On the range of lattice models in high dimensions
Mark Holmes, Edwin Perkins
PROBABILITY THEORY AND RELATED FIELDS | SPRINGER HEIDELBERG | Published : 2020
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 longterm 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, that is for the extrinsic one-arm probability. We show that these conditions are satisfied by the voter model in dimensions d ≥ 2 , sufficiently spread out critical oriented percolation ..View full abstract
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Awarded by Aust. Research Council
M. Holmes is supported by Future Fellowship FT160100166 from the Aust. Research Council. E. Perkins is supported in part by an NSERC Discovery grant.