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

In this paper, we investigate the contact process higher-point functions which denote the probability that the infection started at the origin at time 0 spreads to an arbitrary number of other individuals at various later times. Together with the results of the two-point function in [16], on which our proofs crucially rely, we prove that the higher-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 in dimension at most 4 in a local mean-field setting. The proof is based on the lace expansion for the time-discretized contact process, which is a version of oriented percolation. For ordinary oriented percolation, we thus reprove the results of [20]. The lace expansion coefficients are shown to obey bounds uniformly in the discretization parameter, which allows us to establish the scaling results also for the contact process We also show that the main term of the vertex factor, which is one of the non-universal constants in the scaling limit, is 1 for oriented percolation, and 2 for the contact process, while the main terms of the other non-universal constants are independent of the discretization parameter. The lace expansion we develop in this paper is adapted to both the higher-point functions and the survival probability. This unified approach makes it easier to relate the expansion coefficients derived in this paper and the expansion coefficients for the survival probability, which will be investigated in a future paper [18]