Asymptotic behavior of a stochastic combustion growth process
Summary: We study a continuous time growth process on the \(d\)-dimensional hypercubic lattice \(\mathbb{Z}^d\), which admits a phenomenological interpretation as the combustion reaction \(A+B\to 2A\), 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, nonempty, convex set \(C_d\subset\mathbb{R}^d\) such that for every \(\varepsilon>0\), \(P_d\)-a.s. eventually in \(t\), the set \(S_d^0(t)\) is within an \(\varepsilon t\) distance of the set \([C_dt]\), 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.
scientific article; zbMATH DE number 2102878
scientific article