# 自避行走

## 应用

• 溶剂聚合物
• 蛋白质
• 高分子
• 纽结理论
• 随机漫步
• 保罗·弗洛里学了化学中的自避行走。[1]
• 网络理论[2]
• Gompertz distribution[3]
• ER随机图
• 有数学家认为自避行走的缩放极限是一个κ = 8/3Schramm-Loewner演变[4]

## 介绍

d = 2 4/3
d = 3 5/3
d ≥ 4 2 4是“upper critical dimension”（上面临界维度）

m × n 矩形点阵在只允許選擇減少曼哈頓距離的方向從一角往其對角行走的情況下有

${\displaystyle {m+n \choose m,\ n}}$

## 普遍性

${\displaystyle c_{n}}$是SAW数。这满足${\displaystyle c_{n}c_{m}\leq c_{n+m}}$所以${\displaystyle \log c_{n}}$次可加的以及

${\displaystyle \mu =\lim _{n\to \infty }c_{n}^{1/n}}$

${\displaystyle c_{n}\approx \mu ^{n}n^{11/32}}$

## 参考文献

1. ^ P. Flory. Principles of Polymer Chemistry. Cornell University Press. 1953: 672. ISBN 9780801401343.
2. ^ Carlos P. Herrero. Self-avoiding walks on scale-free networks. Phys. Rev. E. 2005, 71 (3): 1728. Bibcode:2005PhRvE..71a6103H. PMID 15697654. . doi:10.1103/PhysRevE.71.016103.
3. ^ Tishby, I.; Biham, O.; Katzav, E. The distribution of path lengths of self avoiding walks on Erdős–Rényi networks. Journal of Physics A: Mathematical and Theoretical. 2016, 49 (28): 285002. Bibcode:2016JPhA...49B5002T. . doi:10.1088/1751-8113/49/28/285002.
4. Duminil-Copin, Hugo; Smirnov, Stanislav. The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$. arXiv:1007.0575 [math-ph]. 2011-06-27.
5. ^ S. Havlin, D. Ben-Avraham. New approach to self-avoiding walks as a critical phenomenon. J. Phys. A. 1982, 15 (6): L321–L328 [2020-02-10]. Bibcode:1982JPhA...15L.321H. doi:10.1088/0305-4470/15/6/013. （原始内容存档于2020-09-22）.
6. ^ S. Havlin, D. Ben-Avraham. Theoretical and numerical study of fractal dimensionality in self-avoiding walks. Phys. Rev. A. 1982, 26 (3): 1728–1734 [2020-02-10]. Bibcode:1982PhRvA..26.1728H. doi:10.1103/PhysRevA.26.1728. （原始内容存档于2018-11-12）.
7. ^ A. Bucksch, G. Turk, J.S. Weitz. The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion. PLOS ONE. 2014, 9 (1): e85585. Bibcode:2014PLoSO...985585B. . PMID 24465607. . doi:10.1371/journal.pone.0085585.
8. ^ Hayes B. How to Avoid Yourself (PDF). American Scientist. Jul–Aug 1998, 86 (4): 314 [2020-02-10]. doi:10.1511/1998.31.3301. （原始内容存档 (PDF)于2020-09-28）.
9. ^ Liśkiewicz M; Ogihara M; Toda S. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science. July 2003, 304 (1–3): 129–56. doi:10.1016/S0304-3975(03)00080-X.

## 阅读

1. Madras, N.; Slade, G. The Self-Avoiding Walk. Birkhäuser. 1996. ISBN 978-0-8176-3891-7.
2. Lawler, G. F. Intersections of Random Walks. Birkhäuser. 1991. ISBN 978-0-8176-3892-4.
3. Madras, N.; Sokal, A. D. The pivot algorithm – A highly efficient Monte-Carlo method for the self-avoiding walk. Journal of Statistical Physics. 1988, 50 (1–2): 109–186. Bibcode:1988JSP....50..109M. doi:10.1007/bf01022990.
4. Fisher, M. E. Shape of a self-avoiding walk or polymer chain. Journal of Chemical Physics. 1966, 44 (2): 616–622. Bibcode:1966JChPh..44..616F. doi:10.1063/1.1726734.