Asymptotic behavior of a stochastic combustion growth process

Alejandro F. Ramírez ; Vladas Sidoravicius

Journal of the European Mathematical Society, Tome 6 (2004) no. 3, pp. 293-334.

Résumé

We study a continuous time growth process on the d-dimensional hyper-cubic lattice Zd, which admits a phenomenological interpretation as the combustion reaction A+B→2A, where A represent heat particles and B inert particles. This process can be described as an interacting particle system in the following way: at time 0 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 Pd​ the law of such a process and Sd0​(t) the set of visited sites at time t. In this article we prove that there exists a bounded, non-empty, convex set Cd​⊂Rd, such that for every ε>0, Pd​-a.s. eventually in t, the set Sd0​(t) is within an εt neighborhood of the set [Cd​t], where for A⊂Rd we define [A]:=A∩Zd. 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 d large enough, we establish that the set Cd​ is not a ball under the Euclidean norm.

Détail

DOI : 10.4171/jems/11

Classification : 60-XX, 35-XX, 00-XX
Keywords: Random walk, Green function, sub-additivity