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The critical contact process dies out. (English) Zbl 0718.60109

T. E. Harris [Ann. of Probab. 2, 969-988 (1974; Zbl 0334.60052)] introduced the contact process (depending on a parameter \(\lambda >0)\) which is a Markov process whose state space is the set of all subsets of \(Z^ d\). It might be thought of as a model for the spread of infection in a d-dimensional orchard. There exists a critical value \(\lambda_ c=\lambda_ c(d)\) such that, if \(\lambda <\lambda_ c\), the contact process dies out with probability 1, whereas, if \(\lambda >\lambda_ c\), it survives with positive probability. The question of the survival or nonsurvival at the critical value was open until now. In the present paper it is shown that the process dies out if \(\lambda =\lambda_ c(d)\) (d\(\geq 1)\). The authors also obtain the complete convergence theorem for all \(\lambda >0\) and \(d\geq 1\) as well as the shape theorem. The latter says the following. Let t(x) denote the first infection time of \(x\in Z^ d\) (given that at time 0 only the origin is infected). Let \(H_ t\) denote the set of all \(y\in R^ d\) such that, for some \(x\in Z^ d\), t(x)\(\leq t\) and \(\| x-y\| \leq\) \((\| \cdot \|\) denoting the \(L^{\infty}\) norm on \(R^ d)\). Then there exists a convex set \(U\subset R^ d\) such that, for any \(\epsilon >0\), \[ (1- \epsilon)U\subset (1/t)H_ t\subset (1+\epsilon)U\quad eventually, \] almost surely on the event that the contact process survives (given that at time 0 only the origin is infected). The proofs use (of course!) the graphical representation of the contact process, introduced by T. E. Harris [ibid. 6, 355-378 (1978; Zbl 0378.60106)] as well as certain techniques of an unpublished paper of D. J. Barsky, G. R. Grimmett and C. M. Newman entitled “Percolation in half-spaces: Equality of critical probabilities and continuity of the percolation probability” (1989).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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