TOPICS
Search

Self-Avoiding Walk Connective Constant


Let the number of random walks on a d-D hypercubic lattice starting at the origin which never land on the same lattice point twice in n steps be denoted c_d(n). The first few values are

c_d(0)=1
(1)
c_d(1)=2d
(2)
c_d(2)=2d(2d-1).
(3)

In general,

 d^n<=c_d(n)<=2d(2d-1)^(n-1)
(4)

(Pönitz and Tittman 2000), with tighter bounds given by Madras and Slade (1993). Conway and Guttmann (1996) have enumerated walks of up to length 51.

On any lattice, breaking a self-avoiding walk in two yields two self-avoiding walks, but concatenating two self-avoiding walks does not necessarily maintain the self-avoiding property. Let c(n)=c_d(n) denote the number of self-avoiding walks with n steps in a lattice of d dimensions. Then the above observation tells us that c(m+n)<=c(m)c(n), and Fekete's lemma shows that

 mu_d=lim_(n->infty)[c_d(n)]^(1/n),
(5)

called the connective constant of the lattice, exists and is finite. The best ranges for these constants are

mu_2 in [2.62002,2.679192495]
(6)
mu_3 in [4.572140,4.7476]
(7)
mu_4 in [6.742945,6.8179]
(8)
mu_5 in [8.828529,8.8602]
(9)
mu_6 in [10.874038,10.8886]
(10)

(Beyer and Wells 1972, Noonan 1998, Finch 2003). The upper bound of mu_2 improves on the 2.6939 found by Noonan (1998) and was computed by Pönitz and Tittman (2000).

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

 mu=sqrt(2+sqrt(2)),
(11)

(Madras and Slade 1993).

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

 {lim_(n->infty)(c(n))/(mu^nn^(gamma-1))   for d!=4; lim_(n->infty)(c(n))/(mu^nn^(gamma-1)(lnn)^(1/4))   for d=4,
(12)

where the critical exponent gamma=1 for d>4 (Madras and Slade 1993) and it has been conjectured that

 gamma={(43)/(32)   for d=2; 1.162...   for d=3; 1   for d=4.
(13)

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

s(n)=<|omega(n)|^2>
(14)
=1/(c(n))sum_(omega)|omega(n)|^2.
(15)

The following limits are believed to exist and be finite:

 {lim_(n->infty)(s(n))/(n^(2nu))   for d!=4; lim_(n->infty)(s(n))/(n^(2nu)(lnn)^(1/4))   for d=4,
(16)

where the critical exponent nu=1/2 for d>4 (Madras and Slade 1993), and it has been conjectured that

 nu={3/4   for d=2; 0.59...   for d=3; 1/2   for d=4.
(17)

See also

Random Walk, Self-Avoiding Walk

Explore with Wolfram|Alpha

References

Alm, S. E. "Upper Bounds for the Connective Constant of Self-Avoiding Walks." Combin. Probab. Comput. 2, 115-136, 1993.Beyer, W. A. and Wells, M. B. "Lower Bound for the Connective Constant of a Self-Avoiding Walk on a Square Lattice." J. Combin. Th. A 13, 176-182, 1972.Conway, A. R. and Guttmann, A. J. "Square Lattice Self-Avoiding Walks and Corrections to Scaling." Phys. Rev. Lett. 77, 5284-5287, 1996.Finch, S. R. "Self-Avoiding Walk Constants." §5.10 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 331-339, 2003.Madras, N. and Slade, G. The Self-Avoiding Walk. Boston, MA: Birkhäuser, 1993.Noonan, J. "New Upper Bounds for the Connective Constants of Self-Avoiding Walks." J. Stat. Phys. 91, 871-888, 1998.Pönitz, A. and Tittman, P. "Improved Upper Bounds for Self-Avoiding Walks in Z^d." Electronic J. Combinatorics 7, No. 1, R21, 1-19, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r21.html.

Referenced on Wolfram|Alpha

Self-Avoiding Walk Connective Constant

Cite this as:

Weisstein, Eric W. "Self-Avoiding Walk Connective Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Self-AvoidingWalkConnectiveConstant.html

Subject classifications