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3-Dimensional Random Walk

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RandomWalk3DLattice

On a three-dimensional lattice, a random walk has less than unity probability of reaching any point (including the starting point) as the number of steps approaches infinity. The probability of reaching the starting point again is 0.3405373296.... This is one of Pólya's random walk constants.

See also

1-Dimensional Random Walk, 2-Dimensional Random Walk, Pólya's Random Walk Constants

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References

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.Trott, M. "The Mathematica Guidebooks Additional Material: Lattice Sites Visited by Random Walkers." https://www.mathematicaguidebooks.org/additions.shtml#S_2_04.

Cite this as:

Weisstein, Eric W. "3-Dimensional Random Walk." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/3-DimensionalRandomWalk.html

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