The walk that always comes home

The walk that always comes home — until it can fly.

A walker stands on a grid and steps at random for ever. No memory, no destination, no preference. Will it ever stand on its starting square again? On a line, and on a flat grid, the answer is not probably. It is certainly — probability exactly one. Give it one more dimension to wander in and certainty collapses to about one chance in three.

One walker, stepping at random — and how often a walk ever gets home ● where it started — share of walks home by step n — the ceiling for this many dimensions

dimensions— walks that got home— ceiling, given for ever— walks finished—

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How many dimensions it can wander in 2 — a flat grid

How long each walk is given 3,000 steps

↳ your turn — drag how many dimensions from 2 to 3 and watch the cyan curve stop climbing

What you're seeing

The walker above has no plan. At each step it picks one of its neighbouring squares at random — two of them on a line, four on a flat grid, six in the air — and goes there. Its starting square is ringed in amber. Nothing pulls it home and nothing pushes it away.

The chart below keeps score over many such walks. Each finished walk answers one question: did you ever stand on the start again? The cyan curve is the share that did. Every extra step is another chance, so the curve can only climb. The question is what it climbs to.

On a line and on a flat grid it climbs to one. Not to ninety-nine per cent, to one. Give the walks longer and the share home gets closer; give them for ever and coming home is certain.

But watch the pace, because this is the part that usually gets oversold. In one dimension the share still out after n steps shrinks like one over the square root of n. After ten thousand steps, eight walks in a thousand are still wandering. Certain is not the same word as soon.

Now drag the dimensions slider to 3. The curve lifts off, bends, and stops — flat against a grey line at about 34%. That is not impatience. About a third of the walks come home and the other two thirds never do — and the third that make it are quick about it. Half are home by step four, nine in ten inside a hundred. Give them a hundred thousand steps instead of three thousand and the ceiling does not move, because almost everyone who was ever coming back has already arrived.

Here is why the third direction breaks it. Add up the time a walk expects to spend standing on its start, over all steps, and you get either a finite number or an infinite one. On a line and on a grid that sum is infinite, so the walker keeps coming back. In the air it is finite: about 1.52 visits, counting the one it begins with. A walk that expects to stand there 1.52 times cannot be sure of standing there twice.

The more directions, the lonelier it gets: 34% in three, 19% in four, 14% in five. There is more room to be lost in and no more reason to come back.

What this is and isn't

This is a walk on a perfect endless lattice, with fair, independent, memoryless steps. Those words are load-bearing, and the first one most of all. Tilt the coin to 55/45 and even a walk on a line is no longer certain to return: it comes home nine times in ten, which is not the same as always.

It says nothing about a walk with a wall, an edge, or a home that moves. And the figure on screen is a tally of finished walks, not a formula drawn as a curve. You are watching a sample agree with a theorem, which is why it wobbles.

The rule, exactly. A simple random walk on the lattice ℤ^d steps to one of the 2d neighbours with equal probability. Let u be the expected number of times it stands on the origin, counting step zero. Each visit is a fresh independent chance to return, so the number of visits is geometric: P(ever returns) = 1 − 1/u Pólya (1921) showed u is infinite for d = 1 and 2, and finite from d = 3 on. In three dimensions Watson's integral gives u = 1.5163860592, hence P = 0.3405373296. One dimension is exact and needs no simulation at all: the chance of not having returned within 2n steps is C(2n,n)/4ⁿ ≈ 1/√(πn).
Checked before this page was written (improve/verify/130-polya-recurrence.js):

The one-dimensional law confirmed exactly at hand-checkable sizes — 1/2 after two steps, 5/8 after four — and against its 1/√(πn) asymptote to 0.1% out to 100,000 steps.
The identity P = 1 − 1/u checked both ways against the published Watson constant, so “34%” and “1.52 visits” are confirmed to be one fact rather than two.
This page's own walk run in node at d = 3, 4 and 5, landing on the published constants. The tolerance is deliberately one-sided, because a walk cut off after a finite number of steps can only undercount returns.
The threshold tested as a change of shape rather than asserted. Over a hundredfold more steps the two-dimensional share climbs 70% → 80%, while the three-dimensional one moves 33.0% → 33.8% and sits on its ceiling.
Negative control — a “three-dimensional” walk carrying the axis bug that confines it to the plane comes home 77% of the time instead of 34%, and the correct walk on the same seed does not.
Negative control — the exact one-dimensional law halved, tested where the error is large rather than where the claim is loudest. The first version of that control looked at ten thousand steps and passed a page that was wrong.
Negative control — the comfortable “it must come home eventually” stated and refuted, alongside the divergent and convergent visit sums that are the real reason.
Negative control — a 55/45 coin returns with probability 2 min(p,q) = 0.9, not 1, so fairness is shown to carry weight. That closed form was wrong in this proof's first draft, and the simulation is what corrected it.
The claim that the returns come early was measured rather than asserted, after a review flagged it as unanchored: across 30,000 three-dimensional walks the median return happens at step 4, and 91% of all returns have happened by step 100.

Thought Toys · exhibit 130 · built 11 September 2026