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The existence of an intermediate phase for the contact process on trees. (English) Zbl 0878.60061
Summary: Let $$\mathbb{T}_d$$ be a homogeneous tree in which every vertex has $$d$$ neighbors. A new proof is given that the contact process on $$\mathbb{T}_d$$ exhibits two phase transitions when $$d\geq 3$$, a behavior which distinguishes it from the contact process on $$\mathbb{Z}^n$$. This is the first proof which does not involve calculation of bounds on critical values, and it is much shorter than the previous proof for the binary tree, $$\mathbb{T}_3$$. The method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties, for which no such result was previously known.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
contact process; tree; multiple phase transition
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##### References:
 [1] BEZUIDENHOUT, C. and GRIMMETT, G. 1990. The critical process dies out. Ann. Probab. 18 1462 1482. Z. · Zbl 0718.60109 · doi:10.1214/aop/1176990627 [2] DURRETT, R. 1980. On the growth of one-dimensional contact processes. Ann. Probab. 8 890 907. Z. · Zbl 0457.60082 · doi:10.1214/aop/1176994619 [3] DURRET, R. 1988. Lecture Notes on Particle Sy stems and Percolation. Wadsworth, Pacific Grove, CA. Z. [4] 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 [5] HARRIS, T. E. 1978. Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355 378. Z. · Zbl 0378.60106 · doi:10.1214/aop/1176995523 [6] LIGGETT, T. M. 1985. Interacting Particle Sy stems. Springer, New York. Z. [7] LIGGETT, T. M. 1996. Multiple transition points for the contact process on the binary tree. Ann. Probab. 24 1455 1490. Z. · Zbl 0871.60087 · doi:10.1214/aop/1041903202 [8] MADRAS, N. and SCHINAZI, R. 1992. Branching random walks on trees. Stochastic Process. Appl. 42 255 267. · Zbl 0763.60042 · doi:10.1016/0304-4149(92)90038-R [9] 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 [10] PEMANTLE, R. 1992. The contact process on trees. Ann. Probab. 20 2089 2116. Z. · Zbl 0762.60098 · doi:10.1214/aop/1176989541 [11] TRETy AKOV, A. Y. and KONNO, N. 1996. Phase transition of the contact process on the binary tree. Preprint. Z. · Zbl 0972.82510 · doi:10.1143/JPSJ.64.4069 [12] WILLIAMS, D. 1991. Probability with Martingales. Cambridge Univ. Press. · Zbl 0722.60001 · doi:10.1017/CBO9780511813658
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