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Pólya’s Theorem on Random Walks via Pólya’s Urn

David A. Levin and Yuval Peres
The American Mathematical Monthly
Vol. 117, No. 3 (March 2010), pp. 220-231
DOI: 10.4169/000298910x480072
Stable URL:
Page Count: 12
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Pólya’s Theorem on Random Walks via Pólya’s Urn
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Abstract We give a proof of Pólya’s 1921 theorem on the transience of the simple random walk on${\mathbb{Z}^3}$ using the Pólya urn process (Eggenberger and Pólya, 1923; Pólya, 1931). We give a self-contained exposition of the method of flows, which provides a necessary and sufficient condition for transience of a simple random walk on an infinite graph. The key ingredient to our proof of transience of ${\mathbb{Z}^3}$ is the construction of a flow on ${\mathbb{Z}^3}$ using the Pólya urn process. While the transience result is classical and can be proved in many ways, it is particularly satisfying to derive it from Pólya’s urn, a connection which surely was not realized by Pólya himself.

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