# 自避行走

## 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

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

## 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

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## 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.