1 5 N ov 1 99 7 Persistence in the Voter model : continuum reaction - diffusion approach

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[−f 2 (q)(ln t) 2 ] in d = 2; r(q, t) ∼ exp[−f d (q)t (d−2)/2 ] for 2 < d < 4; r(q, t) ∼ exp[−f 4 (q)t/ ln t] in d = 4; and r(q, t) ∼ exp[−f d (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 site takes the opinion of one of its 2d neighbours, also chosen at random. This model is equivalent to a system of coalescing random walkers [1]. Therefore, due to the recurrence properties of random walks, the interactions between 1