Kesten’s theorem on random walks on groups

E839308

mathematical theorem result in geometric group theory result in probability theory

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.

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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 ⓘ

Referenced by (1)

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Harry Kesten → notableWork → Kesten’s theorem on random walks on groups ⓘ