The convergence of the biased annihilating branching process and the double-flipping process in \(\mathbb Z^d\)
The biased annihilating branching process (BABP) is a simple interacting particle system with the state space \(\{0,1\}^{\mathbb Z^d}\) and the flip-rate \[ c(x,\eta) = [\eta (x)+\lambda (1-\eta (x))] \sum _{y\in N_x} \eta (y) \] for each \(x\in \mathbb Z^d\) and \(\eta \in \{0,1\}^{\mathbb Z^d}\), where \(N_x =\{y\in \mathbb Z^d;| y-x| =1\}\) is the neighborhood and \(\lambda \) is the annihilating rate. Thus the adjacent pairs evolve according to \(11 {\rightarrow }_1 10\), \(10 {\rightarrow }_{\lambda } 11\), while in the double-flipping process (DFP) a pair flip together so that \(11 {\rightarrow }_a 00\), \(00 {\rightarrow }_b 11\), \(01 {\leftrightarrow }_c 10\). For a DFP it is shown \(\mu _t \to \nu _{\sqrt {b}/(\sqrt {a}+\sqrt {b})}\) providing the initial distribution \(\mu _0\) is translation invariant. Here \(\mu _t\) is the distribution at time \(t\) and \(\nu _p\) is the product measure with \(\text{Pr}(1)=p\). Thanks to a relationship between BABP and DFP it is proved for a BABP that \(\mu _t \to u \nu _{\lambda /(1+\lambda)} + (1-u) \nu _0\) for some \(u\in [0,1]\) providing \(\mu _0\) is translation invariant; with \(u=1\) if \(\mu _0=\nu_p\), \(p>0\). Moreover, starting from finite initial configuration, the number of occupied sites tends to infinity in probability.
scientific article; zbMATH DE number 1240021