Random walk

A random walk is a mathematical concept that describes a stochastic process where a particle, molecule, or any other entity moves randomly in space and time. The movement is often modeled as a series of discrete steps, where the direction and distance of each step are determined randomly.

Random walks are used in many fields, including physics, chemistry, biology, economics, and finance, to model various phenomena such as diffusion, Brownian motion, and stock market prices. They are also used in computer science and artificial intelligence to simulate random behavior and generate random numbers.

In a random walk, the entity's position at each step depends only on its previous position and the direction and distance of the current step. The probability distribution of the steps can be symmetric or asymmetric, and the step sizes can be constant or variable.

Random walks can be one-dimensional, two-dimensional, or higher-dimensional, depending on the number of dimensions of the space in which the entity moves. They can also be discrete or continuous, depending on the nature of the steps.

The behavior of a random walk depends on the parameters of the model, such as the step size and the probability distribution of the steps. In some cases, the random walk can converge to a steady state or exhibit periodic behavior, while in other cases, it can diverge or exhibit chaotic behavior.