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Pólya's Random Walk Constants

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Pólya's random walk constants p(d) give the probability that a simple symmetric random walk on the integer lattice Z^d, starting at the origin, returns there at a positive time, where d is a positive integer. Each step independently moves to one of the 2d nearest neighbors with equal probability. Pólya (1921) 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))=0.3405373295...
(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 with elliptic modulus k, theta_3(0,q) is a Jacobi theta function, and Gamma(z) is the gamma function.

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.

For integers d>=3, Gaunt et al. (2024) gave the representation

u(d)=F_C^((d))(1,1/2;1,...,1;1/(d^2),...,1/(d^2)),
(13)

where F_C^((d)) is a Lauricella function with d denominator parameters and d arguments.

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

dOEISp(d)
3A0862300.340537
4A0862320.193202
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.Domb, C. "On Multiple Returns in the Random-Walk Problem." Proc. Cambridge Philos. Soc. 50, 586-591, 1954.Finch, S. R. "Pólya's Random Walk Constant." §5.9 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 322-331, 2003.Gaunt, R. E.; Nadarajah, S.; and Pogány, T. K. "On Pólya's Random Walk Constants." Proc. Amer. Math. Soc. 152, 3593-3597, 2024. https://doi.org/10.1090/proc/16854.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. https://doi.org/10.1073/pnas.74.5.1800.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.Pólya, G. "Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz." Math. Ann. 84, 149-160, 1921. https://doi.org/10.1007/BF01458701.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. 10, 266-276, 1939. https://doi.org/10.1093/qmath/os-10.1.266.

Referenced on Wolfram|Alpha

Pólya's Random Walk Constants

Cite this as:

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

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