Kesten’s theorem
E839310
Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kesten’s theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10076758 — 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 Context triple: [Harry Kesten, notableConcept, Kesten’s theorem]
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A.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
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B.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
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C.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
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D.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
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E.
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.
- 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 Target entity description: Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
-
A.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
B.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
C.
de Bruijn–van Aardenne–Ehrenfest theorem
The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
-
D.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in probability theory ⓘ |
| appliesTo |
random walks on graphs
ⓘ
random walks on groups ⓘ |
| assumes |
finitely generated group
ⓘ
symmetric probability measure with finite support on the group ⓘ |
| characterizes |
recurrence of random walks on groups
ⓘ
transience of random walks on groups ⓘ |
| concerns |
escape rate of random walks
ⓘ
long-term behavior of random walks ⓘ return probability decay ⓘ |
| conclusion | amenability is equivalent to spectral radius 1 for the associated random walk ⓘ |
| field |
percolation theory
ⓘ
probability theory ⓘ random walk theory ⓘ |
| formalism |
operator theory on Hilbert spaces
ⓘ
spectral theory of Markov operators ⓘ |
| hasImplicationFor |
growth of groups
ⓘ
isoperimetric inequalities on groups ⓘ percolation on Cayley graphs ⓘ return probabilities of random walks ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | non-amenable groups have random walks with spectral radius less than 1 ⓘ |
| influenced |
geometric group theory
ⓘ
modern theory of random walks on groups ⓘ probabilistic methods in group theory ⓘ study of percolation thresholds ⓘ |
| namedAfter | Harry Kesten ⓘ |
| provenBy | Harry Kesten ⓘ |
| relatedTo |
Cheeger inequalities
ⓘ
Følner’s criterion for amenability ⓘ
linked to:
Tarski’s theorem on amenable groups
isoperimetric constant of a graph ⓘ spectral gap of random walk operators ⓘ |
| relates |
amenability of groups
ⓘ
spectral radius of random walk operator ⓘ |
| statesThat | a finitely generated group is amenable if and only if the spectral radius of a symmetric random walk on the group equals 1 ⓘ |
| typicalAssumption |
finite generating set for the group
ⓘ
symmetric generating measure on the group ⓘ |
| usedIn |
analysis of random walks on non-amenable graphs
ⓘ
analysis of simple random walk on free groups ⓘ study of critical percolation on Cayley graphs ⓘ |
| usesConcept |
Cayley graphs
ⓘ
linked to:
Cayley graph
Markov operators ⓘ amenable groups ⓘ spectral radius ⓘ symmetric random walks ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kesten’s theorem Description of subject: Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.