Pólya (1921): A simple random walk on ℤᵈ returns to the origin with probability 1 in dimensions d = 1 and d = 2 (recurrent), but escapes to infinity with positive probability in d ≥ 3 (transient). The critical probability p_ret = 1 for d ≤ 2 and p_ret ≈ 0.341 for d = 3. As Kakutani noted: "A drunk man will find his way home, but a drunk bird may get lost forever."