Remco van der Hofstad and Akira Sakai
Gaussian scaling for the critical spread-out contact process above the upper critical dimension
(1083K, Postscript)
ABSTRACT. We consider the critical spread-out contact process in the
d-dimensional integer lattice \Zd, whose infection range
is denoted by L. The two-point function \tau_t(x) is the
probability that x \in \Zd is infected at time t by the
infected individual located at the origin at time 0. We
prove Gaussian behavior for the two-point function with
L \geq L_0 for some finite L_0 = L_0(d) for d > 4.
When d \leq 4, we also perform a local mean-field limit to
obtain Gaussian behaviour for \tau_{tT}(x) with t > 0
fixed and T \to \infty when the infection range depends
on T such that L_T = L T^b for any b > (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 mean-field values. The results
in this paper provide crucial ingredients to prove convergence
of the finite-dimensional distributions for the contact process
towards the canonical measure of super-Brownian motion, which
we defer to a sequel of this paper.