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Self-Avoiding Walk

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.


Let the number of Random Walks on a $d$-D lattice starting at the Origin which never land on the same lattice point twice in $n$ steps be denoted $c(n)$. The first few values are

$\displaystyle c_d(0)$ $\textstyle =$ $\displaystyle 1$ (1)
$\displaystyle c_d(1)$ $\textstyle =$ $\displaystyle 2d$ (2)
$\displaystyle c_d(2)$ $\textstyle =$ $\displaystyle 2d(2d-1).$ (3)

The connective constant
\begin{displaymath}
\mu_d\equiv\lim_{n\to\infty} [c_d(n)]^{1/n}
\end{displaymath} (4)

is known to exist and be Finite. The best ranges for these constants are
$\displaystyle \mu_2$ $\textstyle \in$ $\displaystyle [2.62002, 2.6939]$ (5)
$\displaystyle \mu_3$ $\textstyle \in$ $\displaystyle [4.572140, 4.7476]$ (6)
$\displaystyle \mu_4$ $\textstyle \in$ $\displaystyle [6.742945, 6.8179]$ (7)
$\displaystyle \mu_5$ $\textstyle \in$ $\displaystyle [8.828529, 8.8602]$ (8)
$\displaystyle \mu_6$ $\textstyle \in$ $\displaystyle [10.874038, 10.8886]$ (9)

(Finch).


For the triangular lattice in the plane, $\mu<4.278$ (Alm 1993), and for the hexagonal planar lattice, it is conjectured that

\begin{displaymath}
\mu=\sqrt{2+\sqrt{2}}\,
\end{displaymath} (10)

(Madras and Slade 1993).


The following limits are also believed to exist and to be Finite:

\begin{displaymath}
\cases{
\lim_{n\to\infty} {c(n)\over \mu^n n^{\gamma-1}} & ...
...} {c(n)\over \mu^n n^{\gamma-1}(\ln n)^{1/4}} & for $d=4$,\cr}
\end{displaymath} (11)

where the critical exponent $\gamma=1$ for $d>4$ (Madras and Slade 1993) and it has been conjectured that
\begin{displaymath}
\gamma=\cases{
{\textstyle{43\over 32}} & for $d=2$\cr
1.162\ldots & for $d=3$\cr
1 & for $d=4$.\cr}
\end{displaymath} (12)


Define the mean square displacement over all $n$-step self-avoiding walks $\omega$ as

\begin{displaymath}
s(n)\equiv \left\langle{\vert\omega(n)\vert^2}\right\rangle{} = {1\over c(n)}\sum_\omega \vert\omega(n)\vert^2.
\end{displaymath} (13)

The following limits are believed to exist and be Finite:
\begin{displaymath}
\cases{
\lim_{n\to\infty} {s(n)\over n^{2\nu}} & for $d\not...
...n\to\infty} {s(n)\over n^{2\nu}(\ln n)^{1/4}} & for $d=4$,\cr}
\end{displaymath} (14)

where the critical exponent $\nu=1/2$ for $d>4$ (Madras and Slade 1993), and it has been conjectured that
\begin{displaymath}
\nu=\cases{
{\textstyle{3\over 4}} & for $d=2$\cr
0.59\ldots & for $d=3$\cr
{\textstyle{1\over 2}}& for $d=4$.\cr}
\end{displaymath} (15)

See also Random Walk


References

Alm, S. E. ``Upper Bounds for the Connective Constant of Self-Avoiding Walks.'' Combin. Prob. Comput. 2, 115-136, 1993.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cnntv/cnntv.html

Madras, N. and Slade, G. The Self-Avoiding Walk. Boston, MA: Birkhäuser, 1993.



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© 1996-9 Eric W. Weisstein
1999-05-26