Contact process (mathematics)
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The contact process is an example of interacting particle systems. It is a continuous time Markov process with state space {0,1}S, where S is a finite or countable graph, usually Zd. The process is usually interpreted as a model for the spread of an infection: if the state of the process at a given time is η, then a site x in S is "infected" if η(x) = 1 and healthy if η(x) = 0. Infected sites become healthy at a constant rate, while healthy sites become infected at a rate proportional to the number infected neighbors. One can generalize the state space to 
, such is called the multitype contact process. It represents a model when more than one type of infection is competing for space.
More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site x,
where the sum is over all the neighbors in S of x. This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and viceversa).
For each graph S there exists a critical value λc for the parameter λ so that if λ > λc then the 1's survive (that is, if there is at least one 1 at time zero, then at any time there are ones) with positive probability, while if λ < λc then the process dies out. For contact process on the integer lattice, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out at the critical value. Their proof makes use of percolation theory.
[edit] References
- Thomas M. Liggett, "Interacting Particle Systems", Springer-Verlag, 1985.
- Thomas M. Liggett, "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999.
- C. Bezuidenhout and G. Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990), 1462 -- 1482.
