Pólya’s theorem on random walks

E637316

result in random walk theory theorem in probability theory

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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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 ⓘ

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George Pólya → notableIdea → Pólya’s theorem on random walks ⓘ