiently large N and λ = γNEρ , N E (cid:2) λρ λρ (cid:3) > . Therefore, according to Lemma 3.1, λ c ( N ) ≤ γN Eρ for sufficiently large N and hence lim sup N → + ∞ N λ c ( N ) ≤ γEρ . Since γ is arbitrary, let γ → P ( ρ = 1) = 1 − P ( ρ = 0) = p ∈ (0 ,
1] and
N > /p , Lemma 3.1 givesa precise upped bound of λ c ( N ) that λ c ( N ) ≤ N p − , since N E (cid:2) λρ λρ (cid:3) = λNp λ .According to Lemma 3.1, we can also judge whether λ c < + ∞ . Corollary 3.2. If P ( ρ >
0) = 1 , then λ c ( N ) < + ∞ for each N ≥ . If P ( ρ > < , then λ c ( N ) < + ∞ for N > /P ( ρ > and λ c ( N ) = + ∞ for N ≤ /P ( ρ > .Proof. According to Domination Convergence Theorem,lim λ → + ∞ E (cid:2) λρ λρ (cid:3) = P ( ρ > . Therefore, in the case where P ( ρ >
0) = 1 and N ≥ P ( ρ > < N > /P ( ρ > λ → + ∞ N E (cid:2) λρ λρ (cid:3) > λ c ( N ) < λ for sufficiently large λ according to Lemma 3.1. As a result, in these two cases, λ c < + ∞ . For the case where P ( ρ > < N ≤ /P ( ρ > B ( N, P ( ρ > D ( ω ) is finite with probability one and theMarkov process { C Ot } t ≥ is with finite state space { A : A ⊆ D ( ω ) } . Since ∅ is the uniqueabsorption state for { C Ot } t ≥ , the process will be frozen in state ∅ eventually. As a result,for any λ > P ωλ ( ∀ t ≥ , C Ot = ∅ ) = 0for any ω ∈ Ω except a set with probability zero and hence P Nλ ( ∀ t ≥ , C Ot = ∅ ) = 0 . Therefore, λ c > λ for any λ > λ c = + ∞ . At last we give the proof of Lemma 3.1.
Proof of Lemma 3.1.
To control the size of C t from below, we introduce the following SIRepidemic model with random edge weights. Let { ξ t } t ≥ be Markov process with state space7 − , , } T N . At t = 0, ξ ( O ) = 1 and ξ ( x ) = 0 for each other x ∈ T N . For any t ≥
0, wedefine I t = { x ∈ T N : ξ t ( x ) = 1 } , S t = { x ∈ T N : ξ t ( x ) = 0 } ,R t = { x ∈ T N : ξ t ( x ) = − } . Now we can identify ξ t with ( S t , I t , R t ). After the edge weights { ρ ( e ) } e ∈ E N is given, { ( S t , I t , R t ) } t ≥ evolves as follows. For each x ∈ I t , ( S t , I t , R t ) flips to ( S t , I t \ { x } , R t ∪ { x } )with rate 1. For any x, y satisfy that y is a son of x , x ∈ I t and y ∈ S t , ( S t , I t , R t ) flips to( S t \ { y } , I t ∪ { y } , R t ) at rate λρ ( x, y ).Intuitively, 1 , , − y may be infected when and only when its father x is infected. x infects y at rate proportionalto ρ ( x, y ). A removed vertex will stay in the this state forever.For { C t } t ≥ , an infected vertex can infect any healthy neighbor while for { ξ t } t ≥ , aninfected vertex can only infect its sons. For { C t } t ≥ , when an infected vertex become healthy,it may be infected again while for { ξ t } t ≥ , when an infected vertex becomes removed, it willnever be infected again. As a result, according to the approach of basic coupling (see section2.1 of [9]), it is easy to see that I t ⊆ C Ot for any t > λ andedge weights { ρ ( e ) } e ∈ E N . Therefore, P ωλ ( ∀ t, C Ot = ∅ ) ≥ P ωλ ( ∀ t, I t = ∅ )for any ω ∈ Ω and hence P Nλ ( ∀ t, C Ot = ∅ ) ≥ P Nλ ( ∀ t, I t = ∅ ) (3.1)for any t > N ≥ I + ∞ = [ t ≥ I t as the set of vertices which have been infected. I t = ∅ for any t ≥ {∀ t ≥ , I t = ∅} = {| I + ∞ | = + ∞} . (3.2)By (3.1) and (3.2), P Nλ ( ∀ t ≥ , C t = ∅ ) ≥ P Nλ ( | I + ∞ | = + ∞ ) . (3.3)For x ∈ T N and a son y of x , let T be an exponential time with rate λρ ( x, y ) and T be anexponential time with rate 1 and independent of T , then conditioned on x is infected, theprobability that x infects y equals P ( T < T ) = λρ ( x, y )1 + λρ ( x, y ) .
8s a result, under the annealed measure P Nλ , the mean of the number of infected sons of aninfected vertex equals N E (cid:2) λρ λρ (cid:3) . As a result, under the annealed measure P Nλ , I + ∞ forms a Galton-Watson tree with anoffspring distribution with mean N E (cid:2) λρ λρ (cid:3) . According to the extinction criterion of Galton-Watson trees, P Nλ ( | I + ∞ | = + ∞ ) > λ satisfies N E (cid:2) λρ λρ (cid:3) >
1. Lemma 3.1 follows from this fact and (3.3). λ e In this section we will give lower bound of λ e . First, we give a lemma about simple randomwalk on T N for later use.For N ≥
1, we denote by { S Nn } n ≥ simple random walk on T N such that P ( S Nn +1 = y (cid:12)(cid:12) S Nn = x ) = 1deg( x )for each x ∈ T N , each neighbor y of x and n ≥
0. We assume that S N = O . The probabilitymeasure and expectation operator with respect to { S Nn } n ≥ are denoted by e P and e E .We define Γ : T N → Z such that Γ( O ) = 0 and Γ( y ) = Γ( x ) + 1 when y is a son of x . Inother words, for each x ∈ T N there is an unique path p ( O, x ) from O to x which does notbacktrack. Γ( x ) equals the length of p ( O, x ). Lemma 4.1.
For any x ∈ (0 , and n ≥ , e Ex Γ( S Nn ) ≤ (cid:2) N xN + 1 + 1( N + 1) x (cid:3) n . Proof.
According to the definition of S Nn , e P (cid:0) Γ( S Nn +1 ) − Γ( S Nn ) = 1 (cid:12)(cid:12) S Nn = x (cid:1) =1 − e P (cid:0) Γ( S Nn +1 ) − Γ( S Nn ) = − (cid:12)(cid:12) S Nn = x (cid:1) (4.1)= ( x = O, NN +1 if x = O. Let { Z n } n ≥ be a Markov process with state space { . . . , − , − , , , , , . . . } and evolveaccording to { S Nn } n ≥ . In detail, we assume that Z = 0. For n ≥
1, if S Nn = O , then Z n +1 − Z n is independent of S Nn +1 and satisfies P ( Z n +1 − Z n = 1) = 1 − P ( Z n +1 − Z n = −
1) = NN + 1 . If S Nn = O , then Z n +1 − Z n = 1 when Γ( S Nn +1 ) − Γ( S Nn ) = 1 and Z n +1 − Z n = − S Nn +1 ) − Γ( S Nn ) = − n ≥ Z n +1 − Z n ≤ Γ( S Nn +1 ) − Γ( S Nn ). Since Z = Γ( S N ) = 0, Z n ≤ Γ( S Nn )9or each n ≥ x ∈ (0 , e Ex Γ( S Nn ) ≤ Ex Z n . (4.2)By (4.1) and the definition of Z n , it is easy to see that { Z n − Z n − } n ≥ are i. i. d randomvariables such that P ( Z n − Z n − = 1) = 1 − P ( Z n − Z n − = −
1) = NN + 1 . Therefore, Ex Z n = (cid:2) Ex Z − Z (cid:3) n = (cid:2) N xN + 1 + 1( N + 1) x (cid:3) n . (4.3)Lemma 4.1 follows from (4.2) and (4.3).To control P ( C Ot = ∅ ) from above, we introduce the binary contact path process { ζ t } t ≥ with random edge weights on T N . The classic binary contact path process is introduced byGriffeath in [4], which inspires us a lot.The state space of { ζ t } t ≥ is { , , , , . . . } T N . At t = 0, we assume that ζ ( x ) = 1 foreach x ∈ T N .When the edge weights { ρ ( e ) } e ∈ E N are given, { ζ t } t ≥ evolves according to Poisson pro-cesses { N x ( t ) : t ≥ } x ∈ T N and { U ( x,y ) ( t ) : t ≥ } x ∼ y . For any x ∈ T N , N x ( · ) is with rate1. For any x, y such that x ∼ y , U ( x,y ) ( · ) is with rate λρ ( x, y ). Please note that we carethe order of x and y , hence U ( x,y ) = U ( y,x ) . We assume that all these Poisson processes areindependent.For any t > x ∈ T N , we define ζ t − ( x ) = lim s Let { η t } t ≥ be the contact process defined in (1.1) with η ( x ) = 1 for any x ∈ T N .Then, according to an approach of graphical representation introduced in [7], the contactprocess satisfies the dual-relationship that P ωλ ( C Ot = ∅ ) = P ωλ ( η t ( O ) = 1) (4.5)for any ω ∈ Ω and therefore P Nλ ( C Ot = ∅ ) = P Nλ ( η t ( O ) = 1) . (4.6)For readers who are not familiar with the self-duality of contact processes, we give a rigorousproof of (4.5) in the appendix.For any t ≥ x ∈ T N , we define e η t ( x ) = ( ζ t ( x ) ≥ , ζ t ( x ) = 0 . According to the definition of { ζ t } t ≥ , e η ( x ) flips from 1 to 0 at moment s when and onlywhen s is an event time of N x ( · ) and ζ s − ( x ) ≥ 1. So e η ( x ) flips from 1 to 0 at rate 1. e η ( x )flips from 0 to 1 at moment r when and only when ζ r − ( x ) = 0 and r is an event time of U ( x,y ) ( · ) such that y ∼ x and ζ r − ( y ) ≥ 1. Therefore, e η ( x ) flips from 0 to 1 at rate X y : y ∼ x λρ ( x, y )1 { ζ ( y ) ≥ } = X y : y ∼ x λρ ( x, y ) e η ( y ) . As a result, { e η t } t ≥ evolves as the same way as that of { η t } t ≥ .Since η ( x ) = e η ( x ) = 1 for each x ∈ T N , { e η t } t ≥ and { η t } t ≥ have the same probabilitydistribution.Therefore, P Nλ ( η t ( O ) = 1) = P Nλ ( e η t ( O ) = 1) = P Nλ ( ζ t ( O ) ≥ ≤ E Nλ ζ t ( O ) . (4.7)Lemma 4.2 follows from (4.5) and (4.7).Finally, we give the proof of λ e ≥ (cid:0) N Eρ + M Eρ (cid:1) − . Proof of λ e ≥ (cid:0) N Eρ + M Eρ (cid:1) − . It is easy to see that we only need to deal with the casewhere M = 1. For general M > 0, we take e ρ = ρM and denote by e λ e the critical value withrespect to e ρ . Then, λ e = 1 M e λ e ≥ M N E e ρ + E e ρ = (cid:0) N Eρ + M Eρ (cid:1) − . So from now on we assume that P ( ρ ≤ 1) = 1.11ccording to the generator of { ζ t } t ≥ given in (4.4) and Theorem 9.1.27 of [9], for each x ∈ T N and given edge weights { ρ ( e, ω ) } e ∈ E N , ddt E ωλ ζ t ( x ) = − E ωλ ζ t ( x ) + X y : y ∼ x λρ ( x, y, ω ) E ωλ ζ t ( y ) . (4.8)For readers who do not want to check the theorem in [9], an intuitive explanation of (4.8)is that (4.8) is with the form ddt Ef ( ζ t ) = E [ L f ( ζ t )]with f ( ζ ) = ζ ( x ) as an ‘application’ of Hille-Yosida theorem. In fact, Theorem 9.1.27 of [9]is an extension of Hille-Yosida theorem to processes of linear systems.Let G ω be T N × T N matrix such that G ω ( x, y ) = ( λρ ( x, y, ω ) if x ∼ y, I be T N × T N identity matrix, then by (4.8), ddt E ωλ ζ t = ( G ω − I ) E ωλ ζ t . (4.9)Since P ( ρ ≤ 1) = 1 and there are at most N + 1 positive elements in each row of G ω , it iseasy to check that ODE (4.9) satisfies Lipschitz condition under l ∞ norm of R T N and theseries e tG ω = + ∞ X n =0 t n G nω n !converges. Therefore, according to classic theory of linear ODE, the unique solution of ODE(4.9) is E ωλ ζ t = e − t e tG ω ζ . (4.10)Since ζ ( x ) = 1 for each x ∈ T N , by (4.10), E ωλ ζ t ( O ) = e − t + ∞ X n =0 X x : x ∈ T N t n G nω ( O, x ) n ! . (4.11)For n ≥ 1, we say that −→ x = ( x , x , . . . , x n ) ∈ n M j =0 T N is a path starting at O with length n when x = O and x j +1 ∼ x j for 0 ≤ j ≤ n − 1. Pleasenote that a path may backtrack.For n ≥ 1, we denote by L n the set of paths starting at O with length n .Then according to the definition of G ω and (4.11), E ωλ ζ t ( O ) = e − t + ∞ X n =0 t n λ n n ! (cid:16) X −→ x ∈ L n n − Y j =0 ρ ( x j , x j +1 , ω ) (cid:17) , (4.12)12here −→ x = ( x , x , . . . , x n ) and hence E Nλ ζ t ( O ) = e − t + ∞ X n =0 t n λ n n ! (cid:16) X −→ x ∈ L n E n − Y j =0 ρ ( x j , x j +1 , ω ) (cid:17) . (4.13)For −→ x = ( x , x , . . . , x n ) ∈ L n , there is an unique path p ( O, x n ) from O to x n withlength Γ( x n ). In other words, p ( O, x n ) does not backtrack. According to the structure atree, the path −→ x contains all the edges in p ( O, x n ). Since ρ ≤ E n − Y j =0 ρ ( x j , x j +1 , ω ) ≤ E (cid:2) Y e ∈ p ( O,x n ) ρ ( e, ω ) (cid:3) = ( Eρ ) Γ( x n ) . (4.14)Please note that the equation in (4.14) follows from that p ( O, x n ) is formed with Γ( x n )different edges.By (4.13) and (4.14), E Nλ ζ t ( O ) ≤ e − t + ∞ X n =0 t n λ n n ! h X −→ x ∈ L n ( Eρ ) Γ( x n ) i . (4.15)Since each vertex on T N has degree at most N + 1, X −→ x ∈ L n ( Eρ ) Γ( x n ) ≤ ( N + 1) n X −→ x ∈ L n n − Y j =0 x j ) ( Eρ ) Γ( x n ) . (4.16)By the definition of { S Nn } n ≥ , for −→ x = ( x , x , . . . , x n ) ∈ L n , e P ( S Nj = x j , ≤ j ≤ n ) = n − Y j =0 x j )and hence X −→ x ∈ L n n − Y j =0 x j ) ( Eρ ) Γ( x n ) = e E (cid:2) ( Eρ ) Γ( S n ) (cid:3) . (4.17)By (4.16) and (4.17), X −→ x ∈ L n ( Eρ ) Γ( x n ) ≤ ( N + 1) n e E (cid:2) ( Eρ ) Γ( S n ) (cid:3) . (4.18)By (4.15) and (4.18), E Nλ ζ t ( O ) ≤ e − t + ∞ X n =0 t n λ n ( N + 1) n n ! e E (cid:2) ( Eρ ) Γ( S n ) (cid:3) . (4.19)By (4.19) and Lemma 4.1, E Nλ ζ t ( O ) ≤ e − t + ∞ X n =0 t n λ n ( N + 1) n n ! (cid:2) N EρN + 1 + 1( N + 1) Eρ (cid:3) n = exp n t (cid:2) λ ( N Eρ + 1 Eρ ) − (cid:3)o . (4.20)13y Lemma 4.2 and (4.20), P Nλ ( C Ot = ∅ ) ≤ exp n t (cid:2) λ ( N Eρ + 1 Eρ ) − (cid:3)o . Therefore, lim sup t → + ∞ t log P Nλ ( C Ot = ∅ ) ≤ λ ( N Eρ + 1 Eρ ) − < λ < ( N Eρ + 1 Eρ ) − . As a result, λ e ≥ ( N Eρ + 1 Eρ ) − . Now we can complete the proof Theorem 2.1 and Theorem 2.2. Proof of Theorem 2.1. According to Corollary 3.2, we only need to show that λ c > λ c ≥ λ e ≥ ( N Eρ + M Eρ ) − > , the proof is complete. Proof of Theorem 2.2. Since λ e ≥ ( N Eρ + M Eρ ) − ,lim inf N → + ∞ N λ e ( N ) ≥ Eρ . Since λ e ≤ λ c and we have shown thatlim sup N → + ∞ N λ c ( N ) ≤ Eρ in Section 3, lim N → + ∞ N λ e ( N ) = lim N → + ∞ N λ c ( N ) = 1 Eρ and the proof is complete. In this section we discuss the critical value under quenched measure. For later use, weidentify T N with the set { O } [ + ∞ [ m =1 { , , , . . . , N } m . In detail, O is the root of T N . For 1 ≤ j ≤ N , j represents the j th son of O . For m ≥ ≤ j ≤ N and ( k , k , . . . , k m ) ∈ { , , . . . , N } m , ( k , k , . . . , k m , j ) represents the j th son of ( k , k , . . . , k m ). The following picture describesthe first three generations of T . 14 ❍❍❍❍❍❍✟✟✟✟✟✟ ❅❅❅(cid:0)(cid:0)(cid:0)❅❅❅(cid:0)(cid:0)(cid:0)❆❆❆✁✁✁ ❆❆❆✁✁✁ ❆❆❆✁✁✁ ❆❆❆✁✁✁ , 1) (1 , 2) (2 , 1) (2 , , , 1) (1 , , , , , , 2) (2 , , 1) (2 , , 2) (2 , , 1) (2 , , ≤ j ≤ N , we define injection ϕ j : T N → T N such that ϕ j ( O ) = j and ϕ j ( k , k , . . . , k m ) = ( j, k , k , . . . , k m )for each m ≥ k , k , . . . , k m ) ∈ { , , . . . , N } m .For e ∈ E N with endpoints x, y ∈ T N , we denote by e j the edge with endpoints ϕ j ( x )and ϕ j ( y ). For ω ∈ Ω N and j ≥ 1, we denote by ω j the sample point such that ρ ( e, ω j ) = ρ ( e j , ω )for each e ∈ E N . That is to say, if T N is with edge weights { ρ ( e, ω ) } e ∈ E N , then j and itsdescendants form a regular tree which is rooted at j and with edge weights { ρ ( e, ω j ) } e ∈ E N .For any λ > N ≥ ≤ j ≤ N , we define H ( λ, N ) = { ω ∈ Ω N : P ωλ ( ∀ t ≥ , C Ot = ∅ ) = 0 } and H ( λ, N, j ) = { ω ∈ Ω N : P ω j λ ( ∀ t ≥ , C Ot = ∅ ) = 0 } . The following lemma shows that H ( λ, N ) satisfies a zero-one law, which is crucial for us toprove Theorem 2.3. Please note that A N in the lemma is the same as that defined in (2.9). Lemma 5.1. If P ( ρ > < and N > /P ( ρ > , then < µ N ( A N ) < and µ N (cid:0) H ( λ, N ) (cid:1) ∈ { − µ N ( A N ) , } for any λ > .If P ( ρ > 0) = 1 and N ≥ , then µ N (cid:0) H ( λ, N ) (cid:1) ∈ { , } for any λ > .Proof. For any ω ∈ Ω, we define B ( ω ) = { ≤ j ≤ N : ρ ( O, j, ω ) > } as the set of sons which O can infect. 15ccording to the strong Markov property, for 1 ≤ j ≤ N , P ωλ ( ∀ t ≥ , C Ot = ∅ ) ≥ P ωλ ( ∃ t > , j ∈ C Ot ) P ω j λ ( ∀ t ≥ , C Ot = ∅ ) . (5.1)If j ∈ B ( ω ), then P ωλ ( ∃ t > , j ∈ C Ot ) > 0. Therefore, by (5.1), P ωλ ( ∀ t ≥ , C Ot = ∅ ) = 0and j ∈ B ( ω ) implies that P ω j λ ( ∀ t ≥ , C Ot = ∅ ) = 0. As a result, H ( λ, N ) ⊆ { ω : ω ∈ \ j ∈ B ( ω ) H ( λ, N, j ) } . (5.2)Since { ρ ( e ) } e ∈ E N are i.i.d, H ( λ, N, , H ( λ, N, , . . . , H ( λ, N, N ) are independent of B ( ω ) and are i.i.d events which have the same probability distribution as that of H ( λ, N )under µ N . Therefore, by (5.2), µ N (cid:0) H ( λ, N ) (cid:1) ≤ N X k =0 p k h µ N (cid:0) H ( λ, N ) (cid:1)i k , (5.3)where p k = µ N ( ω : | B ( ω ) | = k )= (cid:18) Nk (cid:19) P ( ρ > k (cid:0) − P ( ρ > (cid:1) N − k . For x ∈ [0 , f ( x ) = N X k =0 p k x k . As we have shown in Section 2, D ( ω ) defined in (2.8) is a Galton-Watson tree with binomialoffspring distribution B ( N, P ( ρ > − µ N ( A N ) is the extinction probability of D ( ω ).When P ( ρ > < N > /P ( ρ > B ( N, P ( ρ > − µ N ( A N ) is theunique solution in (0 , 1) to the equation x = f ( x ) and f ( y ) < y for y ∈ (cid:0) − µ N ( A N ) , (cid:1) .By (5.3), µ N (cid:0) H ( λ, N ) (cid:1) ≤ f (cid:16) µ N (cid:0) H ( λ, N ) (cid:1)(cid:17) , hence µ N (cid:0) H ( λ, N ) (cid:1) ∈ [0 , − µ N ( A N )] ∪ { } . (5.4)For any ω ∈ Ω N \ A N , | D ( ω ) | is finite and hence the Markov process { C Ot } t ≥ under themeasure P ωλ is with finite state space { A : A ⊆ D ( ω ) } and unique absorption state ∅ , whichmakes { C Ot } t ≥ frozen in ∅ eventually. As a result, P ωλ ( ∀ t ≥ , C Ot = ∅ ) = 0for any ω ∈ Ω N \ A N and hence µ N (cid:0) H ( λ, N ) (cid:1) ≥ µ N (Ω N \ A N ) = 1 − µ N ( A N ) . (5.5)By (5.4) and (5.5), µ N (cid:0) H ( λ, N ) (cid:1) ∈ { − µ N ( A N ) , } . P ( ρ > 0) = 1 and N ≥ 2, (5.3) turns into µ N (cid:0) H ( λ, N ) (cid:1) ≤ h µ N (cid:0) H ( λ, N ) (cid:1)i N . (5.6)If 0 < µ N (cid:0) H ( λ, N ) (cid:1) < 1, then h µ N (cid:0) H ( λ, N ) (cid:1)i N < µ N (cid:0) H ( λ, N ) (cid:1) since N ≥ 2, which is contradictory to (5.6). Therefore, µ N (cid:0) H ( λ, N ) (cid:1) ∈ { , } . In the case where P ( ρ > 0) = 1 and N = 1, (5.3) turns into µ (cid:0) H ( λ, N ) (cid:1) ≤ µ (cid:0) H ( λ, N ) (cid:1) ,which gives no information. This is why this case should be discussed specially. We proposean open question about the critical value in this case in section 6.Finally we give the proof of Theorem 2.3. Proof of Theorem 2.3. We first consider the case where P ( ρ > 0) = 1 and N ≥ 2. In thiscase, we have shown in the proof of Theorem 2.1 that λ c ( N ) ∈ (0 , + ∞ ) . So we only need to show that b λ c ( ω, N ) = λ c ( N ) with probability one. For m > /λ c ( N ),let λ m = λ c ( N ) − m and β m = λ c ( N ) + m , then according to the definition of λ c ( N ), P Nλ m ( ∀ t ≥ , C Ot = ∅ ) = E Nλ m (cid:2) P ωλ m ( ∀ t ≥ , C Ot = ∅ ) (cid:3) = 0and P Nβ m ( ∀ t ≥ , C Ot = ∅ ) = E Nβ m (cid:2) P ωβ m ( ∀ t ≥ , C Ot = ∅ ) (cid:3) > . Therefore, according to lemma 5.1, µ N (cid:0) H ( λ m , N ) (cid:1) = 1 (5.7)and µ N (cid:0) H ( β m , N ) (cid:1) = 0 . (5.8)Let K N = \ m H ( λ m , N ) \ \ m (cid:0) Ω N \ H ( β m , N ) (cid:1) , then µ N ( K N ) = 1according to (5.7) and (5.8). For ω ∈ K N , P ωλ m ( ∀ t ≥ , C Ot = ∅ ) = 0 , P ωβ m ( ∀ t ≥ , C Ot = ∅ ) > λ m ≤ b λ c ( ω, N ) ≤ β m . m → + ∞ , then we have that b λ c ( ω, N ) = λ c ( N )for ω ∈ K N .Now we deal with the case where P ( ρ > < N > /P ( ρ > λ c ( N ) ∈ (0 , + ∞ )in this case. We also use λ m and β m to denote λ c ( N ) − m and λ c ( N ) + m respectively.According to a similar analysis with that of the first case and Lemma 5.1, µ N (cid:0) H ( λ m , N ) (cid:1) = 1 (5.9)and µ N (cid:0) H ( β m , N ) (cid:1) = 1 − µ N ( A N ) . (5.10)We have shown in the proof of Lemma 5.1 that H ( λ, N ) ⊇ Ω N \ A N for any λ > 0, hence by (5.10), µ N (cid:0) H ( β m , N ) ∩ A N (cid:1) = 0 . (5.11)Let K N = (cid:16) A N \ [ m H ( β m , N ) (cid:17) \ \ m H ( λ, N ) , then K N ⊆ A N and µ N ( A N \ K N ) = 0 according to (5.9) and (5.11).According to a similar analysis with that of the first case, it is easy to see that b λ c ( ω, N ) = λ c ( N )for any ω ∈ K N .For ω ∈ Ω N \ A N , | D ( ω ) | < + ∞ and hence P ωλ ( ∀ t ≥ , C Ot = ∅ ) = 0for any λ > b λ c ( ω, N ) = + ∞ for any ω ∈ Ω N \ A N .For the last case where P ( ρ > < N ≤ /P ( ρ > A N = ∅ according to the extinction criterion of Galton-Watson trees, and hence b λ c ( ω, N ) = + ∞ for any ω ∈ Ω N . 18 An Open question for N = 1 When N = 1, our model turns into the contact process with random edge weights on Z . Wedo not manage to give an criterion to judge whether λ c < + ∞ in this case.There are two trivial cases for this problem. If P ( ρ > < 1, then | D ( ω ) | is finite withprobability one and hence λ c = + ∞ . If P ( ρ > ǫ ) = 1 for some ǫ > 0, then λ c < + ∞ sincethe classic contact process on Z has finite critical value (see [10]). So we only need to dealwith the case where P ( ρ > 0) = 1 but P ( ρ < x ) > x ∈ (0 , P ( ρ > 0) = 1 is sufficient or P ( ρ > ǫ ) = 1 is necessary for finitecritical value. We guess that the probability of { ρ < x } for small x is crucial.To make our question concrete, for α > 0, we assume that P ( ρ < x ) = x α (6.1)for x ∈ (0 , λ c ( α ) the critical value with respect to ρ . Then it is obviouslythat λ c ( α ) ≤ λ c ( α )for α > α .Then it is reasonable to ask the following question. Question 6.1. We assume that N = 1 and ρ has the distribution as that in (6.1) . Then isthere a critical value < α c < + ∞ such that λ c ( α ) < + ∞ for α > α c and λ c ( α ) = + ∞ for α < α c ? If the answer to Question 6.1 is positive, a further problem is how to estimate α c , whichwill bring more interesting work to do. We will work on Question 6.1 as a further study andhope to discuss with readers who are interested in this question. A Appendix Proof of (4.5) . According to the flip rate functions given by (1.1), the Markov process { C t } t ≥ is with state space 2 T N := { A : A ⊆ T N } and has generator given by L f ( A ) = X x : x ∈ A (cid:2) f ( A \ x ) − f ( A ) (cid:3) + X x : x ∈ A X y : y ∼ x λρ ( x, y ) (cid:2) f ( A ∪ { y } ) − f ( A ) (cid:3) (A.1)for f ∈ C (2 T N ) and A ⊆ T N .We define H : 2 T N × T N → { , } that H ( A, B ) = ( A ∩ B = ∅ , A ∩ B = ∅ (A.2)for A, B ⊆ T N . 19y (A.2), H ( A, B ∪ C ) = H ( A, B ) H ( A, C ) (A.3)for A, B, C ⊆ T N .By (A.1), (A.3) and direct calculation, L H ( A, · )( B ) = X x : x ∈ B (cid:2) H ( A, B \ x ) − H ( A, B ) (cid:3) + X x : x ∈ B X y : y ∼ x λρ ( x, y ) (cid:2) H ( A, B ∪ { y } ) − H ( A, B ) (cid:3) = X x : x ∈ B H ( A, B \ x ) (cid:2) − H ( A, { x } ) (cid:3) + X x : x ∈ B X y : y ∼ x λρ ( x, y ) H ( A, B ) (cid:2) H ( A, { y } ) − (cid:3) = X x : x ∈ A ∩ B H ( A, B \ x ) − X x : x ∈ B X y : y ∼ x,y ∈ A λρ ( x, y ) H ( A, B ) (A.4)for A, B ⊆ T N . According to a similar calculation, L H ( · , B )( A ) = X x : x ∈ A ∩ B H ( A \ x, B ) − X y : y ∈ A X x : x ∼ y,x ∈ B λρ ( x, y ) H ( A, B )= X x : x ∈ A ∩ B H ( A \ x, B ) − X x : x ∈ B X y : y ∼ x,y ∈ A λρ ( x, y ) H ( A, B ) . (A.5)It is easy to see that H ( A, B \ x ) = H ( A \ x, B )for x ∈ A ∩ B . Therefore, by (A.4) and (A.5), L H ( · , B )( A ) = L H ( A, · )( B ) (A.6)for A, B ⊆ T N .We write C t as C At when C = A . Then, according to (A.6) and Theorem 3.39 of [12], E ωλ H ( A, C Bt ) = E ωλ H ( C At , B ) (A.7)for A, B ⊆ T N and t ≥ 0. Let A = { O } and B = T N , then (4.5) follows from (A.7). Acknowledgments. The author is grateful to the financial support from the NationalNatural Science Foundation of China with grant number 11171342 and China PostdoctoralScience Foundation (No. 2015M571095). References [1] Bertacchi, D., Lanchier, N. and Zucca, F. (2011). Contact and voter processes on theinfinite percolation cluster as models of host-symbiont interactions. 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