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ID: INPUT{id.yaml}

Title: >
  Pólya's random walk constants

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:
    type: Z
    constraints: $d \geq 1$
  
Comments:
  
Formulas:
  formula-1d-and-2d: >
    $p(1) = p(2) = 1$ CITE{Pol}.
  formula-3d: >
    $p(3) = 1-1/u(3)$ 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 
    HREF{https://en.wikipedia.org/wiki/Gamma_function}[Gamma function] CITE{MCW}.

Programs:

References:
  Pol:
    bib: >
      G. Pólya, "Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz", 
      Mathematische Annalen, Vol. 84, No. 1-2 (1921), 149-160.
    doi: 10.1007/BF01458701
  MCW:
    bib: >
      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: 10.1017/S0370164600020265
    
Links:
  MathWorld:
    title: "MathWorld: Pólya's Random Walk Constants"
    url: https://mathworld.wolfram.com/PolyasRandomWalkConstants.html
  OEIS: >
    OEIS sequences 
    HREF{https://oeis.org/A086230/}[A086230] $(d=3)$,
    HREF{https://oeis.org/A086232/}[A086232] $(d=4)$,
    HREF{https://oeis.org/A086230/}[A086233] $(d=5)$,
    HREF{https://oeis.org/A086230/}[A086234] $(d=6)$,
    HREF{https://oeis.org/A086230/}[A086235] $(d=7)$,
    HREF{https://oeis.org/A086230/}[A086236] $(d=8)$.

Similar tables:
  
Keywords:
- Polya #different spelling as in title
  
Tags:
- probability theory
- random walks

Data properties:
  type: R
  complete: no
  reliability: no error bounds specified for $d \geq 4$ (help needed)
  sources:
    - CITE{OEIS} for $d \geq 4$
  
Display properties:
  number-header: >
    $p(d)$

Numbers: 
  1: 1
  2: 1
  3: 0.34053732955099914282627318443290289671060821712430209776323610537779196945896238464252808188905713099462301010684071008183988196055
  4: 0.193201673224983937341870973329369160575873386450139495835026185709632292495810846029443
  5: 0.135178609820655291047262429569315879691656444189996581804732903253409269458997391491061
  6: 0.104715495628822010826135851106821546566448165078827186663230820073776585684914521029699
  7: 0.0858449341133790091881081347851735566406978963249647696432467321391084259333693123925316
  8: 0.0729126499593839984697453553883073696016118349162713731900079791927230662446014405543597

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Pólya's random walk constants

edit on github

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 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)

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 $(d=8)$.

Data properties

Entries are of type: real number

Table is complete: no

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

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