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Concept

First return

The first moment at which a walk or an orbit comes back to the place it started from.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as reflection principle — the same set of essays touches all of them, so they are one junction rather than several.

024681012-4-2024steps takenpositionstart, 2mirror, −2first touchof the 792 paths from 2 to 4 in 12 steps, 220 touch the axisand 220 is exactly the number of paths from −2 to 4, which is what folding turns them into

The path folded at its first touch

Counting the walks that touch a line looks like a question about a walk's whole history. Fold each one where it first touches, and it becomes a question about where walks end up — which is a binomial coefficient, and is already known.

probability · random walk 0.0%3.1%6.3%9.4%12.5%040steps spent above the axischecked exhaustively against all 4,096 paths of 12 stepsevery walk of 40 steps, weighted by how likely it is — not a simulation, and not a samplean even split is the least likely outcome at 3.1%, and spending the whole walk on one side is the most likely at12.5% each way

Half the time is the rarest answer

In a fair game of many rounds, the fraction of the time one side is ahead is not usually near a half. It is usually near nought or one, and an even split is the single least likely outcome there is.

probability · random walk

Named alongside it

The objects these essays reach for when they reach for this one.

Random walkReflection principleArcsine lawBallot problemBijectionBinomial coefficientCentral limitDistributionFairnessLattice walk

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