The contact process in high dimensions (Q1105513)

The author proves an improved version of the complete convergence theorem for the contact process in high dimensions. Let \(\xi_ t^{\mu}\) denote the contact process on the lattice \({\mathbb{Z}}^ d \)with initial distribution \(\mu\) and infection parameter \(\lambda\) (i.e. the speed at which the infection propagates), and \(\lambda_ c^{(k)}\) denote the critical value of the infection parameter in dimension k. It is shown that if \(d\geq 3k+1\) and \(\lambda >\lambda_ c^{(k)}\), then \[ \xi_ t^{\mu}=>P(\tau^{\mu}<\infty)\delta_{\emptyset}+P(\tau^{\mu}=\infty) \mu_{\lambda}\quad as\quad t\to \infty, \] where \(\tau^{\mu}\) is the hitting time of the empty set \(\emptyset\) starting from \(\mu\), and \(\mu_{\lambda}\) is the upper invariant measure of the process corresponding to the case \(\lambda >\lambda_ c^{(d)}\).