Satunnaiskävely

Satunnaiskävely, the Finnish term for random walk, is a mathematical model used to describe a path consisting of successive random steps. In its simplest one‑dimensional form, a random walker starts at the origin of the integer number line and at each discrete time step moves either one unit to the left or one unit to the right with equal probability. The model generalizes to higher dimensions, where steps can be taken in any direction, and to continuous time or space, leading to the concepts of Brownian motion and continuous‑time random walks.

The basic properties of a random walk include the distribution of its position after \(n\) steps. For a symmetric walk in one dimension, the probability of being at position \(k\) after \(n\) steps is given by the binomial distribution \(\binom{n}{(n+k)/2}\,2^{-n}\) when \(n+k\) is even. The expected displacement from the origin remains zero at all times, while the expected squared displacement grows linearly with \(n\), evidencing diffusive behaviour. A key concept is the walk’s recurrence or transience: in one and two dimensions a simple random walk returns to its starting point with probability one, whereas in three or more dimensions it may escape to infinity.

Random walks appear throughout science and engineering. In physics they model particle diffusion and polymer chains; in biology they describe foraging behaviour and genetic drift; in finance they underlie models of asset price fluctuations; and in computer science they support algorithms for network exploration and randomized search. Analytical results for random walks relate to various areas of mathematics, including combinatorics, potential theory, and spectral graph theory. The random walk remains a foundational tool for studying stochastic processes and their real‑world manifestations.