attractive systems and the basic coupling
Attractiveness (monotonicity) is the order structure that makes many interacting particle systems tractable even when they cannot be solved exactly, and the basic coupling is the device that realises it. The configuration space {0,1}^S carries a natural partial order: eta <= zeta means eta(x) <= zeta(x) at every site (zeta has particles wherever eta does, and maybe more). A system is attractive if this order is preserved by the dynamics: starting two copies with eta <= zeta, you can run them so that eta_t <= zeta_t for all time. Intuitively, 'more particles now' can only lead to 'more particles later'. This single property unlocks comparison arguments, monotone limits, and the existence of extremal invariant measures, replacing exact formulas with inequalities.
The basic coupling is the canonical way to run two (or more) copies of the same system together on the same probability space so that monotonicity is automatic. Use the same Poisson clocks for both copies: whenever a clock rings at a site, attempt the same update in both configurations, accepting it according to the shared randomness and each copy's local state. If the rates have the right monotone structure — for a spin-flip system, the rate to flip up is increasing in the configuration and the rate to flip down is decreasing — then this common-clock coupling keeps eta_t <= zeta_t whenever eta_0 <= zeta_0, because the lower copy never overtakes the upper one. Formally, attractiveness is equivalent to a monotonicity condition on the rates, and is also characterised by the property that the semigroup P_t maps increasing functions to increasing functions (preserves the stochastic order on measures, written mu_1 <= mu_2 in the sense that they agree on the integral of every increasing function). A key consequence: starting from the all-1 configuration (the top), the law decreases monotonically and converges to the upper invariant measure nu-bar; starting from all-0 (the bottom) it increases to the lower invariant measure; the system is ergodic iff these two coincide.
This machinery is why the contact process, the exclusion process, and many growth models can be analysed at all: monotonicity in lambda or in the initial condition, comparison with oriented percolation, and FKG-type correlation inequalities all flow from attractiveness via the basic coupling. The honest caveats: not every interesting system is attractive — for instance, dynamics with competition or with rates that are non-monotone in the configuration (some Glauber dynamics at certain interactions, the antivoter model) are not, and then the basic coupling fails to preserve order and one loses these tools. Also, the basic coupling preserves order but is not a successful coupling for mixing-time purposes by itself; showing two copies actually meet (coalesce) requires extra work. Attractiveness gives existence and monotone convergence of the extremal invariant measures, but uniqueness (ergodicity) is a separate, often hard, question.
Run two contact processes from configurations eta_0 <= zeta_0 with the same graphical construction (the same infection arrows and recovery marks). At every infection arrow, the lower copy gets infected only if its source is infected, the upper copy whenever its source is — so the upper copy is infected wherever the lower one is; at every recovery mark both recover together. Hence eta_t <= zeta_t for all t, proving the contact process is attractive and that survival is monotone in the initial set and in lambda.
Shared Poisson clocks couple two ordered copies and keep the order forever — the basic coupling realising attractiveness.
Not every IPS is attractive (the antivoter model and some competitive dynamics are not), and where it fails the basic coupling no longer preserves order. Attractiveness yields existence and monotone convergence of the extremal (upper/lower) invariant measures, but ergodicity — that they coincide — is a separate, often hard, question.