Bramson, Maury; Cox, J. Theodore; Le Gall, Jean-François Super-Brownian limits of voter model clusters. (English) Zbl 1029.60078 Ann. Probab. 29, No. 3, 1001-1032 (2001). 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 S. Sawyer, J. Appl. Probab. 16, 482-495 (1979; Zbl 0433.92017) and M. Bramson and 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 J. T. Cox, R. Durrett and E. Perkins [Ann. Probab. 28, 185-234 (2000)]. Reviewer: Mihai Gradinaru (Nancy) Cited in 2 ReviewsCited in 15 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F05 Central limit and other weak theorems 60G57 Random measures 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) Keywords:voter model; super-Brownian motion; coalescing random walk Citations:Zbl 0433.92017; Zbl 0417.60097 PDFBibTeX XMLCite \textit{M. Bramson} et al., Ann. Probab. 29, No. 3, 1001--1032 (2001; Zbl 1029.60078)