“Self-Avoiding Walk”的概念、定义、习题解题方法及翻译-数学辞典

单词

Self-Avoiding Walk

释义

Self-Avoiding Walk

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

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

			(1)
			(2)
			(3)

The connective constant

(4)

is known to exist and be Finite. The best ranges for these constants are

			(5)
			(6)
			(7)
			(8)
			(9)

(Finch).

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

(10)

(Madras and Slade 1993).

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

(11)

where the critical exponent

for

(Madras and Slade 1993) and it has been conjectured that

(12)

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

(13)

The following limits are believed to exist and be Finite:

(14)

where the critical exponent

for

(Madras and Slade 1993), and it has been conjectured that

(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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