Asymptotic behavior of a stochastic combustion growth process

作者：Ramirez AF^*; Sidoravicius V

来源：Journal of the European Mathematical Society, 2004, 6(3): 293-334.

• random walk
• Green function
• subadditivity

We study a continuous time growth process on the d-dimensional hypercubic lattice Z(d), which admits a phenomenological interpretation as the combustion reaction A + B --> 2A, where A represents 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 hypercubic 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 P-d be the law of such a process and S-d(0)(t) the set of sites visited at time t. We prove that there exists a bounded, non-empty, convex set C-d subset of R-d such that for every epsilon > 0, P-d-a.s. eventually in t, the set S-d(0)(t) is within an epsilont distance of the set [C(d)t], where for A subset of R-d we define [A] := A boolean AND Z(d). 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 C-d is not a ball under the Euclidean norm.

• 出版日期2004-7