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Serguei Popov
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As for (3) (recurrence in $d\leq 2$ and transience in $d\geq 3$ of simple random walk), there are "electric networks"-proofs of these facts. See the classical book of Doyle and Snell "Random walks and electric networks", http://arxiv.org/abs/math/0001057

Serguei Popov
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  • 12
  • 21