*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

(1) | |||

(2) | |||

(3) |

The connective constant

(4) |

(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) |

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

(11) |

(12) |

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

(13) |

(14) |

(15) |

**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.

© 1996-9

1999-05-26