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

We consider the critical spread-out contact process in Z with d ≥ 1, whose infection range is denoted by L ≥ 1. In this paper, we investigate the higher-point functions τ (r) ~t (~x) for r ≥ 3, where τ (r) ~t (~x) is the probability that, for all i = 1, . . . , r − 1, the individual located at xi ∈ Z is infected at time ti by the individual at the origin o ∈ Z at time 0. Together with the results of the 2-point function in [15], 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 ≤ 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 in Zd× εZ+, where ε ∈ (0, 1] is the time unit. For ordinary oriented percolation (i.e., ε = 1), we thus reprove the results of [19]. The lace expansion coefficients are shown to obey bounds uniformly in ε ∈ (0, 1], which allows us to establish the scaling results also for the contact process (i.e., ε ↓ 0). We also show that the main term of the vertex factor V , which is one of the non-universal constants in the scaling limit, depends explicitly on the time unit as 2 − ε, while the main terms of the other non-universal constants are independent of ε. The lace expansion we develop in this paper is adapted to both the r-point function 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 reported in Part II [17].