The science
A drunk man taking random steps on a city grid is certain to get home eventually, but his average time to get there is infinite. Let him move up and down as well and the odds change: a random walker in three dimensions ever gets back to its start with probability only about 34% (0.3405…). George Pólya proved the difference between two and three dimensions in 1921; he pictured a stroller in a square street network.
Everything here is simulated: the man and the bird are single random walks, and the flock is 200,000 simulated birds (4,000 drawn), 34.0% of which had been home within 100,000 steps.
Sources
- Pólya, G. (1921). Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz. Mathematische Annalen 84, 149–160. https://doi.org/10.1007/BF01458701
- Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Vol. 1, ch. XIV §7 and ch. XV §5.
- Glasser, M. L. & Zucker, I. J. (1977). Extended Watson integrals for the cubic lattices. PNAS 74(5), 1800–1801. https://doi.org/10.1073/pnas.74.5.1800
- OEIS A086230: the 3D return probability, 0.34053732955…
- Durrett, R. (2019). Probability: Theory and Examples, 5th ed., §5.4, p. 289 (Kakutani's joke).
