Random Walk - Applications

Applications

The following are some applications of random walk:

• In economics, the "random walk hypothesis" is used to model shares prices and other factors. Empirical studies found some deviations from this theoretical model, especially in short term and long term correlations. See share prices.
• In population genetics, random walk describes the statistical properties of genetic drift
• In physics, random walks are used as simplified models of physical Brownian motion and diffusion such as the random movement of molecules in liquids and gases. See for example diffusion-limited aggregation. Also in physics, random walks and some of the self interacting walks play a role in quantum field theory.
• In mathematical ecology, random walks are used to describe individual animal movements, to empirically support processes of biodiffusion, and occasionally to model population dynamics.
• In polymer physics, random walk describes an ideal chain. It is the simplest model to study polymers.
• In other fields of mathematics, random walk is used to calculate solutions to Laplace's equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics.
• In computer science, random walks are used to estimate the size of the Web. In the World Wide Web conference-2006, bar-yossef et al. published their findings and algorithms for the same.
• In image segmentation, random walks are used to determine the labels (i.e., "object" or "background") to associate with each pixel. This algorithm is typically referred to as the random walker segmentation algorithm.

In all these cases, random walk is often substituted for Brownian motion.

• In brain research, random walks and reinforced random walks are used to model cascades of neuron firing in the brain.
• In vision science, fixational eye movements are well described by a random walk.
• In psychology, random walks explain accurately the relation between the time needed to make a decision and the probability that a certain decision will be made.
• Random walks can be used to sample from a state space which is unknown or very large, for example to pick a random page off the internet or, for research of working conditions, a random worker in a given country.

• When this last approach is used in computer science it is known as Markov Chain Monte Carlo or MCMC for short. Often, sampling from some complicated state space also allows one to get a probabilistic estimate of the space's size. The estimate of the permanent of a large matrix of zeros and ones was the first major problem tackled using this approach.

• Random walks have also been used to sample massive online graphs such as online social networks.
• In wireless networking, a random walk is used to model node movement.
• Motile bacteria engage in a biased random walk.
• Random walks are used to model gambling.
• In physics, random walks underlie the method of Fermi estimation.