|
Hokkaido University Collection of Scholarly and Academic Papers >
Graduate School of Science / Faculty of Science >
Peer-reviewed Journal Articles, etc >
Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for d >= d(c)
| Title: | Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for d >= d(c) |
| Authors: | Chen, Lung-Chi Browse this author | | Sakai, Akira Browse this author →KAKEN DB |
| Issue Date: | Dec-2019 |
| Publisher: | Springer |
| Journal Title: | Communications in mathematical physics |
| Volume: | 372 |
| Issue: | 2 |
| Start Page: | 543 |
| End Page: | 572 |
| Publisher DOI: | 10.1007/s00220-019-03385-9 |
| Abstract: | Consider the long-range models on Z(d) of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as |x|(-d-alpha) for some alpha > 0. In the previous work (Chen and Sakai in Ann Probab 43:639-681, 2015), we have shown in a unified fashion for all alpha not equal 2 that, assuming a bound on the "derivative" of the n-step distribution (the compoundzeta distribution satisfies this assumed bound), the critical two-point function G(pc) (x) decays as |x|(alpha boolean AND 2-d) above the upper-critical dimension d(c) = (alpha boolean AND 2)m, where m = 2 for self-avoiding walk and the Ising model and m = 3 for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the n-step distribution, that G(pc) (x) for the marginal case alpha = 2 decays as |x|(2-d)/ log |x| whenever d >= d(c) (with a large spread-out parameter L). This solves the conjecture in Chen and Sakai (2015), extended all the way down to d = d(c), and confirms a part of predictions in physics (Brezin et al. in J Stat Phys 157:855-868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections. |
| Rights: | This is a post-peer-review, pre-copyedit version of an article published in Communications in mathematical physics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00220-019-03385-9 |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/79837 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
|
Submitter: 坂井 哲
|
