Akira Sakai
Mean-field Critical Behavior for Contact Processes
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ABSTRACT.  We prove the triangle condition for the basic contact process 
in sufficiently high dimensions and for the spatially spread-out 
contact process in spatial dimension $d > 4$. 
Combining with the result of Barsky and Wu in 1998, 
we prove the mean-field critical behavior for these processes. 
Our results imply that the upper critical dimension for the 
contact processes is four.