03-371 Remco van der Hofstad and Akira Sakai

Gaussian scaling for the critical spread-out contact process above the upper critical dimension (1083K, Postscript) Aug 15, 03

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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.

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