Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees

TL;DR

This paper establishes bounds on all spatial moments for the critical contact process in high dimensions, confirming a key condition for the scaling limit of the process's range to be super-Brownian motion.

Contribution

It provides a simple proof of all moments estimates for the critical contact process in dimensions greater than four, extending previous results and applying to related models.

Findings

All moments estimates are proved for the critical contact process in high dimensions.

The method applies also to oriented percolation and lattice trees.

Upper bounds on moments lead to explicit asymptotic formulas for the range.

Abstract

Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.