Gaussian Scaling for the Critical Spread-out Contact Process above the Upper Critical Dimension
Remco van der Hofstad, Eindhoven University of Technology, The Netherlands
Akira Sakai, EURANDOM, The Netherlands
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 τt(x)
is the probability that x in Zd
is infected at time t by the infected individual located at the origin o
in Zd at time 0. We prove Gaussian behaviour 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 τtT(x) with t>0
fixed and T tending to infinity when the infection range depends on T
in such a way that LT=LTb
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 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
[20] and extend the results to the local mean-field setting described above
when d≤4.
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