Super-Brownian limits of voter model clusters (Q1872213)

The authors investigate the spatial structure of two kinds of sets of sites in the voter model after large times in dimension larger than 2 and show that, after suitable normalization, there is convergence to some quantities associated with super-Brownian motion. The sets of sites sharing the same opinion as site 0 and of sites having opinion that was originally at 0 are considered [see also \textit{S. Sawyer}, J. Appl. Probab. 16, 482-495 (1979; Zbl 0433.92017) and \textit{M. Bramson} and \textit{D. Griffeath}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 53, 183-196 (1980; Zbl 0417.60097) for results on the sizes of these sets]. For instance, in the two type voter model one denotes by \(\xi_{t}^{0}\) the set of sites with opinion 1, starting from a single 1 at site 0 at time 0; then the law of \(\xi_{t}^{0}\) conditioned on nonextinction and viewed as a measure converges to a quantity related to the canonical measures of super-Brownian motion having some branching rate and some diffusion coefficient. Moreover, \(\xi_{t}^{0}/\sqrt{t}\) under \(P(\cdot\mid\xi_{t}^{0}\neq\emptyset)\) converges in distribution in the Hausdorff metric. Similar results are obtained for multitype voter model. Previous results are reinterpreted in terms of coalescing random walks. The behaviour for the one-dimensional case is briefly discussed. An important tool in the proofs is the main theorem of \textit{J. T. Cox}, \textit{R. Durrett} and \textit{E. Perkins} [Ann. Probab. 28, 185-234 (2000)].