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IOP Publishing, Journal of Physics A: Mathematical and General, 11(31), p. L209-L215

DOI: 10.1088/0305-4470/31/11/001

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Persistence in the Voter model: continuum reaction-diffusion approach

Journal article published in 1997 by M. Howard ORCID, C. Godreche
This paper is available in a repository.
This paper is available in a repository.

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Abstract

We investigate the persistence probability in the Voter model for dimensions d ≥ 2. This is achieved by mapping the Voter model onto a continuum reaction–diffusion system. Using path integral methods, we compute the persistence probability r(q,t), where q is the number of “opinions ” in the original Voter model. We find r(q,t) ∼ exp[−f2(q)(ln t) 2] in d = 2; r(q,t) ∼ exp[−fd(q)t (d−2)/2] for 2 < d < 4; r(q,t) ∼ exp[−f4(q)t/ln t] in d = 4; and r(q,t) ∼ exp[−fd(q)t] for d> 4. The results of our analysis are checked by Monte Carlo simulations. The Voter model is a simple stochastic model which exhibits interesting dimension dependent properties [1]. Whilst, in one dimension, it is equivalent to the Glauber-Ising model at zero temperature, its properties depart from this model in higher dimensions. On each site of a d−dimensional lattice, opinions of a voter, or values of a spin σ = 1, 2,..., q, are initially distributed randomly. Between t and t+dt a site is picked at random. The voter on this