The two-type Richardson model with unbounded initial configurations (Q2475036)

An extension of the Richardson model [see \textit{D. Richardson}, Proc. Camb. Philos. Soc. 74, 515--528 (1973; Zbl 0295.62094)] describing two entities making a competition on \(\mathbb{Z}^d\) is studied. The purpose of the authors is to investigate the two-type Richardson model with unbounded initial configuration, namely, whether both types simultaneously infect infinitely many sites. A nontrivial situation occurs when (say) type 1 starts with infinitely many sites and type 2 with finitely many sites. Set \(\mathcal{H} =\{x\in \mathbb{Z}^d: x_1=0\}\) and \(\mathcal{L}=\{x\in\mathbb{Z}^d: x_1 \leq 0\) and \(x_i=0\) for all \(i\geq 2\}\). Introduce \(I(\mathcal{H})\) meaning that all points in \(\mathcal{H} \setminus \{\mathbf{0}\}\) are type 1 infected and \({\mathbf 0}\) is type 2 infected, and \(I(\mathcal{L})\) configuration with all points in \(\mathcal{L} \setminus \{\mathbf{0}\}\) type 1 infected and \({\mathbf 0}\) being type 2 infected. Let \(P_{\mathcal{H},{\mathbf 0}}^{\lambda_1,\lambda_2}\) (resp. \(P_{\mathcal{L},{\mathbf 0}}^{\lambda_1,\lambda_2}\)) be probability measures associated with two-type process started from configuration \(I(\mathcal{H})\) (resp. \(I(\mathcal{L})\)) where \(\lambda_1\) and \(\lambda_2\) stand for intensities of the infections. We denote by \(G_2\) the event that the type 2 infection reaches sites arbitrary far from the origin. The main result states that for \(d\geq 2\) one has \(P_{\mathcal{H},{\mathbf 0}}^{\lambda_1,\lambda_2}(G_2) >0\) iff \(\lambda_2 > \lambda_1\) whereas \(P_{\mathcal{L},{\mathbf 0}}^{\lambda_1,\lambda_2}(G_2) >0\) iff \(\lambda_2 \geq \lambda_1\).