N.B. A detailed on-line essay by S. Finch was the starting point for this entry.
Let be the probability that a Random Walk on a -D lattice returns to the origin. Pólya (1921) proved
that
(1) |
(2) |
(3) |
(4) | |||
(5) | |||
(6) | |||
(7) | |||
(8) |
(9) |
(10) |
4 | 0.20 |
5 | 0.136 |
6 | 0.105 |
7 | 0.0858 |
8 | 0.0729 |
See also Random Walk
References
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/polya/polya.html
Domb, C. ``On Multiple Returns in the Random-Walk Problem.'' Proc. Cambridge Philos. Soc. 50, 586-591, 1954.
Glasser, M. L. and Zucker, I. J. ``Extended Watson Integrals for the Cubic Lattices.'' Proc. Nat. Acad. Sci. U.S.A. 74,
1800-1801, 1977.
McCrea, W. H. and Whipple, F. J. W. ``Random Paths in Two and Three Dimensions.'' Proc. Roy. Soc. Edinburgh 60, 281-298, 1940.
Montroll, E. W. ``Random Walks in Multidimensional Spaces, Especially on Periodic Lattices.'' J. SIAM 4, 241-260, 1956.
Watson, G. N. ``Three Triple Integrals.'' Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.
© 1996-9 Eric W. Weisstein