the voter model
The voter model is the simplest interacting particle system of consensus formation. Picture a voter at each site x of Z^d holding one of two opinions, eta(x) in {0,1}. At rate 1 each voter wakes up, picks a neighbour uniformly at random, and copies that neighbour's opinion. The question it answers is the most basic question of social dynamics: when does a population of imitating agents reach unanimous agreement (consensus), and when do conflicting opinions persist forever (coexistence)? It is also a model for spatial competition between two biological species or alleles where the descendant of whoever dies inherits the site.
Formally the flip rate is c(x, eta) = (fraction of nearest neighbours of x that disagree with x), so a voter in a region of agreement rarely changes while one at a boundary between opinions changes often. The generator is (L f)(eta) = sum_x [ (1/2d) sum_{y ~ x} 1{eta(y) != eta(x)} ] [ f(eta^x) - f(eta) ]. The model's defining feature is its exact duality with coalescing random walks: to find the joint law of opinions at sites x_1, ..., x_k at time t, run independent random walks backward in time from those sites, let them coalesce whenever two meet, and assign the original (time-0) opinion at each surviving ancestral location. Because two independent random walks on Z^d meet with probability 1 if and only if d <= 2 (recurrence), the model shows a sharp dimension dependence: in d = 1 and d = 2 the dual walks coalesce, every finite set of voters traces back to a single ancestor, and the system clusters toward consensus — locally everyone eventually agrees (though for an infinite lattice this is local consensus on ever-growing patches, not a single global flip). In d >= 3 the dual walks are transient and may never meet, so opinions coexist: there is a one-parameter family of nontrivial extremal invariant measures mu_rho, translation-invariant and ergodic, indexed by the density rho of opinion 1, under which both opinions survive forever.
The voter model is a touchstone because its full equilibrium picture can be computed exactly through duality, a rarity in the field. It also illustrates an important honesty point: the dichotomy is between clustering / consensus (low dimension) and coexistence (high dimension), and the borderline d = 2 clusters but only logarithmically slowly. Note that 'consensus' on the infinite lattice does not mean the whole world flips to one opinion at a finite time; rather, the system is not in a nontrivial stationary state, and any two given sites eventually almost surely agree. The voter model is linear (its duality is with a single coalescing system), which makes it analytically special and also somewhat fragile — adding even mild nonlinearity to the imitation rule, as in threshold or nonlinear voter models, can change the phase structure.
Start the 1D voter model with opinion 1 on the negative integers and 0 on the nonnegative ones. The boundary between the two opinions performs a symmetric random walk, which is recurrent, so the interface fluctuates and any fixed site flips infinitely often; the system clusters, and any two sites eventually agree. In d >= 3, starting from product measure of density rho, the system converges to the nontrivial stationary state mu_rho with both opinions present forever.
Consensus in d <= 2 (recurrent dual walks coalesce) versus coexistence in d >= 3 (transient dual walks may never meet).
On the infinite lattice 'consensus' means clustering — the absence of a nontrivial stationary state and eventual local agreement — not a single global flip at a finite time. The clean phase diagram relies on the model's linearity; nonlinear voter rules can behave very differently.