Pólya's random walk constants

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Numbers

$d$

$p(d)$

0.34053732955099914282627318443290289671060821712430209776323610537779196945896238464252808188905713099462301010684071008183988196055

0.193201673224983937341870973329369160575873386450139495835026185709632292495810846029443

0.135178609820655291047262429569315879691656444189996581804732903253409269458997391491061

0.104715495628822010826135851106821546566448165078827186663230820073776585684914521029699

0.0858449341133790091881081347851735566406978963249647696432467321391084259333693123925316

0.0729126499593839984697453553883073696016118349162713731900079791927230662446014405543597

Definition

For $d\geq 1$, $p(d)$ is the probability that the simple random walk on $\mathbb{Z}^d$, started at the origin and taking each step uniformly among the $2d$ nearest neighbours, ever 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 $u(3)=\frac{\sqrt{6}}{32\pi^3}\Gamma(\frac{1}{24})\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})\Gamma(\frac{11}{24})$; the factors $\Gamma(1/24)$, $\Gamma(5/24)$, $\Gamma(7/24)$ and $\Gamma(11/24)$ are entries of Values of the Gamma function at rational numbers [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)

Links

[3]

MathWorld: Pólya's Random Walk Constants

[4]

OEIS sequences A086230 $(d=3)$, A086232 $(d=4)$, A086233 $(d=5)$, A086234 $(d=6)$, A086235 $(d=7)$, A086236 $(d=8)$.

Similar tables

Values of the Gamma function at rational numbers — contains $\Gamma(1/24)$, $\Gamma(5/24)$, $\Gamma(7/24)$ and $\Gamma(11/24)$, the gamma values in the closed form for $p(3)$

Data properties

Entries are of type: real number

Table is complete: no (it holds $1\leq d\leq8$, the dimensions covered by the cited closed forms and OEIS sources)

Reliability: no error bounds specified for $d \geq 4$ (help needed)

Sources of data: [4] for $d \geq 4$

How well the digits are known: heuristic

How they were obtained:

Mixed, and labelled by its weakest entries. The values for $d=1,2,3$ are exact or have a closed form: $d=3$ is Watson's integral, which ball arithmetic confirms. The rest were transcribed; the script's own attempt to compute them carries the comment that it does not work.