On the range of lattice models in high dimensions
We investigate the scaling limit of the range (the set of visited vertices) for a general class of critical lattice models, starting from a single initial particle at the origin. Conditions are given on the random sets and an associated ``ancestral relation" under which, conditional on long-term survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball. Applications include voter models, contact processes, oriented percolation and lattice trees. This is joint work with Mark Holmes and also features work of Akira Sakai and Gord Slade.