Abstract
What is in this chapter? The contact process has the same birth and death rates as a particular branching Markov chain for which births occur only on nearest neighbor sites. The difference between the two models is that there is at most one particle per site for the contact process while there is no bound in the number of particles per site for branching Markov chains. So, branching Markov chains and the contact process may be thought of as two extreme points in the same class of models.
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Notes and references
The contact process is an example of an interacting particle system. There are two excellent books on this subject, Liggett (1985) and Durrett (1988). The contact process (on Zd) was first introduced by Harris (1974). See Durrett (1991) for a recent account of the contact process on Zd. Pemantle (1992) started the study of the contact process on trees and proved that there are two phase transitions for all d ≥ 4. Very recently Liggett (1996) and Stacey (1996) have independently proved that there are two phase transitions for d = 3 as well.
Observe that our comparisons with branching Markov processes do not allow us to get an upper bound for λ2 but it is possible to do so (see Pemantle (1992)).
Theorem VII.2.1 is due to Madras and Schinazi (1992), Theorems VII.2.2 and VII.2.3 are due to Morrow, Schinazi and Zhang (1994).
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© 1999 Springer Science+Business Media New York
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Schinazi, R.B. (1999). The Contact Process on a Homogeneous Tree. In: Classical and Spatial Stochastic Processes. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1582-0_7
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DOI: https://doi.org/10.1007/978-1-4612-1582-0_7
Publisher Name: Birkhäuser, Boston, MA
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