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Crossover phenomena in the critical behavior for long-range models with power-law couplings
| Title: | Crossover phenomena in the critical behavior for long-range models with power-law couplings |
| Authors: | Sakai, Akira Browse this author →KAKEN DB |
| Keywords: | Random walk | | percolation | | percolation | | Ising model | | power-law coupling | | critical two-point function | | critical dimension | | crossover phenomena | | lace expansion |
| Issue Date: | Apr-2020 |
| Publisher: | 京都大学数理解析研究所 |
| Journal Title: | 数理解析研究所講究録別冊 |
| Journal Title(alt): | RIMS Kokyuroku Bessatsu |
| Volume: | B79 |
| Start Page: | 51 |
| End Page: | 62 |
| Abstract: | This is a short review of the two papers [9, 10] on the x-space asymptotics of the critical two-point function Gpc (x) for the long-range models of self-avoiding walk, percolation and the Ising model on Zd, defined by the translation-invariant power-law step-distribution/coupling D(x)|x|-d-αor some α > 0. Let S1(x) be the random-walk Green function generated by D. We have shown that S1(x) changes its asymptotic behavior from Newton (α > 2) to Riesz (α < 2), with log correction at α = 2; Gpc(x) - A/pc S1(x) as|x| → ∞ in dimensions higher than (or equal to, if α = 2) the upper critical dimension dc (with sufficiently large spread-out parameter L). The model-dependent A and dc exhibit crossover at α = 2. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of S1, (ii) bounds on convolutions of power functions (with log corrections, if α = 2) to optimally control the lace-expansion coefficients π(n)p, and (iii) probabilistic interpretation (valid only when α ≤ 2) of the convolution of D and a function Πp of the alternating series Σ∞n=0(-1)nπ(n)p. We outline the proof, emphasizing the above key elements for percolation in particular. |
| Publisher URI: | https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/260644 |
| Type: | article |
| URI: | http://hdl.handle.net/2115/85983 |
| 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)
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Submitter: 坂井 哲
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