Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions

TL;DR

This paper proves that the critical finite-range contact process's higher-point functions converge to super-Brownian motion measures above the upper critical dimension, extending previous 2-point function results.

Contribution

It establishes convergence of r-point functions for the critical contact process to super-Brownian motion measures, including new results for higher points and dimensions.

Findings

r-point functions converge to super-Brownian motion measures

Results extend previous 2-point function analysis

Partial results obtained for dimensions below or equal to 4

Abstract

We consider the critical spread-out contact process in Z^d with d\ge1, whose infection range is denoted by L\ge1. In this paper, we investigate the r-point function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that, for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time t_i by the individual at the origin o\in Z^d at time 0. Together with the results of the 2-point function in [van der Hofstad and Sakai, Electron. J. Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper-critical dimension 4. We also prove partial results for d\le4 in a local mean-field setting.