The survival probability and r-point functions in high dimensions

Abstract

In this paper we investigate the survival probability, n, in high-dimen- sional statistical physical models, where n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satisfied, then nn → 2=(AV ), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4 + 1 dimensions, and the spread-out contact process above 4 + 1 dimensions. In the case of..