Pólya's Random Walk Constants

Let  be the probability that a random walk on a -D lattice returns to the origin. In 1921, Pólya proved that

(1)

but

(2)

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

(3)

(OEIS A086230), where

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant,  is a complete elliptic integral of the first kind,  is a Jacobi theta function, and  is the gamma function.

Closed forms for  are not known, but Montroll (1956) showed that for ,

(10)

where

			(11)
			(12)

and  is a modified Bessel function of the first kind.

Numerical values of  from Montroll (1956) and Flajolet (Finch 2003) are given in the following table.

	OEIS
3	A086230	0.340537
4	A086232	0.193206
5	A086233	0.135178
6	A086234	0.104715
7	A086235	0.0858449
8	A086236	0.0729126

REFERENCES:

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.

Finch, S. R. "Pólya's Random Walk Constant." §5.9 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 322-331, 2003.

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.

Sloane, N. J. A. Sequences A086230, A086231, A086232, A086233, A086234, A086235, and A086236 in "The On-Line Encyclopedia of Integer Sequences."

Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.