Stand at a corner of an infinite grid and start to wander — at every tick, one step in a direction chosen by a fair coin, no memory of where you have been. Question with a yes-or-no answer and a famous one: are you certain to come back to the corner you started from?
On a line, yes. On a flat plane, also yes — return is certain, probability exactly one, though you may have to wait a very long time. Add a third dimension — let the walker fly, up and down as well as the four compass steps — and the certainty breaks. The flying walker comes home only about 34% of the time. Two times in three it drifts off and never returns. George Pólya proved this in 1921: the simple symmetric random walk is recurrent in one and two dimensions and transient in three and above. The plane keeps you; volume lets you escape. The difference is one dimension too many.
Watch one walker
In 1-D and 2-D the teal flashes keep coming, forever — leave it running and the return count never stops climbing. In 3-D the walker usually crosses home a few times early, then wanders out of reach and the flashes stop. One walk can't prove anything (a 3-D walk might get lucky) — for that, run a crowd of them below.
Run a crowd of them
Certainty is a statement about the whole ensemble, so let many walkers run and count what fraction ever touches home again within a fixed horizon. A finite horizon can only undercount — some walkers would have returned later — so these numbers sit a little below the true probabilities. But the shape is unmistakable.
Gold line = the true return probability. As you extend the horizon, the 1-D and 2-D bars climb toward 1 (slowly in 2-D — the borderline case, see below), while the 3-D bar stalls just under 0.34 no matter how long you wait. That stall is transience: the missing two-thirds are gone for good.
Why two is the borderline
The proof is a single sum. Let p₂ₙ be the probability the walker is back at the origin after 2n steps (only even times can return). The walk is recurrent exactly when the expected number of returns is infinite — when Σ p₂ₙ diverges. A local central-limit estimate pins the size of each term:
So the terms shrink like n−d/2. In 1-D that is n−1/2 and the sum diverges; in 2-D it is n−1 — the harmonic series, which still just barely diverges; in 3-D it is n−3/2, and that sum converges. Two dimensions is the knife-edge: the walker comes home with probability one, but the expected wait is infinite. Three dimensions tips over the edge, and the total number of returns becomes finite.
The exact number
When the sum converges, it converges to something nameable. Its value u(3) = Σ p₂ₙ is the expected number of visits to the origin (counting the start), and the return probability is p = 1 − 1/u. In 1921 this was a limit nobody could evaluate; in 1939 G. N. Watson cracked the triple integral that u(3) hides inside, and found a closed form in Gamma functions:
So the drunk bird's odds of ever getting home are about 34.05% — a number with a Gamma-function pedigree, not a round one. The same machinery gives the odds in higher dimensions, and they keep falling: more room to get lost in means less chance of stumbling back.
| dimension | return prob p(d) | verdict | chance of coming home |
|---|
1-D and 2-D return with certainty (p = 1); from 3-D on, the bird escapes. The d≥4 values have no simple closed form — they are computed numerically (Finch, Mathematical Constants, §5.9). Sources for every figure are in the colophon.
Where the picture bends, honestly
The lattice is a model. Pólya's theorem is about steps on a perfect integer grid. The same recurrent-below-3, transient-at-3 threshold holds for Brownian motion (the continuous limit) and for any walk with finite-variance, mean-zero steps — the dimension is what matters, not the grid — but a walk with heavy-tailed jumps or a drift can behave differently. The claim is exactly: simple symmetric walk, finite variance, no drift.
The Monte-Carlo bars under-count. A walker that hasn't come home by step 4,000 might still return at step 40,000; with a finite horizon you never see those, so the measured fractions sit below the true 1, 1, and 0.3405. Extend the horizon and 1-D/2-D climb; 3-D does not, because its missing mass is genuinely at infinity, not merely late.
The aphorism is folklore. The drunk-man / drunk-bird line is universally attributed to Kakutani and the wording above is the standard one, but it carries no citation to a paper or dated talk — it is best read as traditionally attributed, not sourced. The mathematics under it is exact; the quotation's provenance is not, and saying so is part of the job.
Two dimensions is certain but cruel. "Recurrent" means return with probability one — it does not mean soon. In 2-D the expected time to return is infinite (that harmonic-series knife-edge again). You will get home; do not hold your breath.
- 1-D, 2-D, 3-D return terms p₂ₙ by exact BigInt counting, matched against brute-force path enumeration
- the local-CLT size 2(d/4πn)d/2 confirmed against the exact terms (ratio → 1)
- u(3) three independent ways: exact series + asymptotic tail, Watson's triple integral by quadrature, and the Gamma closed form — all agree on 1.5163860592
- p(3) = 1 − 1/u(3) = 0.3405373296; higher-d constants p(4)≈0.193, p(5)≈0.135 by lattice quadrature
- Monte-Carlo: 3-D return fraction lands near a third; 1-D far higher — recurrence vs transience seen, not assumed
29/29 checks pass in research/polya-recurrence/verify.mjs — pure Node, no dependencies, run with node research/polya-recurrence/verify.mjs.
One dimension is a tightrope you keep falling back onto. Two is a floor that always, eventually, catches you. Three is open sky — and in open sky, most things that leave do not come back. The whole theorem is in the gap between those last two, and the gap is exactly one coordinate wide.