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Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: The higher-point functions
| Title: | Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: The higher-point functions |
| Authors: | Sakai, Akira Browse this author →KAKEN DB | | Hofstad, Remco van der Browse this author |
| Keywords: | contact process | | mean-field behavior | | critical exponents | | super-Brownian motion |
| Issue Date: | 17-Jun-2010 |
| Journal Title: | Electronic Journal of Probability |
| Volume: | 15 |
| Start Page: | 801 |
| End Page: | 894 |
| Publisher DOI: | 10.1214/EJP.v15-783 |
| Abstract: | We consider the critical spread-out contact process in Z^d with d ≥ 1, whose infection range is denoted by L ≥ 1. In this paper, we investigate the higher-point functions τ^(r)_ t^^→ (x^^→) for r ≥ 3, where τ^(r)_ t^^→ (x^^→) is the probability that, for all i = 1, . . . , r − 1, the individual located at x_i ∈ Z^d is infected at time t_i by the individual at the origin o ∈ Z^d at time 0. Together with the results of the 2-point function in [16], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper critical dimension 4. We also prove partial results for d ≤ 4 in a local mean-field setting. The proof is based on the lace expansion for the time-discretized contact process, which is a version of oriented percolation in Z^d × ԑZ_+, where ԑ ∈ (0,1] is the time unit. For ordinary oriented percolation (i.e., ԑ = 1), we thus reprove the results of [20]. The lace expansion coefficients are shown to obey bounds uniformly in ԑ ∈ (0,1], which allows us to establish the scaling results also for the contact process (i.e., ǫ ↓ 0). We also show that the main term of the vertex factor V, which is one of the non-universal constants in the scaling limit, is 2 − ԑ (= 1 for oriented percolation, = 2 for the contact process), while the main terms of the other non-universal constants are independent of ԑ. |
| Type: | article |
| URI: | http://hdl.handle.net/2115/44119 |
| Appears in Collections: | 創成研究機構 (Creative Research Institution) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 坂井 哲
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