Pólya,s Random Walk Constants
المؤلف:
Borwein, J. and Bailey, D
المصدر:
Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.
الجزء والصفحة:
...
24-3-2021
2215
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
(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 
,
![p(d)=1-[u(d)]^(-1),](https://mathworld.wolfram.com/images/equations/PolyasRandomWalkConstants/NumberedEquation4.gif) |
(10)
|
where
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.
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