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 lattice starting at the Origin which never land on the same lattice point twice in steps be denoted . 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

(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 -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

(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.