⟨r²(t)⟩ = 2dDt ⟨|r(t)|⟩ ≈ √(2dt) d = spatial dimension
Polya's theorem: a random walk is recurrent (returns to origin w.p.1) in d≤2 but transient in d≥3. In 1D: returns every time. In 2D: "will return but might take a long time." In 3D: only 34% chance of ever returning to origin. RMS displacement grows as √t in all dimensions.