## Pólya's Random Walk Constants

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)

but
 (2)

for . Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that
 (3)

where

 (4) (5) (6) (7) (8)

where is a complete Elliptic Integral of the First Kind and is the Gamma Function. Closed forms for are not known, but Montroll (1956) showed that
 (9)

where

 (10)

and is a Modified Bessel Function of the First Kind. Numerical values from Montroll (1956) and Flajolet (Finch) are

 4 0.20 5 0.136 6 0.105 7 0.0858 8 0.0729

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.