Pólya's Theorem — 2D vs 3D Random Walk Recurrence

Pólya (1921): a simple random walk on ℤ² returns to the origin with probability 1 (recurrent). On ℤ³, return probability ≈ 0.3405 (transient). "A drunk man will find his way home, but a drunk bird may get lost forever." Watch live walks — 2D almost always returns, 3D often escapes.

2D Walk (ℤ²) — Recurrent

0
returns to origin
0
total steps
P(return | ℤ²) = 1 — proven via divergence of harmonic series
Mean return time = ∞ (null-recurrent)

3D Walk (ℤ³) — Transient

0
returns to origin
0
total steps
P(return | ℤ³) ≈ 0.3405 (Watson 1939)
Exact: 3√6/(96π³) × Γ(1/4)⁴ ≈ 0.3405

Return Rate vs Steps (running average)