duality with coalescing random walks
Duality with coalescing random walks is the exact analytic mechanism that makes the voter model solvable. The idea answers a hard forward question — what is the configuration of opinions at a large time t? — by translating it into an easier backward question about a system of random walks that merge when they meet. It is the prototypical example of duality in interacting particle systems and the reason the voter model's entire equilibrium structure can be written down.
The precise statement uses a duality function. For the voter model the relevant function is H(eta, A) = 1{ eta is identically 1 on the finite set A } (equivalently one studies indicators of agreement on finite sets). The claim is that E_eta[ H(eta_t, A) ] = E_A[ H(eta, A_t) ], where on the left eta_t is the voter model started from configuration eta and A is a fixed finite set, and on the right A_t is a system of coalescing random walks started from the set A and run for time t while eta is held fixed at time 0. In words: to know the opinions at sites A at time t, send a random walker backward in time from each site of A; each walker follows the imitation arrows in reverse; whenever two walkers land on the same site they coalesce into one (because both copied from the same source); after time t the surviving walkers sit on the ancestral sites, and the time-t opinions on A are exactly the time-0 opinions at those ancestors. The number of distinct ancestors can only decrease, which is the coalescence. Duality holds because the voter generator L, applied in the eta variable to H(eta, A), equals the coalescing-walk generator applied in the A variable to the same H — an algebraic identity one checks term by term.
The power of the method is that recurrence-versus-transience of the dual walks dictates the phase diagram of the original system: coalescence with probability 1 (d <= 2) forces consensus, while positive survival of multiple distinct ancestors (d >= 3) gives coexistence and a continuum of invariant measures. The honest caveat is that this beautiful self-duality is special: it works because the voter model is linear (its rates are linear in the configuration through the fraction of disagreeing neighbours). Most interacting particle systems are not self-dual; they may have a different dual (the contact process is dual to itself in a different, monotone sense; the exclusion process is self-dual via a different duality function), or no useful dual at all. Coalescing-walk duality is also closely tied to the dual notion of the stepping-stone and Moran models in population genetics, where the backward genealogy is literally the coalescent.
To compute, in the voter model started from product measure of density rho, the probability that two sites x and y agree at time t, run two coalescing random walks from x and y. They agree if they have coalesced by time t (then they share one ancestor and so one opinion), or if they have not coalesced but happen to start from like opinions. As t grows in d <= 2 the coalescence probability tends to 1, so P(x and y agree) -> 1: clustering.
Forward opinions at time t are decoded by walking ancestral lineages backward and merging them — the coalescent genealogy of the voter model.
Self-duality with coalescing walks is a luxury of the voter model's linearity; do not assume an arbitrary IPS has such a dual. The dual walks run backward in time, and recurrence/transience of two independent walks (d<=2 versus d>=3) is precisely what controls consensus versus coexistence.