Gaussian scaling for the critical spread-out contact process above the upper critical dimension

2004/02/04 by Remco van der Hofstad, Akira Sakai, van der Hofstad, Remco +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR

paper · pdf · doi:10.48550/arxiv.math/0402049

50 pages, 5 figures

arxiv created 2004/09/24 · arxiv updated 2009/12/01

Abstract

We consider the critical spread-out contact process in \Zd with d≥ 1, whose infection range is denoted by L≥1. The two-point function τ_t(x) is the probability that x∈\Zd is infected at time t by the infected individual located at the origin o∈\Zd at time 0. We prove Gaussian behavior for the two-point function with L≥ L₀ for some finite L₀=L₀(d) for d>4. When d≤ 4, we also perform a local mean-field limit to obtain Gaussian behaviour for τ_tT with t>0 fixed and T→ ∞ when the infection range depends on T such that L_T=LT^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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