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Pólya's Random Walk Constants

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

p(1)=p(2)=1,
(1)

but

p(d)<1
(2)

for d>2. Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that

p(3)=1-1/(u(3))=0.3405373296...
(3)

(OEIS A086230), where

u(3)=3/((2pi)^3)int_(-pi)^piint_(-pi)^piint_(-pi)^pi(dxdydz)/(3-cosx-cosy-cosz)
(4)
=(12)/(pi^2)(18+12sqrt(2)-10sqrt(3)-7sqrt(6)){K[(2-sqrt(3))(sqrt(3)-sqrt(2))]}^2
(5)
=3(18+12sqrt(2)-10sqrt(3)-7sqrt(6))[1+2sum_(k=1)^(infty)exp(-k^2pisqrt(6))]^4
(6)
=3(18+12sqrt(2)-10sqrt(3)-7sqrt(6))theta_3^4(0,e^(-pisqrt(6)))
(7)
=(sqrt(6))/(32pi^3)Gamma(1/(24))Gamma(5/(24))Gamma(7/(24))Gamma((11)/(24))
(8)
=1.5163860592...
(9)

(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant, K(k) is a complete elliptic integral of the first kind, theta_3(0,q) is a Jacobi theta function, and Gamma(z) is the gamma function.

Closed forms for d>3 are not known, but Montroll (1956) showed that for d>3,

p(d)=1-[u(d)]^(-1),
(10)

where

u(d)=d/((2pi)^d)int_(-pi)^piint_(-pi)^pi...int_(-pi)^pi_()_(d)(d-sum_(k=1)^(d)cosx_k)^(-1)dx_1dx_2...dx_d
(11)
=int_0^infty[I_0(t/d)]^de^(-t)dt,
(12)

and I_0(z) is a modified Bessel function of the first kind.

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

dOEISp(d)
3A0862300.340537
4A0862320.193206
5A0862330.135178
6A0862340.104715
7A0862350.0858449
8A0862360.0729126

See also

Random Walk, Watson's Triple Integrals

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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.

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Pólya's Random Walk Constants

Cite this as:

Weisstein, Eric W. "Pólya's Random Walk Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolyasRandomWalkConstants.html

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