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Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions
| Title: | Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions |
| Authors: | Hofstad, Remco van der Browse this author | | Sakai, Akira Browse this author →KAKEN DB |
| Issue Date: | 2005 |
| Publisher: | Springer |
| Journal Title: | Probability Theory and Related Fields |
| Volume: | 132 |
| Issue: | 3 |
| Start Page: | 438 |
| End Page: | 470 |
| Publisher DOI: | 10.1007/s00440-004-0405-4 |
| Abstract: | We consider self-avoiding walk and percolation in Zd , oriented percolation in Zd × Z+, and the contact process in Zd , with pD( · ) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probabilitypD(y −x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point pc. We investigate the value of pc when d > 6 for percolation and d > 4 for the other models, and L 1. We prove in a unified way that pc = 1 + C(D) + O(L −2d ), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D) = O(L −d ) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that pc = 1 + cL −d + O(L −d−1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models. |
| Rights: | "The original publication is available at www.springerlink.com" |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/44916 |
| 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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