Kesten’s theorem on random walks on groups
E839308
Kesten’s theorem on random walks on groups is a fundamental result in probability theory that characterizes amenability of groups via the spectral radius of associated random walks.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kesten’s theorem on random walks on groups canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10076743 — 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: Kesten’s theorem on random walks on groups Context triple: [Harry Kesten, notableWork, Kesten’s theorem on random walks on groups]
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A.
Pólya’s theorem on random walks
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.
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B.
Kac walk
The Kac walk is a probabilistic model introduced by mathematician Mark Kac to study the approach to equilibrium in kinetic theory via a simplified random process.
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C.
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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D.
Kakutani’s random ergodic theorem
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
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E.
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.
- 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: Kesten’s theorem on random walks on groups Target entity description: Kesten’s theorem on random walks on groups is a fundamental result in probability theory that characterizes amenability of groups via the spectral radius of associated random walks.
-
A.
Pólya’s theorem on random walks
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.
-
B.
Kac walk
The Kac walk is a probabilistic model introduced by mathematician Mark Kac to study the approach to equilibrium in kinetic theory via a simplified random process.
-
C.
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.
-
D.
Kakutani’s random ergodic theorem
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in geometric group theory ⓘ result in probability theory ⓘ |
| appliesTo |
countable discrete groups
ⓘ
finitely generated groups ⓘ |
| assumes |
probability measure with finite support
ⓘ
symmetric probability measure on a group ⓘ |
| characterizes | amenability of groups ⓘ |
| concerns |
convolution operators on groups
ⓘ
left-regular representation of a group ⓘ |
| equivalenceStatement | a group is amenable if and only if the spectral radius of a symmetric, finitely supported random walk on it equals 1 ⓘ |
| field |
geometric group theory
ⓘ
harmonic analysis on groups ⓘ probability theory ⓘ random walk theory ⓘ |
| hasConsequence |
amenability is equivalent to absence of spectral gap for all symmetric, finitely supported random walks
ⓘ
non-amenable groups admit a spectral gap for some symmetric random walk ⓘ |
| historicalPeriod | 20th century ⓘ |
| implies | non-amenable groups have spectral radius strictly less than 1 for some symmetric, finitely supported random walk ⓘ |
| influenced |
random walks on discrete groups and graphs
ⓘ
theory of expander graphs ⓘ |
| involvesConcept |
Følner condition
ⓘ
amenable group ⓘ random walk on a group ⓘ return probabilities of random walks ⓘ spectral radius of a bounded linear operator ⓘ |
| involvesObject |
Cayley graph of a group
ⓘ
Markov operator on ℓ² of the group ⓘ |
| mathematicsSubjectClassification |
43A07
ⓘ
60B15 ⓘ 60J10 ⓘ |
| namedAfter | Harry Kesten NERFINISHED ⓘ |
| providesCriterionFor | amenability via spectral radius ⓘ |
| relatedTo |
Day’s fixed point characterization of amenability
NERFINISHED
ⓘ
Følner’s characterization of amenability ⓘ spectral gap criteria for expansion ⓘ |
| relates |
amenability of a group
ⓘ
exponential decay of return probabilities ⓘ spectral radius of the associated Markov operator ⓘ |
| states | for non-amenable groups, return probabilities of a symmetric, finitely supported random walk decay exponentially fast ⓘ |
| typicalFormulation | for a finitely generated group with symmetric, finitely supported generating measure μ, the group is amenable iff the spectral radius of the associated convolution operator on ℓ²(G) is 1 ⓘ |
| usedIn |
analysis of random walks on Cayley graphs
ⓘ
ergodic theory on groups ⓘ operator algebra approaches to group theory ⓘ study of growth of groups ⓘ |
| usesConcept | spectral radius of a random walk ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Kesten’s theorem on random walks on groups Description of subject: Kesten’s theorem on random walks on groups is a fundamental result in probability theory that characterizes amenability of groups via the spectral radius of associated random walks.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.