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Absorbing State

A state that, once entered, is never left, its transition probability to itself is exactly 1.

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In Plain English

An absorbing state is a trap door: once the chain steps into it, it's stuck there forever. "Game over" in a board game, "bankrupt" in a gambler's-ruin model, or "death" in a simple epidemic model are all absorbing states, there's no transition out. A chain with at least one absorbing state, and a way to reach it from every other state, will eventually be absorbed with probability 1, which is exactly the kind of long-run guarantee that makes absorbing Markov chains useful for modeling processes with a natural endpoint.

Definition

A state $i$ in a Markov chain is absorbing if $p_{ii} = 1$ , equivalently, $p_{ij} = 0$ for all $j \neq = i$ . Once the chain enters an absorbing state it remains there for all future time steps. A Markov chain containing at least one absorbing state, in which every non-absorbing (transient) state has a positive-probability path to some absorbing state, is called an absorbing Markov chain. For such chains, standard results give closed-form expressions (via the fundamental matrix $N = (I - Q)^{- 1}$ , where $Q$ is the sub-matrix of transition probabilities among transient states) for quantities like the expected number of steps before absorption and the probability of ending up in each particular absorbing state.

Formula

p_{ii} = 1 ⟺ p_{ij} = 0 for all j \neq = i

Notation

p_{ii}

—Self-transition probability of state $i$; equals 1 exactly when $i$ is absorbing

N = (I - Q)^{- 1}

—Fundamental matrix of an absorbing chain, $Q$ = transition probabilities among transient states

Properties

Absorbing states model natural terminal outcomes cleanly: bankruptcy in the classic gambler's ruin problem, death or full recovery in simple compartmental disease models, and 'accept' or 'reject' outcomes in sequential decision processes are all standard examples.
A chain can have multiple absorbing states simultaneously, e.g. both 'ruin' and 'target wealth reached' in gambler's ruin, in which case the analysis asks not just when absorption happens but which absorbing state is reached, computed from the fundamental matrix.
Absorbing Markov chain theory gives exact, closed-form answers (not just simulation estimates) for expected time to absorption and absorption probabilities, one of the cleanest applications of the transition-matrix machinery to a concrete, interpretable question.

At a Glance

Defining condition

p_{ii} = 1

: the chain never leaves once it arrives

Contrastopposite in spirit to a transient state, which the chain eventually leaves for good

Absorbing chainone with at least one absorbing state reachable from every other state

Key computable quantityexpected steps to absorption, via the fundamental matrix

N = (I - Q)^{- 1}

Last updated September 8, 2026← Back to the Dictionary