The drunkard and the bird
It is the hour when the lamps are the only sober things on the street. A man leans in the doorway of the tavern, listing gently, deciding which of the four ways out of the square is his. On the sill above him a starling has been drinking the spilled beer from the gutter and is, in its own smaller way, also listing.
THE MAN. Don’t look at me like that. I’ll get home.
THE BIRD. I didn’t say you wouldn’t.
THE MAN. You’ve got that look. The look that says he’ll be face-down in the fountain by two. But I want you to understand something, bird. I couldn’t tell you my own name right now. I couldn’t point to home if you paid me. And I will still get there. Watch. I’ll stumble off in some direction, and at the next corner I’ll forget why, and pick another, and forget that too, all night — and it doesn’t matter. The city will hand me back to my own door. It always does.
THE BIRD. On the ground, yes. I’ll grant you the ground.
THE MAN. What’s that supposed to mean.
THE BIRD. It means the ground is a plane, and a plane is kind. Take a step you choose at random — north, south, east, west, a coin for each — and keep doing it forever, no memory, no aim, the way you’re about to. Do you know how many times, over an endless night, such a walker passes back through his own doorway?
THE MAN. Once. If he’s lucky.
THE BIRD. Infinitely many times. Not “he gets home” — he gets home, and leaves, and gets home again, and again, without end. Certainty isn’t even the surprising part. The surprising part is that once is impossible. A wandering man on a flat earth visits every doorway, including his own, an unlimited number of times. Return isn’t likely. It’s guaranteed, and then guaranteed again.
THE MAN. (straightening, pleased) Then what are we arguing about. I’ll get home.
THE BIRD. We aren’t the same, is what we’re arguing about. You have two directions to be lost in. I have three.
THE MAN. Up.
THE BIRD. Up. When you forget yourself at a corner you pick one of four ways. When I forget myself in the air I pick one of six — your four, and up, and down. And that one extra freedom is not a small thing, drunkard. It is the difference between a night that ends at your own bed and a night that never ends at all.
THE MAN. Come off it. Random is random. What difference could a little headroom make to a bird too sozzled to use it.
THE BIRD. Here is the difference, and it is exact, so listen past the beer. Whether a wanderer is sure to come home turns on a single question: over all of time, how many times would you expect him to be standing on the very spot he started? Add up, step by step, the chance he’s home right now, plus the chance he’s home two steps from now, plus four, plus six — the whole infinite ledger. If that sum runs off to infinity, he comes home for certain, over and over. If the sum settles to a finite number, he does not.
THE MAN. And?
THE BIRD. And on your flat ground the chance of being home at step number n fades slowly — like one-over-n. And the sum of one-over-n, drunkard, is the one every schoolboy is shown never stops growing. Slowly, grudgingly, logarithmically — the faintest possible growth that is still growth — but it never stops. So the ledger is infinite. So the plane keeps you. So you go home.
THE MAN. (smug) I go home.
THE BIRD. In the air the chance I’m home at step n fades faster — like one over n-and-a-half-roots-of-n, because the third direction gives my luck one more way to leak away and not come back. And that sum — I’ve added it, I’ve nothing else to do up here — that sum stops. It comes to rest at about one and a half. One and a half. Over my entire endless life I should expect to stand on my starting perch about three halves of a time, counting the once I start there — and then, most likely, never again.
THE MAN. So you get home once and a half.
THE BIRD. So I get home with probability about a third. Roll it out: barely more than a one-in-three chance I ever touch that perch again. Two times in three, drunkard, the bird does not come home. Not “comes home late.” Does not come home. Wanders outward into the dark and the sum of all its chances to return adds up to less than one, and the missing part — the part that would have made it certain — is simply gone into the third dimension.
(The man is quiet a moment. Somewhere a clock gives up counting.)
THE MAN. That can’t be right. We take the same drunken steps. You just have more room.
THE BIRD. More room is exactly the curse. You keep hearing “more room” as “more ways to get lucky.” It’s the opposite. More room means more ways for a path to thread past its old home without ever crossing it again. On a line you can hardly avoid your doorway — you can only be to the left of it or the right, and to get from one side to the other you must step on it. On a plane it’s harder to miss but you still can’t, not forever. In the air there’s so much emptiness to fall through that a path, once it drifts off, has room to drift forever, and the door it left behind is a single point in an ocean of directions it will never happen to pick again.
THE MAN. (slowly) So it was never about me. Whether I get home.
THE BIRD. It was never about you. Same steps, same drunkenness, same blind luck — and one of us is certain and one of us is a coin weighted against him, and the only thing that decided it was how many directions the night had. The homecoming is a fact about the room, drunkard. Not about the walker. You could be the wisest sober man alive; in my three dimensions you’d still likely never see your bed. I could be blind drunk, which I am; on your two, I’d be certain of mine.
THE MAN. Then trade me. You take the ground.
THE BIRD. (a dry little laugh) I would. But I’ll tell you the cruelty they don’t put on the pretty side of this. Even yours — even the sure homecoming — comes with a catch, and it’s a mean one. You will get home. But if you ask me when — how long on average your stagger will take — the answer is: forever. The average is infinite. Certain to arrive, and no finite time you can name for it. You’ll make it. I just can’t promise it inside any number of nights. The plane keeps its word and takes its time, all the time there is.
THE MAN. Certain, but the wait is forever.
THE BIRD. Certain, but the wait is forever. That’s the fine print on the good outcome.
THE MAN. (pushing off the doorframe, aiming himself at a gap between the lamps) One more thing, bird, before I go be a proof of your theorem. If it’s the room and not the walker — could I lose the guarantee? On my own two dimensions?
THE BIRD. In a heartbeat. Tilt the ground. Put the faintest slope under you — a breeze, a limp, a street that runs downhill — so your coin isn’t fair, so on average each step carries you a hair’s-breadth one way. Then even on a line you’re my kind of lost: you drift off down the slope and never come back. The guarantee was never really the plane’s. It was the fairness of the plane — that your blind steps had no lean in them, that they cancelled. Take away the balance and two dimensions won’t save you either. It isn’t the room alone. It’s the room, and that nothing was tilting you.
THE MAN. (from the middle of the square now, turning, forgetting why he turned) Fair steps on flat ground. That’s the whole of it?
THE BIRD. That’s the whole of it. Go on. You’re certain. Take your infinite time about it.
He picks a direction he will not remember choosing. The bird watches him fold into the dark and come out the far side and fold in again, twice, three times, crossing his own old footprints — the plane already handing him back to himself. Then the bird finishes the last of the gutter-beer, and lifts, because that is what it has, into the one extra direction that will most likely cost it everything, and is gone, upward, into the enormous arithmetic of the night.
The drunk man’s return is certain (probability 1) and the drunk bird’s is not (about 0.34) — and the only thing that differs is the number of dimensions. In one and two, the sum of a walker’s chances of being home diverges, so return is sure and endless; in three it converges (to 1.516, the Watson number), so a walker keeps, on average, only about half a homecoming and most likely none. The certainty is a property of the space, not the traveller — and it survives only while the steps are fair. (Pearson named the random walk in Nature, 1905; Rayleigh had it as random sound-phases in 1880; Pólya proved the dimension law in 1921; Kakutani gave it the drunk man and the bird. All numbers rebuilt from scratch — research/home-depends-on-the-room/, 22/22.)
