Asymptotic behavior of a stochastic combustion growth process

Ramírez, Alejandro, and Sidoravicius, Vladas. "Asymptotic behavior of a stochastic combustion growth process." Journal of the European Mathematical Society 006.3 (2004): 293-334. <http://eudml.org/doc/277677>.

@article{Ramírez2004,
abstract = {We study a continuous time growth process on the $d$-dimensional hypercubic lattice $\mathcal \{Z\}^d$, which admits a phenomenological interpretation as the combustion reaction $A+B\rightarrow 2A$, 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 $S^0_d(t)$ 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 $\epsilon >0$, $P_d$-a.s. eventually in $t$, the set $S^0_d(t)$ is within an $\epsilon t$ 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.},
author = {Ramírez, Alejandro, Sidoravicius, Vladas},
journal = {Journal of the European Mathematical Society},
keywords = {random walk; Green function; sub-additivity; random walk; Green function; subadditivity},
language = {eng},
number = {3},
pages = {293-334},
publisher = {European Mathematical Society Publishing House},
title = {Asymptotic behavior of a stochastic combustion growth process},
url = {http://eudml.org/doc/277677},
volume = {006},
year = {2004},
}

TY - JOUR
AU - Ramírez, Alejandro
AU - Sidoravicius, Vladas
TI - Asymptotic behavior of a stochastic combustion growth process
JO - Journal of the European Mathematical Society
PY - 2004
PB - European Mathematical Society Publishing House
VL - 006
IS - 3
SP - 293
EP - 334
AB - We study a continuous time growth process on the $d$-dimensional hypercubic lattice $\mathcal {Z}^d$, which admits a phenomenological interpretation as the combustion reaction $A+B\rightarrow 2A$, 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 $S^0_d(t)$ 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 $\epsilon >0$, $P_d$-a.s. eventually in $t$, the set $S^0_d(t)$ is within an $\epsilon t$ 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.
LA - eng
KW - random walk; Green function; sub-additivity; random walk; Green function; subadditivity
UR - http://eudml.org/doc/277677
ER -