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July 1996 The complete convergence theorem of the contact process on trees
Yu Zhang
Ann. Probab. 24(3): 1408-1443 (July 1996). DOI: 10.1214/aop/1065725187

Abstract

Consider the contact process on a homogeneous tree with degree $d \geq 3$. Denote by

$$\lambda_c = \inf\{\lambda : P(o \in \xi^o_t \text{i.o.}) > 0\}$$

the critical value of local survival probability, where o is the root of the tree. Pemantle and Durrett and Schinazi both conjectured that the complete convergence theorem should hold if $\lambda >\lambda_c$. Here we answer the conjecture affirmatively. Furthermore, we will show that

$$P(o \in \xi^o_t \text{i.o.}) = 0 \quad \text{at $\lambda_c}.$$

Therefore, the conclusion of the complete convergence theorem cannot hold at $\lambda_c$

Citation

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Yu Zhang. "The complete convergence theorem of the contact process on trees." Ann. Probab. 24 (3) 1408 - 1443, July 1996. https://doi.org/10.1214/aop/1065725187

Information

Published: July 1996
First available in Project Euclid: 9 October 2003

zbMATH: 0876.60092
MathSciNet: MR1411500
Digital Object Identifier: 10.1214/aop/1065725187

Subjects:
Primary: 60K35

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 3 • July 1996
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