Random walk — the series
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A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
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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.
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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.
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Two barriers and a fair game
A fair walk between two absorbing barriers is ruined with a probability that is a straight line in the starting stake, and lasts for a number of steps that is the product of what each side can lose. Both facts come from the same two-line recurrence, and both are bad news for the smaller player.
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The walk that becomes a curve
Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.
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The ground a walk covers
A random walk of a thousand steps visits far fewer than a thousand places: in one dimension about fifty, in the plane about four hundred, in space about six hundred and sixty. The share of steps that land on new ground is exactly the chance of never coming home — so the number that decides whether a walker returns also decides how much of the world it sees.
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A walk that may not step where it has been
Forbid a walk on the square grid from ever revisiting a site and the number of possible n-step walks grows like 2.638ⁿ instead of 4ⁿ — a number nobody can write down exactly. On the honeycomb it is exactly √(2 + √2), proved in 2010. And the walks spread out like n to the three-quarters, faster than any ordinary walk, which physicists have used since 1949 and mathematicians still cannot prove.
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A walk that follows its own footsteps
Let a walk step up with probability equal to the share of up-steps it has taken so far, and every count of up-steps after twenty steps is exactly equally likely. The walk leaves along a straight line at a speed chosen by its first few steps, and although it is expected to come back to its start infinitely often, almost every run comes back only a few times.
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An urn forgets its start only below one half
Let each draw from an urn add balls of both colours in fixed amounts, and the long run depends on a single ratio of two eigenvalues. Below one half the urn behaves like a coin, its fluctuations spread like the square root of the draws and settle into a bell. Above one half the first few draws decide most of the outcome, the spread grows faster, and the shape that results is not a bell and depends on how the urn began.
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Two losing games that win together
Game A is a coin that wins 49.5% of the time. Game B tosses a bad coin when the capital is a multiple of three and a good one otherwise, and it loses too, because the capital spends more than a third of its time on multiples of three. Choose between the two games at random and the walk drifts upwards by 0.0157 a round; play A, B, B over and over and it gains 0.0574. Nothing is wrong with the arithmetic. The losing coin A wins by knocking the capital off the bad remainder.
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Two losing games with a memory
Parrondo's first paradox needed a game that watched the player's capital. This one watches nothing but the last two results: a coin that wins nine times in ten after two losses, one time in four after a loss and a win, seven times in ten after two wins. Alone it loses, and so does a slightly unfair coin; played at random together they win. The walk treats every position alike, and the paradox survives — because it was never about position.