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Concept

Absorbing state

A state of a Markov chain that cannot be left once entered, so every walk reaching it stays there. Their presence replaces the question of where a chain settles with two others — how long until it stops, and where — and both have exact answers rather than limits.

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

A walk with a barrier at each end. Three games played to absorption on a table of 12, beside the chance of ruin from each starting stake — a straight line, because the walk is fair.

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.

probability · Random walk A game that stops, over totals 0 to 5. States in a row with arrows up and down between them and the two ends absorbing, above a table of the expected number of steps and the chance of ending at the top from each start.

The chain that stops

Give a chain a state it cannot leave and there is no long run to find — every walk ends. What is worth computing instead is how long it lasts and where it finishes, and both are exact answers to a linear system rather than limits of anything.

probability · Markov chains

Named alongside it

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

ExpectationMarkov chainRandom walkGamblers ruinInvariantLinear systemMartingaleMatrixProbabilityRecurrence relation

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