survival versus extinction in the contact process
Survival versus extinction is the central dichotomy of the contact process, and the prototype of a phase transition for a stochastic spatial system. Started from a single infected site, the contact process either goes extinct — reaching the all-healthy absorbing state in finite time — or it survives, the infection persisting for all time. Which happens is governed by the infection rate lambda, and the sharp threshold separating the two regimes is the critical value lambda_c. This is the rigorous spatial counterpart of the epidemic threshold (the basic reproduction number R0 crossing 1) familiar from mean-field epidemiology.
There are several distinct notions to keep straight. Extinction means the process from a finite seed reaches the empty configuration almost surely. Weak survival (or global survival) means the total number of infected sites stays positive forever with positive probability. Strong (local) survival means a fixed site is reinfected infinitely often. On Z^d these coincide and define a single critical lambda_c in (0, infinity): for lambda < lambda_c the infection dies out almost surely; for lambda > lambda_c it survives with positive probability. The proof of a nontrivial transition uses two comparisons: a comparison with a subcritical branching process or a small-cluster bound shows lambda_c > 0 (low infection rates die out), while a block / oriented-percolation construction shows lambda_c < infinity (high infection rates survive). Monotonicity in lambda (more infection can only help, via the basic coupling) makes lambda_c well defined as a genuine threshold. By self-duality, survival of the infection from a single site is equivalent to the existence of a nontrivial upper invariant measure, the limit started from all-infected, which is nonzero exactly when lambda > lambda_c.
This transition matters as the cleanest example where one can prove rigorously that local interaction rules produce a macroscopic threshold phenomenon, and it is the workhorse for testing techniques like the contour / Peierls argument adapted to dynamics. Several honesty points: lambda_c depends on the dimension and the graph (on trees one even has two distinct critical values, lambda_1 for global and lambda_2 for local survival, with lambda_1 < lambda_2 — global survival without local survival is possible on a tree). The value of lambda_c is not known exactly in any dimension d >= 1; it is bounded and estimated. Most importantly, at exactly lambda = lambda_c the process dies out (Bezuidenhout and Grimmett proved the critical contact process on Z^d goes extinct), so unlike some equilibrium models there is no surviving phase precisely at criticality — a deep result, not a definition.
On a regular tree (Bethe lattice) the contact process famously has two thresholds: for lambda between lambda_1 and lambda_2 the infection survives globally — the total number infected stays positive forever — yet dies out locally, so any fixed vertex is eventually healthy forever as the infection escapes to infinity along the tree. On Z^d the two thresholds coincide into a single lambda_c.
Global survival and local survival can split apart on a tree; on the integer lattice they merge into one critical infection rate.
Survival from a finite seed equals the existence of a nontrivial upper invariant measure, by self-duality. Crucially the critical process itself dies out (a hard theorem), and lambda_c is not known exactly in any dimension; on trees global and local survival have genuinely different thresholds.