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Vol. 15 (2010), Paper no. 73, pages 2220–2260. Journal URL http://www.math.washington.edu/~ejpecp/
Multitype Contact Process on Z: Extinction and Interface Daniel Valesin 1
Abstract We consider a two-type contact process on Z in which both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval [−L, L] and the other type occupies infinitely many sites both in (−∞, L) and (L, ∞). Additionally, we show that if both types are present in finite number in the initial configuration, then there is a positive probability that they are both present for all times. Finally, it is shown that, starting from the configuration in which all sites in (−∞, 0] are occupied by type 1 particles and all sites in (0, ∞) are occupied by type 2 particles, the process ρ t defined by the size of the interface area between the two types at time t is tight.
Key words: Interacting Particle Systems, Interfaces, Multitype Contact Process. AMS 2000 Subject Classification: Primary 60K35. Submitted to EJP on May 11, 2010, final version accepted November 30, 2010.
1
Institut de Mathématiques, Station 8, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland,
[email protected], http://ima.epfl.ch/prst/daniel/index.html Acknowledgements. The author would like to thank Thomas Mountford, Augusto Teixeira, Johel Beltran and Renato Santos for helpful discussions and the anonymous referee for his careful and detailed comments on the first version of this paper.
2220
1 Introduction The contact process on Z is the spin system with generator X Ω f (ζ) = ( f (ζ x ) − f (ζ)) c(x, ζ); x
ζ ∈ {0, 1}Z
where ¨
ζ x ( y) = ζ( y) if x 6= y; ζ x (x) = 1 − ζ(x);
¨ c(x, ζ) =
λ
P
1 if ζ(x) = 1; ζ( y) · p( y − x) if ζ(x) = 0; y
for λ > 0 and p(·) a probability kernel. We take p to be symmetric and to have finite range R = max{x : p(x) > 0}. The contact process is usually taken as a model for the spread of an infection; configuration ζ ∈ {0, 1}Z is the state in which an infection is present at x ∈ Z if and only if ζ(x) = 1. With this in mind, the dynamics may be interpreted as follows: each infected site waits an exponential time of parameter 1, after which it heals, and additionally each infected site waits an exponential time of parameter λ, after which it chooses, according to the kernel p, some other site to which the infection is transmitted if not already present. We refer the reader to [13] for a complete account of the contact process. Here we mention only the most fundamental fact. Let ζ¯ and 0 be the configurations identically equal to 1 and 0, respectively, S(t) the semi-group associated to Ω, Pλ the probability measure under which the process has rate λ and ζ0t the configuration at time t, started from the configuration where only the origin is infected. There exists λc , depending on p, such that • if λ ≤ λc , then Pλ (ζ0t 6= 0 ∀t) = 0 and δζ¯ S(t) → δ0 ; • if λ > λc , then Pλ (ζ0t 6= 0 ∀t) > 0 and δζ¯ S(t) converges, as t → ∞, to some non-trivial invariant measure. Again, see [13] for the proof. Throughout this paper, we fix λ > λc . The multitype contact process was introduced in [15] as a modification of the above system. Here we consider a two-type contact process, defined as the particle system (ξ t ) t≥0 with state space {0, 1, 2}Z and generator X Λ f (ξ) = ( f (ξ x,0 ) − f (ξ))+ x:ξ(x)6=0
X x: