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Multiple transition points for the contact process on the binary tree. (English) Zbl 0871.60087
Let $$T^d$$ denote the homogeneous (connected) tree in which each vertex has $$d+1$$ neighbours and let $$(A_t)$$ be the finite contact process on $$T_d$$. By definition, we have the transitions $$A\to A\setminus\{x\}$$ for $$x\in A$$ at rate 1, and $$A\to A\cup\{x\}$$ for $$x\notin A$$ at rate $$\lambda\cdot\#\{y\in A: |y-x|=1\}$$ $$(|y-x|$$ denoting the distance between $$x, y\in T^d)$$. $$(A_t)$$ is said to survive strongly if $$P^{\{x\}}(x\in A_t$$ for arbitrarily large $$t) > 0$$. On the other hand, $$(A_t)$$ survives if $$P^{\{x\}}(A_t\neq 0,$$ $$t\geq 0)>0$$. One says that $$(A_t)$$ dies out if it does not survive, and that it survives weakly if it survives, but does not survive strongly. Critical values $$\lambda_1\leq \lambda_2$$ are defined by the requirement that $$(A_t)$$ survives strongly for $$\lambda_1>\lambda_2$$, survives weakly for $$\lambda_1<\lambda< \lambda_2$$ and dies out for $$\lambda < \lambda_1$$. In the case $$d\geq 3$$, R. Pemantle [ibid. 20, No. 4, 2089-2116 (1992; Zbl 0762.60098)] obtained upper bounds on $$\lambda_1$$ and lower bounds on $$\lambda_2$$ implying that $$\lambda_1<\lambda_2$$. In the present paper it is shown that (for homogeneous trees) in the case $$d=2$$, $$\lambda_1\leq 0.605$$, $$\lambda_2\geq 0.609$$ which implies $$\lambda_1< \lambda_2$$.
Reviewer: K.Schürger (Bonn)

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
contact process; survive strongly; survives weakly; lower bounds
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##### References:
 [1] DURRETT, R. and SCHINAZI, R. 1995. Intermediate phase for the contact process on a tree. Ann. Probab. 23 668 673. Z. · Zbl 0830.60093 · doi:10.1214/aop/1176988283 [2] GRILLENBERGER, C. and ZIEZOLD, H. 1988. On the critical infection rate of the one dimensional basic contact process: numerical results. J. Appl. Probab. 25 1 8. Z. JSTOR: · Zbl 0643.60095 · doi:10.2307/3214228 · links.jstor.org [3] HOLLEY, R. and LIGGETT, T. M. 1978. The survival of contact processes. Ann. Probab. 6 198 206. Z. · Zbl 0375.60111 · doi:10.1214/aop/1176995567 [4] LIGGETT, T. M. 1985. Interacting Particle Sy stems. Springer, New York. Z. [5] LIGGETT, T. M. 1995. Improved upper bounds for the contact process critical value. Ann. Probab. 23 697 723. Z. · Zbl 0832.60093 · doi:10.1214/aop/1176988285 [6] LIGGETT, T. M. 1997. Branching random walks and contact processes on homogeneous trees. Probab. Theory Related Fields. To appear. Z. · Zbl 0867.60092 · doi:10.1007/s004400050073 [7] MADRAS, N. and SCHINAZI, R. 1992. Branching random walks on trees. Stochastic Process. Appl. 42 255 267. Z. · Zbl 0763.60042 · doi:10.1016/0304-4149(92)90038-R [8] MORROW, G., SCHINAZI, R. and ZHANG, Y. 1994. The critical contact process on a homogeneous tree. J. Appl. Probab. 31 250 255. Z. JSTOR: · Zbl 0798.60091 · doi:10.2307/3215251 · links.jstor.org [9] PEMANTLE, R. 1992. The contact process on trees. Ann. Probab. 20 2089 2116. Z. · Zbl 0762.60098 · doi:10.1214/aop/1176989541 [10] STACEY, A. M. 1996. The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1491 1506. Z. · Zbl 0878.60061 · doi:10.1214/aop/1041903203 [11] WU, C. C. 1995. The contact process on a tree: behavior near the first phase transition. Stochastic Process. Appl. 57 99 112. Z. · Zbl 0821.60093 · doi:10.1016/0304-4149(94)00080-D [12] ZHANG, Y. 1996. The complete convergence theorem of the contact process on trees. Unpublished manuscript. · Zbl 0876.60092 · doi:10.1214/aop/1065725187 [13] LOS ANGELES, CALIFORNIA 90024 E-MAIL: tml@math.ucla.edu
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