COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS ON Td Z

2004,24B(4):513-518M ahentaCta 2intia数学物理学报COMPLETE CONVERGENCE THEORY OF THECONTACT PROCESS ON Tax Z 1Jia Shumei(贾淑梅)School of Mathematical Sciences, Peking University, Beijing 100871, ChinaE-mail: shmjia@hotmail.comAbstractThe author considers the contact process on a branching plane Td x Z, whichis the product of a regular tree Ta and the line Z. It is shown that above the second criticalpoint, the complete convergence theory holds.Key words Contact process, Ta x Z, complete convergence theory2000 MR Subject Classification60G071 Introduction and Main ResultsBefore 1990, the contact process was mostly studied on the d-dimensional cubic latticeZd. The ergodic behavior of the contact process on Zd was completely characterized in thefundamental paper by Bezuidenhout and Grimmett[1, Interest in the behavior of the contactprocess on trees was raised by Pemantle[2]. It was shown by Pemantle[2} and Stacy[3] that thecontact process on homogeneous trees Ta(d 2 2), where every vertex has exactly d+ 1 neighbors,has at least two diferent critical points and between them the system can survive in a globalsense, but not in a local sense. From that papers on, a lot of papers on the problems on treesappears. In particular, Zhangl4] and Salzano & Schonmann[5] solved the complete convergencetheorem on trees with two diferent methods. But little has been known for graphs other thantrees and lattices. Cayley graph is another graph we focused on. Some conjectures on thecritical points and the complete convergence theorem have been made on it. But none of themhas been proved. These are the motivations for us to study the contact process on the branchingplane Tdx Z, for it is a Cayley graph but different from Td and Zd.A vertex in Tdx Z is denoted as v = (u,z), v∈Ta,z∈Z. Two vertices v = (v,z) andv' = (v",z') are connected by an edge, if one of the following satisfies:1) U and v' are connected by an edge in Td; z= z';2) v=v'; z and z' are connected by an edge in Z.The distance |v1 - v2| between two vertices v1 and V2 is defined to be the number of edgesof the shortest path between them. If |v1 - v2| = 1, we call V1 and V2 are neighbors. Forsimplicity; let |v| denote the distance between v and the root. Let (0,0) be a distinguishedvertex of Ta x Z, which we call the root. Correspondi中国煤化工!rootofTa..1 Received July 16, 2001; revised June 11, 2003. Research waTYHCNMHG; G1999075106 fromthe Ministry of Science and Technology of China.514ACTA MATHEMATICA SCIENTIAVol.24 Ser.BLet G be any connected infinite subgraph ofTa x Z. Without confusion, we shall also use Gto denote the set of vertices of the graph G. The contact process on the graph G with infectionparameter λ > 0 is a continuous time Markov Process with state space {0, 1}G. Elements ofthis state space are called configurations. We will denote by (5G. : t≥0) the contact processon the graph G starting from the configuration n. The vertices in ξc .t are thought of as infectedand the system evolves as follows:(1)Ifv∈ξG.t then v becomes healthy at rate 1.(2) If v∈ζGe, then v is infected at rate入times the number of occupied neighbors.IfG = Tdx Z, we denote ft,t = 542. More specially, we are interested in the process ξt andξ{0.0), where ξt is the process with ζb = Tax Z and≤{00) is the process with (0.0) = {(0,0)}.Setλ1 = inf{\:P(lξ{°)|≥1 for allt>0)>0},λz = inf{\: P(0)((0,0)=1 for an arbitrary larget) > 0}, .where入1 and入2 are the critical values for the global survival and local survival of the contactprocess, respectively.Let 8o be the measure concentrated on the empty configuration, that is, all the vertices inTd x Z are uninfected. Clearly δo is a stationary measure. Following from a simple argument(see Liggtt(6l), we know that ξt > ξb, where→denote weak convergence, and ξ is anotherstationary measure. It is another stationary measure. We call it the upper invariant measure.For both Zd and Ta, δ and are the only two extremal invariant measures when λ >入2;and there are graphs whose extremal invariant measures are more than δo and ξ战when入>入2(M. Salzano & R.H.Schomann[7). For the branching planes, we showTheoremWhen λ > λ2 the complete convergence theorem holds, i.e., for any A∈Tax Z,ξ$→P(r^<∞)+ P(r^=∞)5 as t→∞,where rA = inf{t:5A= }.Before our proofs, we shall make some introduction to the standard graphical constructionof the contact process. We associate each site v∈Ta x Z with d + 4 independent Poissonprocesses, one with rate 1, and d + 3 others with rate入. Let {Ty,k :n≥1},k = 0,1,2,...,d+3be the arrival times of these d + 4 processes, respectively; the process {TY,0 : n≥1} has rate1, the others have rate 入. For each v and n≥1 we write a δ mark at the point (v,Ty,0) toindicate that a death will occur if v is occupied; while ifk≥1 we draw arrows from (v, Tyk)to (v(6), Ty*) to indicate if v is occupied then there will be a birth from v to v(k), wherev(k),k: = 1,2...d + 3 are the neighbors of v arranged in some arbitrary order. We say thatthere is a path from (v, s) to (y,t) if there is a sequence of timess= So < 81... < Sn < Sn+1 =tand spatial locations v = Vo,V1..,Vn = y so that for i = 1,2,..,n there is an arrow fromVi-1 to Vi at time 8i and the vertical segments {v;} x_ (Si,Si+1) do not contain any δ for. i= 0,1,...n. We will say that the path is inside G中国煤化工utained in G.Given a tree Ta, we inmerse the graph Z+YHCN MH G2,} and edgesconnecting points which differ by one unit into the tree. The vertex i in z+ is an arbitraryneighbor of i- 1, and diferent from {0,..,i - 1}. Removing one of the neighbors of the root,No.4Jia: COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS515we define the remaining connected component which contains the root as Td . We supposethat the removed vertex is not the vertex 1, so the set of sites {0,1,2,..} is contained in Tt .For any U∈Tt, if we remove the vertex who is adjacent to u and stays nearer to the rootthan U, Td is broken into two parts; we denote the connected part which contains U as Td (v).The distance |U1 一v2| between two vertices U1,V2 ∈Ta is defined to be the number of edgesof the shortest path between them. And |u| is the distance between U and the root 0. LetB(x,N)={v∈Tu:|u-x|≤N}. We use the abbreviation B(N) = B(0, N). For a branchingplane Tax Z, we use B(x,N) to denote B(x, N) x Z, correspondingly B(N)= B(N)x Z.2 Complete Convergence TheoremDeine Un = P((O)((n,0)) = 1 for somet≥0). From the inequality Un+m≥UnUm; itfollows that limn- +oo(un)1/n = a= a(λ) exists.Define Yn, = P(there is a path from ((0, 0),0) to ((n, );s) inside Tt x Z).Lemma 1 There exists a sequence (s(n))n21 such that limn- +o(Yn,s(n)1/n= a.Proof It is clear that for any sequence s(n),lim sup(Yn,s(n)1/n≤lim sup(un)1/n = a.n→∞Define .Vm,k = P(There is a path from ((0,0),0) to ((m,0),t) for some t≤k inside B(k))= P():((m,0))=1 for somet≤k).Clearlylim Vm,k = Um.(2.1)k→∞oNext defineWn,k = P(there is a path from ((0, 0),0)to ((n, 0),t) for somet≤kn inside Td x Z)= P(Ex:((n,0))=1for somet≤kn).We will argue nextWn,k≥Cm,(Vm,k)[n/m]-1,(2.2)where Cm,k is a positive quantity which does not depend on n. To prove this, we set I =min{i∈Z+ : im > k} = min{i∈Z+ : B(im,k) C Tt). Consider now the sequence of sitesx1 = Im,x2= (I+ 1)m,..,xJ =「n/m lm. Define Ti as the first time that there is a path from(xi-1,0) to (x;,0) after Ti-1; Let πo= 1. Clearly, the random times {Ti : i≥1} are stopping .times. Suppose that all of the following events occur:A. There is a path go from ((0,0),0) to ((x1 ,0), 1) without exiting Tt x Z.B. For each i = 0,...,J- 1, there is a path from (x;,0) to (xi+1,0) beginning at time Tiand ending at time Ti+1 that does not exit B(x;,k), al中国煤化工Since the times Ti are stopping times and the ev:0HCNMH(_non-overlappingparts of the percolation structure, it follows by the Ul alslaoll livarlauice of the graphicalconstruction and the strong Markov property of the underlying Poisson processes that the516.ACTA MATHEMATICA SCIENTIAVol.24 Ser.Bprobability of the existence of a path from the root (0,0) to (xJ, 0) is at least Cm,e(Vm,k)[n/m1-',where Cm,k is the probability that A occurs. HenceWn,k= P(0:0) ((n,)= 1 for somet s kn),T xZ;t2 P(°2;(([n/m]m,))= 1 for somet≤krn),≥Cm,(nm,x)Tn/m1-1.DefineWn,k=。max. P(there is a path from ((0, 0),0) to ((n,),t)0≤jSnk-1inside Tt x Z for somet∈[i,j + 1])=。max. , P(2(,0))=1 for somet∈[,j+1).0 1/va, then inft>o P(5(0.0)sL:2:(0))=1)>0.Proof Suppose that a(A) > 1/vd. Then according to Lemma 1, we can take n and 8such that(Yn,)2/n=a> 1/va.(2.4)We will show that for a proper choice of a positive integer l,= in.. P(gE2x2(0,0))=1)>0.(0.0)(2.5)This clearly suffices for our purposes, since thenp1 ((0,0)int P(5yzxz;(0,0))=1)≥e-2le i=on... P(rxZx:(0,0))=1)>0.In order to prove (2.5), we consider now the following modification of the contact process onTt x Z. Until time s, we run the usual contact process on Tt x Z, starting from a single particleat the root. At time s, we remove all particles except those which are in (B(n)\ B(n- 1))x {0};from this time on, we keep the set B(n - 1) free of particle bhnt. nIntil time 2s we let the systemevolve in the remaining vertices with the usual conta中国煤化工ne 2s we removeall particles except those which are in (B(2n) \ B(2nMHCNMHGimeon,wekeepthe set B(2n- 1) free of particles, but until time 3s we let the system evolve in the remainingvertices with the usual contact process rules. The modification should now be clear. For eachNo.4Jia: COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS517j, at time js, we remove all particles except those which are in (B(jn) \ B(jn- 1))x {0}; fromtime js on, we keep the set B(jn - 1) free of particles, but until time (j + 1)s, we let the systemevolve in the remaining vertices with the usual contact process rules.If we only pay our attention to this modified process at time js, and define the modifiedprocess at time js to be a process {Zj，j = 0,1,2,-}. It is clearly that all the particlesof Zj are in (B(jn) \ B(jn- 1))x {0}. Let |Zj;| be the number of particles in this modifiedprocess at time js. We can see that {|Zj|, j = 0,1,2,.} is a branching process with meanoffspring number d"Yn,s = (da)". Since the offspring distribution has a finite support (namely{0, 1,., d"}), it follows from standard branching process theory that for some random variableXwithmeanEX=1,|Z;|lim;=Xa.s..j→∞(da)inIn particular there ise > 0 such that(da)n.P(|Z|≥~)≥e,(2.6)for all large enough l.Choose l large enough for (2.6) to hold, and also so that(1- an()da)"/≤t(2.7)This last requirement can be fulflled because(da)"1→∞lim(1- =an1)a)"/z≤lim exp(-Cana F)= lim exp(-(da2)川)= 0,I→∞since da2 > 1, by (2.4).Now defineri= P(xz2i((0,0)) = 1). We shall show inductively in i thatr;≥fan",(2.8)verifying therefore the validity of (2.5). For i = 0 inequality (2.8) is clearly true, and we willshow that if it is true for i, it is true for i + 1. From the Markov property and attractivenessof the contact process we haveri+1 = P(rTzx2:2(+1e(0,0)=1)≥P(5rt x:(+1)o((x,0)) =1 for somex∈B(nl)\ B(nl - 1))YIn,lo≥P(x(+1)2((x,)) = 1 for some x∈B(nl)\ B(nl - 1)(Yn,o).It is clear that the contact process on Tt x Z dominates {Zj;j≥1} in the usual sense that ithas a particle at any space-time location in which this other one has a particle. The same istrue if we consider a somewhat different modification of the contact process on Td x Z, in whichuntil time ls we make the same modifications as before, but from this time on we keep only thesites in B(nl- 1) free of particles, and we let the syst中国煤化工ing vertices withthe usual contact process rules. Using these observati:YHCNMHGveyieldsri+1≥P(Z\>(ag)-).(1-(1-r))-(n.o)'≥ξa",(da)nl518ACTA MATHEMATICA SCIENTIAVol.24 Ser.Bwhere in the second inequality we have used (2.6), the induction hypothesis (2.8), (2.7) and(2.4).Lemma 3(Salzano and Schonmann5l) For each infinite connected graph G and eachvalue λ > 0, let 0 be a distinguished vertex of G, which we call the root. For the contact processon G, limn- +∞lim inft→∞P(Efo,M∩R(o,N)≠φ) = 1, is equivalent to having simultaneously(a) and (b), where(a) P(NA)>0,SA= {w:o∈ξA,i.o. };b) For any finite set A C G, ξA→(1一P(NA))8φ + P(NA)vr, where vr satisfies vr(5 :5∩A≠刺= P(N4);R(o,N) is the ball of center o and radius N in G .Lemma 4(Salzano and Schonmann5l) Suppose that for a graph G, there exist aδ > 0,sothatforeveryx∈G,P(o∈5ξforsomet≥0)≥δ>0,ThenP(o∈5A,i.o.}=P(Vt>0,|5I≥1) hold.Proof of Theorem Whenλ> 入z, we have a(\)= 1>六So(fP(2((0)=1)>0(2.9)by Lemma 3. Using (2.9) for anyx∈Tdx Z,P(ξ((0,0))= 1 for somet> 0)≥inftzo P(ξ*(x)= 1) .= inf2o P(90)((0,0)) = 1)(2.10)≥inf2zo P(S:2:((0,0))=1)> 0.Considering the contact process starting from B(0,N) x {0}, we havelimN-+∞lim inf- +.o P(sB(0,N)x{0}∩(B(0,N)x {0})≠电)≥limN→∞lim inf; +. P(3v∈B(0,N),≤ζuv0)xzx∩{(v,0)}≠ 動(2.11)≥limn. +∞1-(1 - inf2o P(q27,((0,0))= 1)~= 1.The last equality of (2.11) comes from (2.9). Finally Theorem comes from (2.10), (2.11), Lemma3 and Lemma 4.References1 Bezuidenhout C, Grimmett G. The critical contact process dies out. Ann Probab, 1990, 18: 1462- 14822 Pemantle R. The contact process on trees. Ann Probab, 1992, 20: 2089-21163 Stacy A M. The existence of an intermediate phase for contact process on trees. Ann Probab, 1996, 24:1711-17264 Zhang Y. The complete convergence theorem of the contact process on trees. Ann Probab, 1996, 24:1408-14435 Salzano M, Schonmann R H. A new proof that for the contact process on homogenous trees local surviveimplies complete convergence. Ann Probab, 1998, 26:中国煤化工6. Liggett T M. Interacting Partical Systems. New York:MH7 Salzano M, Schonmann R H. The second lowest extremaCN M H Gontact proess. AnnProbab, 1997, 25: 1846-18718 Durrett R. Probability: Theory and examples. Menlo Park, CA: Wadsworth and Brooks Cole, 1991