On the range of lattice models in high dimensions

Abstract

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 and critical contact processes in dimensions

d > 4

, and sufficiently spread out critical lattice trees in dimensions

d > 8

. The latter result proves Conjecture 1.6 of van der Hofstad et al. (Ann Probab 45:278-376, 2017) and also has important consequences for the behaviour of random walks on lattice trees in high dimensions.