the complete convergence theorem
The complete convergence theorem answers the question that survival-versus-extinction leaves open: granted that the contact process can survive, what does its law actually converge to in the long run, and how does the answer depend on the starting configuration? It is the definitive description of the asymptotic behaviour of the (supercritical) contact process, and the reason we can speak of its equilibrium at all. The name comes from the completeness of the description: it pins down the limit law for every initial configuration, not just for special ones.
Consider the contact process on Z^d with infection rate lambda > lambda_c, started from an arbitrary initial configuration eta. The theorem states that the distribution of eta_t converges weakly, as t -> infinity, to a mixture of just two invariant measures: the trivial measure delta_0 concentrated on the empty (all-healthy) configuration, and the unique nontrivial upper invariant measure nu (the limit started from all sites infected). Precisely, the law of eta_t converges to alpha(eta) * delta_0 + (1 - alpha(eta)) * nu, where alpha(eta) = P(the process started from eta eventually dies out). So the only thing the initial condition controls in the limit is the probability of extinction; conditioned on survival the system always relaxes to the same equilibrium nu. The proof combines self-duality (to identify the surviving part with the upper invariant measure) with sharp estimates on the supercritical contact process, in particular that survival implies the infected region grows linearly and fills space at the equilibrium density (a consequence of the Bezuidenhout-Grimmett restart / block construction).
Its importance is conceptual: it shows that a contact process has exactly two extremal invariant measures in the supercritical regime, delta_0 and nu, and the whole forward evolution is a competition between dying out (frozen at delta_0) and surviving (relaxing to nu). This is a much stronger and more useful statement than mere existence of an invariant measure. The honest caveats are that the clean two-measure form is a feature of attractive, self-dual systems and is genuinely a theorem with hypotheses — establishing it required the deep supercritical estimates, and analogous complete-convergence statements for other or less symmetric systems are harder or false. Also note the theorem describes weak convergence of the law; the realised path does not converge (a surviving configuration keeps fluctuating forever, it just looks locally like a sample from nu).
Take the supercritical contact process started from a single infected site. With probability alpha it dies out and is thereafter frozen at delta_0; with probability 1 - alpha it survives, and on the survival event the configuration around any fixed site looks, for large t, like a draw from the upper invariant measure nu. The limiting law is alpha * delta_0 + (1 - alpha) * nu.
Every starting configuration relaxes to the same two-point mixture: extinct (delta_0) or, conditioned on survival, the universal equilibrium nu.
Only the extinction probability alpha(eta) depends on the start; conditioned on survival every configuration relaxes to the same nu. It is convergence of the law, not the path — a surviving realisation fluctuates forever — and the clean two-measure form leans on the deep supercritical block estimates and on attractiveness/self-duality.