Asymptotic behavior of a stochastic combustion growth process

Abstract

We study a continuous time growth process on the $d$-dimensional hyper-cubic lattice ${\mathbb{Z}}^d$, which admits a phenomenological interpretation as the combustion reaction $A+B\to 2A$, 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 $P_d$ the law of such a process and $S_d^0(t)$ the set of visited sites at time $t$. In this article we prove that there exists a bounded, non-empty, convex set $C_d\subset {\mathbb{R}}^d$, such that for every $ϵ>0$, $P_d$-a.s. eventually in $t$, the set $S_d^0(t)$ is within an $ϵt$ neighborhood of the set $[C_{d}t]$, where for $A\subset {\mathbb{R}}^d$ we define $[A]:=A\cap {\mathbb{Z}}^d$. 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.