Random walks model a host of phenomena and find applications in virtually all sciences. With only minor adjustments they may represent the thermal motion of electrons in a metal, or the migration of holes in a semiconductor. The continuum limit of the random walk model is known as “diffusion”. It may describe Brownian motion of a particle immersed in a fluid, as well as heat propagation, the spreading of a drop of dye in a glass of still water, bacterial motion and other types of biological migration, or the spreading of diseases in dense populations. Random-walk theory is useful in sciences as diverse as thermodynamics, crystallography, astronomy, biology, and even economics, in which it models fluctuations in the stock market.

The simple random walk

A random walk is a stochastic process defined on the points of a lattice. Usually, the time variable is considered discrete. At each time unit the “walker” steps from its present position to one of the other sites of the lattice according to a prescribed random rule. This rule is independent of the history of the walk, and so the process is Markovian.

In the simplest version of a random walk, the walk is performed in a hypercubic d-dimensional lattice of unit lattice spacing. At each time step the walker hops to one of its nearest-neighbor sites, with equal probabilities.