Survival of the weak in hyperbolic spaces, a remark on competition and geometry (Q2759020)

Let \(G\) be a graph, let \(m\geq 2\) be given, and let \(x_0,y_0\) be distinct vertices. Set \(X_0=\{x_0\}\), \(Y_0:=\{y_0\}\), \(X_{j+1}\) the set of vertices \(x\) with \(d(x,X_j)\leq m\) which are not in \(Y_j\), \(Y_{j+1}\) the set of vertices \(y\) with \(d(y,Y_j)\leq 1\) which are not in \(X_{j+1}\), \(X=\cup_jX_j\), and \(Y=\cup_jY_j\). On \(Z^d\) with \(d>1\), the cluster \(X_j\) eventually surrounds \(Y_j\) and thus \(Y\) is finite. The author shows that on any Gromov hyperbolic graph which contains a bi-infinite geodesic, both \(X\) and \(Y\) are infinite - coexistence pertains - provided the initial positions are chosen properly.