Open Access
October, 1992 The Contact Process on Trees
Robin Pemantle
Ann. Probab. 20(4): 2089-2116 (October, 1992). DOI: 10.1214/aop/1176989541


The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter $\lambda$ is varied. For small values of $\lambda$ a single infection eventually dies out. For larger $\lambda$ the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of $\lambda$, and the proof of this is much easier than it is for the contact process on $d$-dimensional integer lattices.) For still larger $\lambda$ the infection converges in distribution to a nontrivial invariant measure. For any $n$-ary tree, with $n$ large, the first of these transitions occurs when $\lambda \approx 1/n$ and the second occurs when $1/2\sqrt{n} < \lambda < e/\sqrt{n}$. Nonhomogeneous trees whose vertices have degrees varying between 1 and $n$ behave essentially as homogeneous $n$-ary trees, provided that vertices of degree $n$ are not too rare. In particular, letting $n$ go to $\infty$, Galton-Watson trees whose vertices have degree $n$ with probability that does not decrease exponentially with $n$ may have both phase transitions occur together at $\lambda = 0$. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.


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Robin Pemantle. "The Contact Process on Trees." Ann. Probab. 20 (4) 2089 - 2116, October, 1992.


Published: October, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0762.60098
MathSciNet: MR1188054
Digital Object Identifier: 10.1214/aop/1176989541

Primary: 60K35

Keywords: contact process , Galton-Watson tree , homogeneous tree , multiple phase transition , periodic tree , tree

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 4 • October, 1992
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