Kesten’s theorem

E839310

mathematical theorem result in probability theory

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.

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All labels observed (1)

Label	Occurrences
Kesten’s theorem canonical	1

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 NERFINISHED ⓘ
provenBy	Harry Kesten NERFINISHED ⓘ
relatedTo	Cheeger inequalities NERFINISHED ⓘ Følner’s criterion for amenability NERFINISHED ⓘ 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 NERFINISHED ⓘ Markov operators ⓘ amenable groups ⓘ spectral radius ⓘ symmetric random walks ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Harry Kesten → notableConcept → Kesten’s theorem ⓘ