We consider the critical spread-out contact process in $Z^d$ with $d\geq1$, whose infection range is denoted by $L\geq1$. The two-point function $\tau_t(x)$ is the probability that $x\in Z^d$ is infected at time $t$ by the infected individual located at the origin $o\in Z^d$ at time 0. We prove Gaussian behaviour for the two-point function with $L\geq L_0$ for some finite $L_0=L_0(d)$ for $d \gt 4$. When $d\leq4$, we also perform a local mean-field limit to obtain Gaussian behaviour for $\tau_{ tT}(x)$ with $t \gt 0$ fixed and $T\to\infty$ when the infection range depends on $T$ in such a way that $L_{T}=LT^b$ for any $b \gt (4-d)/2d$.

The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on their respective mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards those for the canonical measure of super-Brownian motion, which we defer to a sequel of this paper.

The results in this paper also apply to oriented percolation, for which we reprove some of the results in van der Hofstad and Slade (2003) and extend the results to the local mean-field setting described above when $d\leq4$.