A species of voter model driven by immigration (Q734696)

An interactive particle system is considered on a metric space \(E\) with two types of particles: Black and White. At the initial moment the Blacks and Whites occupy sets \(B_0\subset E\) and \(W_0\subset E\) respectively. Then new particles arrive one by one, each choosing an independent point in \(E\) with distribution \(\mu\) and each adopting the color of its nearest neighbor from previously derived particles. Let \(B_\infty\) and \(W_\infty\) be the sets of resulting Black and White configurations, \(B'\) and \(W'\) be the sets of their accumulation points. Then the final Grey set is \(G=B'\cap W'\) and \(B=B'\setminus G\), \(W=W'\setminus G\). The author considers mainly the case \(E=\mathbb R^2\), \(B_0={0}\), \(W_0\) being the unit circumference. It is shown that in this case \(B\) and \(W\) are open sets, \(B_\infty\subset B\), \(W_\infty\setminus W_0\subset W\), \(G=\partial B\) is nonempty, closed dense in itself and nowhere dense, \(\mu(G)=0\), \(B\), \(W\) and \(G\) are disconnected and each has infinity of connected components.