We study a continuous time growth process on the -dimensional hyper-cubic lattice , which admits a phenomenological interpretation as the combustion reaction , where represent heat particles and inert particles. This process can be described as an interacting particle system in the following way: at time a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hyper-cubic lattice; then, as soon as any random walk visits a site previously unvisited by any other random walk, it creates a new independent simple symmetric random walk starting from that site. Let us call the law of such a process and the set of visited sites at time . In this article we prove that there exists a bounded, non-empty, convex set , such that for every , -a.s. eventually in , the set is within an neighborhood of the set , where for we define . Furthermore, answering questions posed by M. Bramson and R. Durrett, we prove that the empirical density of particles converges weakly to a product Poisson measure of parameter one, and moreover, for large enough, we establish that the set is not a ball under the Euclidean norm.
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Alejandro F. Ramírez, Vladas Sidoravicius, Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. 6 (2004), no. 3, pp. 293–334DOI 10.4171/JEMS/11