In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975.
One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, one of the chosen voter's neighbors is chosen according to a given set of probabilities and that neighbor’s opinion is transferred to the chosen voter. An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation. Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains.
Definition A voter model is a (continuous time) Markov process η t {\displaystyle \eta _{t}} with state space S = { 0 , 1 } Z d {\displaystyle S=\{0,1\}^{Z^{d}}} and transition rates function c ( x , η ) {\displaystyle c(x,\eta )} , where Z d {\displaystyle Z^{d}} is a d-dimensional integer lattice, and c ( {\displaystyle c(} •,• ) {\displaystyle )} is assumed to be nonnegative, uniformly bounded and continuous as a function of η {\displaystyle \eta } in the product topology on S {\displaystyle S} . Each component η ∈ S {\displaystyle \eta \in S} is called a configuration. To make it clear that η ( x ) {\displaystyle \eta (x)} stands for the value of a site x in configuration η ( . ) {\displaystyle \eta (.)} ; while η t ( x ) {\displaystyle \eta _{t}(x)} means the value of a site x in configuration η ( . ) {\displaystyle \eta (.)} at time t {\displaystyle t} . The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at x {\displaystyle \scriptstyle x} from 0 to 1 or vice versa is given by a function c ( x , η ) {\displaystyle c(x,\eta )} of site x {\displaystyle x} . It has the following properties:
c ( x , η ) = 0 {\displaystyle c(x,\eta )=0} for every x ∈ Z d {\displaystyle x\in Z^{d}} if η ≡ 0 {\displaystyle \eta \equiv 0} or if η ≡ 1 {\displaystyle \eta \equiv 1}
c ( x , η ) = c ( x , ζ ) {\displaystyle c(x,\eta )=c(x,\zeta )} for every x ∈ Z d {\displaystyle x\in Z^{d}} if η ( y ) + ζ ( y ) = 1 {\displaystyle \eta (y)+\zeta (y)=1} for all y ∈ Z d {\displaystyle y\in Z^{d}}
c ( x , η ) ≤ c ( x , ζ ) {\displaystyle c(x,\eta )\leq c(x,\zeta )} if η ≤ ζ {\displaystyle \eta \leq \zeta } and η ( x ) = ζ ( x ) = 0 {\displaystyle \eta (x)=\zeta (x)=0}
c ( x , η ) {\displaystyle c(x,\eta )} is invariant under shifts in Z d {\displaystyle \scriptstyle Z^{d}}
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