Random walk

A random walk is a sequence of steps whose directions or sizes are chosen randomly. It models evolving positions, values, and other quantities through chance-driven change.

Simple random walk: A walk on a graph or lattice that chooses each next step according to fixed transition probabilities. The classic version takes independent steps, often one unit left or right with equal probability.
Probability distribution: A mathematical description assigning probabilities to the possible values of a random variable. A step's distribution specifies which moves are possible and how likely each one is.
Gambler's ruin: A probability model in which a gambler's capital changes through repeated wins and losses until a boundary is reached. It is a finite walk whose absorbing boundaries represent bankruptcy or a target fortune.
Deterministic dynamical system: A system whose future state is uniquely determined by its current state and governing rules. Unlike a random walk, it has no chance-based choice between possible next steps.
Markov chain: A stochastic process whose next state depends on its current state, possibly through fixed transition probabilities. A random walk is a Markov chain when its current position determines its next-step distribution.
Independent and identically distributed random variables: Random variables that are mutually independent and share the same probability distribution. Many basic walks are sums of repeated steps with this common independence structure.
Diffusion: The net spread of particles or quantities from regions of higher concentration toward lower concentration. Random microscopic motion can generate macroscopic diffusion.
Lévy flight: A random process with step lengths drawn from a heavy-tailed distribution, allowing occasional very long jumps. Its jump distribution can produce extreme displacements unlike bounded-step walks.
Transition matrix: A matrix whose entries give probabilities of moving between states in one step. It encodes the allowed moves and their probabilities for walks on finite state spaces.
Expectation: The probability-weighted average value of a random variable. Expected step size determines whether a walk has a directional drift.

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