Probability theory and stochastic processes Special processes Example

Voter model

Example

Let $d \ge 1$ . The voter model on $\Z^d$ is the spin system on $\set{0, 1}^{\Z^d}$ with flip rates $c(x, \eta) = \frac{1}{2d} \, \#\set{y \in \Z^d : \abs{y - x} = 1, \ \eta(y) \ne \eta(x)}.$ Here $\eta(x)$ is the opinion of the voter at site $x$ . At rate $1$ , each voter picks one of its $2d$ neighbours uniformly at random and adopts that neighbour's opinion. The rates lie in $[0, 1]$ and depend on $2d + 1$ sites, so it is an interacting particle system.

Remarks

The two constant configurations, all $0$ and all $1$ , are absorbing. The model is dual to a system of coalescing random walks. Tracing back whose opinion a voter holds gives a random walk path, and two such paths merge when they meet. Because of this duality, the long-time behaviour depends on whether simple random walk on $\Z^d$ is recurrent, which by Pólya's recurrence theorem holds exactly for $d \le 2$ . Started from independent opinions with density $\rho \in (0, 1)$ , for $d \le 2$ any two sites tend to agree, $P(\eta_t(x) \ne \eta_t(y)) \to 0$ as $t \to \infty$ . For $d \ge 3$ there is a one-parameter family of invariant measures in which both opinions keep a positive density.

Tag	`YBAE5QCH1A`
URI	`m2://YBAE5QCH1A`
Kind	example
Writing	W0 stub
MSC	60K35, 82C22
Instance of	Interacting particle system
Same as	Wikidata Q7942158 Wikipedia Voter model
Changed	2 November 2026

Voter model

Example

Remarks

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