Pólya’s theorem on random walks
E637316
Pólya’s theorem on random walks is a fundamental result in probability theory stating that simple random walks on one- and two-dimensional lattices are recurrent (almost surely return to the starting point infinitely often), while in three or more dimensions they are transient.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Pólya’s theorem on random walks canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030869 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Pólya’s theorem on random walks Context triple: [George Pólya, notableIdea, Pólya’s theorem on random walks]
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A.
Random Walk and the Theory of Brownian Motion
"Random Walk and the Theory of Brownian Motion" is a mathematical work by Mark Kac that rigorously develops the connection between discrete random walks and continuous Brownian motion within probability theory.
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B.
Sylvester’s theorem on partitions
Sylvester’s theorem on partitions is a result in number theory that provides a systematic way to count integer partitions subject to certain congruence or restriction conditions, forming part of the foundational work in partition theory.
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C.
Harmonic Analysis and the Theory of Probability
Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
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D.
Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
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E.
Erdős–Rényi law of large numbers
The Erdős–Rényi law of large numbers is a refinement of the classical law of large numbers that provides precise asymptotic behavior and convergence rates for sums of independent random variables, developed by mathematicians Pál Erdős and Alfréd Rényi.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Pólya’s theorem on random walks Target entity description: Pólya’s theorem on random walks is a fundamental result in probability theory stating that simple random walks on one- and two-dimensional lattices are recurrent (almost surely return to the starting point infinitely often), while in three or more dimensions they are transient.
-
A.
Random Walk and the Theory of Brownian Motion
"Random Walk and the Theory of Brownian Motion" is a mathematical work by Mark Kac that rigorously develops the connection between discrete random walks and continuous Brownian motion within probability theory.
-
B.
Sylvester’s theorem on partitions
Sylvester’s theorem on partitions is a result in number theory that provides a systematic way to count integer partitions subject to certain congruence or restriction conditions, forming part of the foundational work in partition theory.
-
C.
Harmonic Analysis and the Theory of Probability
Harmonic Analysis and the Theory of Probability is a seminal mathematical monograph that connects Fourier-analytic methods with probabilistic concepts, helping to lay the foundations of modern probability theory.
-
D.
Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
-
E.
Erdős–Rényi law of large numbers
The Erdős–Rényi law of large numbers is a refinement of the classical law of large numbers that provides precise asymptotic behavior and convergence rates for sums of independent random variables, developed by mathematicians Pál Erdős and Alfréd Rényi.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in random walk theory
ⓘ
theorem in probability theory ⓘ |
| appliesTo |
lattice random walk
ⓘ
simple random walk ⓘ |
| assumes |
independent and identically distributed steps
ⓘ
simple symmetric step distribution ⓘ steps to nearest neighbors on the lattice ⓘ |
| characterizes | dimension dependence of recurrence and transience for simple random walks ⓘ |
| concernsProperty |
recurrence
ⓘ
transience ⓘ |
| conclusionForRecurrentCase | the random walk returns to its starting point infinitely often with probability 1 ⓘ |
| conclusionForTransientCase | the random walk returns to its starting point only finitely many times with probability 1 ⓘ |
| dimensionThreshold | 2 ⓘ |
| domain | integer lattice Z^d ⓘ |
| field |
probability theory
ⓘ
random walk theory ⓘ stochastic processes ⓘ |
| historicalPeriod | early 20th century ⓘ |
| implies |
in dimensions three and higher the probability of ever returning to the origin is less than 1
ⓘ
one- and two-dimensional simple random walks are null recurrent ⓘ |
| influenced |
development of modern Markov process theory
ⓘ
later work on random walks on graphs ⓘ |
| involves |
infinite time horizon behavior of random walks
ⓘ
probability of return to the origin ⓘ |
| mathematicalClassification | 0-1 law type result ⓘ |
| namedAfter | George Pólya NERFINISHED ⓘ |
| quantifier | almost surely ⓘ |
| recurrenceHoldsForDimension |
1
ⓘ
2 ⓘ |
| relatedConcept |
Green’s function of a random walk
ⓘ
Markov chain recurrence ⓘ Markov chain transience ⓘ harmonic functions on lattices ⓘ |
| relatedTo |
central limit theorem for random walks
NERFINISHED
ⓘ
law of large numbers for random walks ⓘ |
| states |
simple random walks on integer lattices of dimension three or higher are transient
ⓘ
simple random walks on one-dimensional integer lattices are recurrent ⓘ simple random walks on two-dimensional integer lattices are recurrent ⓘ |
| transienceHoldsForDimension |
3
ⓘ
4 ⓘ 5 ⓘ |
| usedIn |
Markov chain theory
NERFINISHED
ⓘ
percolation theory ⓘ potential theory on lattices ⓘ random walk models in physics ⓘ statistical physics ⓘ |
How these facts were elicited
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Input
Subject: Pólya’s theorem on random walks Description of subject: Pólya’s theorem on random walks is a fundamental result in probability theory stating that simple random walks on one- and two-dimensional lattices are recurrent (almost surely return to the starting point infinitely often), while in three or more dimensions they are transient.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.