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 (1) sharing the same opinion as the site
0, and (2) having the opinion that was originally at 0. Results on the sizes of
these sets were given by Sawyer (1979) and Bramson and Griffeath (1980).
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 Cox, Durrett and Perkins
(2000) serves as an important tool for these results.