Concept

First return

The first moment at which a walk or an orbit comes back to the place it started from. On the line it happens with certainty and takes infinitely long on average, which is a pairing worth holding on to.

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

A path folded about the first time it touches. A walk from 2 to 4 that touches the axis, with the part before its first touch reflected. The reflection is a path from the mirrored start to the same endpoint, and the correspondence is exact.

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

Time spent on one side of the axis. The exact distribution of the number of steps a 40-step fair walk spends above the axis. It is U-shaped: the extremes are the likeliest outcomes and an even split is the rarest.

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 walk Reflection principle Arcsine law Ballot problem Bijection Binomial coefficient Central limit Distribution Fairness Lattice walk

All concepts