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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
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(1) |
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(2) |
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(3) |
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(4) |
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(5) |
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(6) |
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(7) |
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(8) |
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(9) |
For the triangular lattice in the plane, (Alm 1993), and for the hexagonal planar lattice, it is conjectured that
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(10) |
The following limits are also believed to exist and to be Finite:
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(11) |
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(12) |
Define the mean square displacement over all -step self-avoiding walks
as
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(13) |
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(14) |
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(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