Voter model
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.
Suppose two nations control the areas (sets of nodes) labelled 0 or 1.
Problems involving the voter model will often be recast in terms of the dual system[clarification needed] of coalescing[clarification needed] Markov chains.
Frequently, these problems will then be reduced to others involving independent Markov chains.
A voter model is a (continuous time) Markov process
is assumed to be nonnegative, uniformly bounded and continuous as a function of
Therefore, a voter model has two trivial extremal stationary distributions, the point-masses
The main question to be discussed is whether or not there are others, which would then represent coexistence of different opinions in equilibrium.
It is said coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0's and 1's.
, then transition rates are simply: A process of coalescing random walks
denotes the set of sites occupied by these random walks at time
with unit exponential holding times and transition probabilities
At that time, the two that meet coalesce into one particle, which continues to move like a random walk with transition probabilities
The concept of Duality is essential for analysing the behavior of the voter models.
forms a coalescing random walks described at the end of section 2.1.
is transient, thus there is a positive probability that the random walks never hit, and hence for
In particular, Remarks: To contrast this with the behavior of the threshold voter models that will be discussed in next section, note that whether the linear voter model clusters or coexists depends almost exclusively on the dimension of the set of sites, rather than on the size of the range of interaction.
is any translation spatially ergodic and invariant probability measure on the state space
This dichotomy is closely related to the fact that simple random walk on
The aim of this section is to give a more precise description of this clustering.
Proposition 2.3 Suppose the voter model is with initial distribution
is a translation invariant probability measure, then Define the occupation time functionals of the basic linear voter model as: Theorem 2.4 Assume that for all site x and time t,
proof By Chebyshev's inequality and the Borel–Cantelli lemma, there is the equation below: The theorem follows when letting
is assumed to be a finite set that is symmetric with respect to all reflections and irreducible (i.e. the group it generates is
The intuition is that having a small threshold makes it easy for flips to occur, so it is likely that there will be a lot of both 0's and 1's around at all times.
Remarks: (a) Threshold models in higher dimensions do not necessarily cluster if
The fact that this random walk is recurrent implies that every site flips infinitely often.
The threshold model has a drift toward the "local minority", which is not present in the linear case.
has a nontrivial invariant measure, then the threshold voter model coexists.
The proof of this theorem is given in a paper named "Coexistence in threshold voter models" by Thomas M. Liggett.
