Convergence of critical oriented percolation to super-Brownian motion above 4 + 1 dimensions

We consider oriented bond percolation on Zd × N, at the critical occupation density pc, for d > 4. The model is a “spread-out” model having long range parameterised by L. We consider configurations in which the cluster of the origin survives to time n, and scale space by n1/2. We prove that for L sufficiently large all the moment measures converge, as n →∞, to those of the canonical measure of super-Brownian motion. This extends a previous result of Nguyen and Yang, who proved Gaussian behaviour for the critical two-point function, to all r-point functions (r ≥ 2). We use lace expansion methods for the two-point function, and prove convergence of the expansion using a general inductive method that we developed in a previous paper. For the r-point functions with r ≥ 3, we use a new expansion method. Subject classifications: 60K35, 82B43.