page_title: Random walk
section_title: Applications
hierarchical_section_title: Random walk / Applications
caption_attribution_description: A less well known sculpture by a famous artist, Antony Gormley. Added to 'cream of the crop' as best of 2004Millenium Man
context_page_description: In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
  An elementary example of a random walk is the random walk on the integer number line, which starts at 0 and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality.
  As illustrated by those examples, random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of π can be approximated by the use of a random walk in an agent-based modeling environment.
context_section_description: As mentioned the range of natural phenomena which have been subject to attempts at description by some flavour of random walks is considerable, in particular in physics and chemistry, materials science, biology and various other fields. The following are some specific applications of random walk:
  In financial 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 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, ocular drift tends to behave like a random walk. According to some authors, fixational eye movements in general are also 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.
  On the web, the Twitter website uses random walks to make suggestions of whom to follow
  Dave Bayer and Persi Diaconis have proven that 7 riffle shuffles are enough to mix a pack of cards (see more details under shuffle). This result translates to a statement about random walk on the symmetric group which is what they prove, with a crucial use of the group structure via Fourier analysis.
caption: Antony Gormley's Quantum Cloud sculpture in London was designed by a computer using a random walk algorithm.
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