Random walk
Introduction
A random walk is a mathematical object, known as a stochastic or random process, that describes a path consisting of a succession of random steps. In simple terms, it is a sequence of steps, each chosen randomly, and can be used to model phenomena in many scientific fields including physics, chemistry, biology, and economics Learn more about Physics, Learn more about Chemistry, Learn more about Biology, Learn more about Economics.
Mathematical Definition
In mathematical terms, a random walk is a sequence of random variables where each variable represents the next step in the walk. The steps are often mapped onto a mathematical space, which could be one-dimensional, two-dimensional, or even multidimensional. The steps themselves can be drawn from any random distribution, but are often chosen to be uniformly or normally distributed.
Types of Random Walks
Random walks can be classified into several types, depending on the constraints and characteristics of the steps. Some of the most common types include:
Simple Random Walk
A simple random walk is the most basic type of random walk, where each step is equally likely to go in any direction. In a one-dimensional simple random walk, each step is either +1 or -1 with equal probability.
Symmetric Random Walk
A symmetric random walk is a type of random walk where the step distribution is symmetric about zero. This means that the walk is equally likely to move in any direction at each step.
Asymmetric Random Walk
An asymmetric random walk is a type of random walk where the step distribution is not symmetric. This means that the walk has a tendency to move in one direction more than the others.
Continuous-Time Random Walk
A continuous-time random walk is a generalization of the simple random walk where the time between steps is also a random variable. This type of random walk is often used to model waiting times in queueing systems or the time between events in a Poisson process Learn more about Queueing Theory, Learn more about Poisson Process.
Applications of Random Walks
Random walks have found numerous applications in various fields of science and technology. Some of the most notable applications include:
Physics
In physics, random walks are used to model the motion of particles in fluids or gases, a process known as Brownian motion. They are also used in the study of polymers, where the random walk represents the movement of a polymer chain.
Biology
In biology, random walks are used to model the movement of animals, cells, or molecules. They are also used in the study of neural networks, where the random walk represents the propagation of signals through the network.
Economics
In economics, random walks are used to model the behavior of stock prices, exchange rates, and other financial variables. The random walk hypothesis is a theory that stock market prices are a random walk and cannot be predicted.
Computer Science
In computer science, random walks are used in the design of algorithms for tasks such as web page ranking and network routing.