Random Walk Return Probability

Pólya's theorem (1921): A symmetric random walk on ℤ^d is recurrent (returns to origin with probability 1) in d=1,2, but transient (escapes forever) in d≥3. The return probability in d=3 is ≈ 0.3405.

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Pólya's Theorem: P(return | d=1) = 1, P(return | d=2) = 1, P(return | d≥3) < 1
Proof idea: generating function G(z) = Σ p_{2n} z^n diverges at z=1 iff d≤2.
d=1: p_{2n} ~ 1/√(πn), series diverges (harmonic-like)
d=2: p_{2n} ~ 1/(πn), series diverges (marginally!)
d=3: p_{2n} ~ C/n^{3/2}, series converges → escape probability > 0