Super-Brownian Limits of Voter Model Clusters

The voter model is one of the standard interacting particle systems. Two related problems for this process are to analyze its behavior, after large times t, for the sets of sites (a) sharing the same opinion as the site 0, and (b) having the opinion that was originally at 0. Results on the sizes of these sets were given in [Sa79] and [BG80]. Here, we investigate the spatial structure of these sets in d ≥ 2, which we show converge to quantities associated with super-Brownian motion, after suitable normalization. The main theorem from [CDP98] serves as an important tool for these results. 1. Introduction. The voter model was introduced independently by Clif-ford and Sudbury in [CS73] (where it was called the invasion process) and by Holley and Liggett in [HL75]. It is one of the simplest interacting particle systems (see [Gr79] and [Li85]), but one which exhibits a wide range of interesting phenomena. The process is easily described. One supposes that at each site x of the d-dimensional integer lattice Z d there is a voter who randomly changes opinion. In the two-type model, each voter holds one of two opinions, say 0 or 1, and at rate-1 exponential random times, selects a neighbor at random according to a given jump kernel p(x, y), and adopts the opinion of the neighbor at the chosen site. (Note that no change occurs if the two opinions are the same.) All voting times and neighbor selections are independent of one another. We denote the process by ξ t , where ξ t (x) is the opinion at site x at time t, and will adopt the convention of identifying the configuration ξ t with {x : ξ t (x) = 1}, the set of sites with opinion 1. For x ∈ Z d , ξ