The survival probability and r-point functions in high dimensions

TL;DR

This paper studies the survival probability in high-dimensional models, showing it scales predictably under certain conditions and applying to models like lattice trees and percolation, with implications for their scaling limits.

Contribution

It establishes the asymptotic behavior of survival probability in high-dimensional models using weak convergence, extending previous results to new models and conditions.

Findings

nθ_n converges to 2/(AV) under specified conditions

Results apply to high-dimensional lattice trees, percolation, and contact process

Derives consequences for the scaling limit of the number of particles

Abstract

In this paper we investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_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 a certain self-repellence condition is satisfied, then n\theta_n\ra 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 oriented percolation, this reproves a result by the first author, den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.