自避行走

在数学中，自避行走（简称：SAW，Self-Avoiding Walk）是一种格点上的随机漫步，但是不会多次访问同一点。所以SAW不是一种马尔可夫链。SAW模型在物理学、化学、生物学中有很多应用。

这是自避行走

这不是自避行走

应用

溶剂和聚合物
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 $κ = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}8/3$ 的Schramm-Loewner演变。^[4]

介绍

自避行走是一个分形。^[5]^[6] 例如，^[7]


维度d	分形维数
$d = 2$	4/3
$d = 3$	5/3
$d \geq 4$	2	4是“upper critical dimension”（上面临界维度）

没有已知的公式来计算给格子的SAW数。^[8]^[9]

$m \times n$ 矩形点阵在只允许选择减少曼哈顿距离的方向从一角往其对角行走的情况下有

{m+n \choose m,\ n}

个SAW。

普遍性

主要条目：普遍性 (物理学)

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

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

存在。格点六角形（hexagonal lattice）的 $\mu ={\sqrt {2+{\sqrt {2}}}}$ 。^[4]（斯坦尼斯拉夫·斯米尔诺夫）

有猜想说：当 $n\to \infty$ 的时候

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

上面的 $\mu$ 依赖格点，但是11/32这个数是普遍的。

参见

临界现象
普遍性 (物理学)
随机走

参考文献

^ P. Flory. Principles of Polymer Chemistry. Cornell University Press. 1953: 672. ISBN 9780801401343.
^ Carlos P. Herrero. Self-avoiding walks on scale-free networks. Phys. Rev. E. 2005, 71 (3): 1728. Bibcode:2005PhRvE..71a6103H. PMID 15697654. arXiv:cond-mat/0412658  . doi:10.1103/PhysRevE.71.016103.
^ 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. arXiv:1603.06613  . doi:10.1088/1751-8113/49/28/285002.
^ ^4.0 ^4.1 Duminil-Copin, Hugo; Smirnov, Stanislav. The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$. arXiv:1007.0575 [math-ph]. 2011-06-27.
^ 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）.
^ 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）.
^ 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. PMC 3899046  . PMID 24465607. arXiv:1304.3521  . doi:10.1371/journal.pone.0085585.
^ 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）.
^ 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

Madras, N.; Slade, G. The Self-Avoiding Walk. Birkhäuser. 1996. ISBN 978-0-8176-3891-7.
Lawler, G. F. Intersections of Random Walks. Birkhäuser. 1991. ISBN 978-0-8176-3892-4.
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.
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.

[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. arXiv:cond-mat/0412658  . 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. arXiv:1603.06613  . doi:10.1088/1751-8113/49/28/285002.

[:0-4] 4.0 ^4.1 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. PMC 3899046  . PMID 24465607. arXiv:1304.3521  . 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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]