Satunnaiskulkuun

Satunnaiskulkuun, known in English as random walk, is a mathematical object that describes a path consisting of a succession of random steps. Imagine a point moving on a grid or in a continuous space. At each step, the point's direction and length are determined by some random process. A simple example is a one-dimensional random walk where a particle on a line can move one step to the right or one step to the left with equal probability. This concept has broad applications in various fields.

In physics, random walks are used to model phenomena like Brownian motion, the erratic movement of particles suspended in a fluid. They also appear in the study of diffusion processes. In computer science, random walks are employed in algorithms for searching and sampling, such as in Markov chain Monte Carlo methods. The financial world uses random walks to model stock price fluctuations, although the assumption of true randomness is often debated.

The mathematical properties of a random walk depend heavily on the dimension of the space it inhabits and the nature of the steps. For instance, in one or two dimensions, a simple random walk is recurrent, meaning it will eventually return to its starting point with probability one. In three or more dimensions, however, the walk is transient; it is unlikely to return to the origin. The study of random walks involves concepts like expected displacement, variance, and the probability of reaching certain points.