Total variation
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
How long until it forgets
The essays before this one settle where a chain ends up and how much time it spends there, and none of them asks how long the settling takes. That question has an exact answer, it is a single number, and it is the only thing any practical use of a chain depends on.
The forgetting that happens all at once
A single small chain forgets its start gradually, a little more with every step. A family of large ones can do something different — stay almost perfectly informed about where it began, and then lose all of it inside a window far shorter than the wait. That cliff is the cutoff phenomenon, and it is why "seven shuffles" is an answer rather than a convention.
A perfect coin and a biased shuffle
Give a program a perfect source of random numbers and ask it to shuffle a deck by swapping each card with a card chosen from anywhere in the deck. The result is biased, provably and for every deck of three or more cards: there are nⁿ equally likely runs and n! orders, and the first number is never a multiple of the second. In a deck of 52 some card-and-place pairs come up a third more often than they should. Swapping only with cards not yet placed fixes it; swapping only with later cards makes nothing but single cycles.
Named alongside it
The objects these essays reach for when they reach for this one.
Markov chainMixing timePermutationSpectral gapConvergenceCoupon collectorCycleDivisibilityEigenvalueHypercubeIterationRandom permutation