Pólya's random walk constants
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Numbers
$d$
$p(d)$
1:
1
2:
1
3:
0.34053732955099914282627318443290289671060821712430209776323610537779196945896238464252808188905713099462301010684071008183988196055
4:
0.193201673224983937341870973329369160575873386450139495835026185709632292495810846029443
5:
0.135178609820655291047262429569315879691656444189996581804732903253409269458997391491061
6:
0.104715495628822010826135851106821546566448165078827186663230820073776585684914521029699
7:
0.0858449341133790091881081347851735566406978963249647696432467321391084259333693123925316
8:
0.0729126499593839984697453553883073696016118349162713731900079791927230662446014405543597
Definition
For $d\geq 1$, $p(d)$ is the probability that a random walk on a $d$-dimensional lattice returns to the origin.
Parameters
$d$
—   integer ($d \geq 1$)
Formulas
(1)
$p(1) = p(2) = 1$ [1].
(2)
$p(3) = 1-1/u(3)$ where $u(3)=\frac{\sqrt{6}}{32\pi^3}\Gamma(\frac{1}{24})\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})\Gamma(\frac{11}{24})$ and $\Gamma$ is the Gamma function [2].
References
[1]
G. Pólya, "Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz", Mathematische Annalen, Vol. 84, No. 1-2 (1921), 149-160. (doi)
[2]
W. H. McCrea and F. J. W. Whipple, "Random Paths in Two and Three Dimensions", Proceedings of the Royal Society of Edinburgh, Vol. 60, No. 3 (1940), 281-298. (doi)
OEIS sequences A086230 $(d=3)$, A086232 $(d=4)$, A086233 $(d=5)$, A086234 $(d=6)$, A086235 $(d=7)$, A086236 $(d=8)$.
Reliability: no error bounds specified for $d \geq 4$ (help needed)
Sources of data: [4] for $d \geq 4$