JournalsjemsVol. 6 , No. 3DOI 10.4171/jems/11

Asymptotic behavior of a stochastic combustion growth process

  • Alejandro F. Ramírez

    Pontificia Universidad Católica de Chile, Santiago, Chile
  • Vladas Sidoravicius

    Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Asymptotic behavior of a stochastic combustion growth process cover

Abstract

We study a continuous time growth process on the dd-dimensional hyper-cubic lattice Zd\Bbb Z^d, which admits a phenomenological interpretation as the combustion reaction A+B2AA+B\to 2A, where AA represent heat particles and BB inert particles. This process can be described as an interacting particle system in the following way: at time 00 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 PdP_d the law of such a process and Sd0(t)S^0_d(t) the set of visited sites at time tt. In this article we prove that there exists a bounded, non-empty, convex set CdRdC_d\subset{\Bbb R}^d, such that for every ϵ>0\epsilon>0, PdP_d-a.s. eventually in tt, the set Sd0(t)S_d^0(t) is within an ϵt\epsilon t neighborhood of the set [Cdt][C_dt], where for ARdA\subset{\Bbb R}^d we define [A]:=AZd[A]:=A\cap {\Bbb 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 dd large enough, we establish that the set CdC_d is not a ball under the Euclidean norm.