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Random Walk

A random walk is a sequence of discrete steps in which each step is randomly taken subject to some set of restrictions in allowed directions and step lengths. Random walks may be taken along a line, in the plane, in space, or in other specified domains. Self-avoiding walks are walks (random or otherwise) in which previous steps may not be taken and/or previous portions of the walk may not be "crossed."

Random walks have interesting mathematical properties that vary greatly depending on the dimension in which the walk occurs and whether it is confined to a lattice.

Physically, random thermal perturbations in a liquid are responsible for a random walk phenomenon known as Brownian motion, and the collisions of molecules in a gas are a random walk responsible for diffusion.

See also

Galton Board, Markov Chain, Martingale, Percolation Theory, Random Walk--1-Dimensional, Random Walk--2-Dimensional, Random Walk--3-Dimensional, Self-Avoiding Walk, Self-Avoiding Walk Connective Constant

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References

Barber, M. N. and Ninham, B. W. Random and Restricted Walks: Theory and Applications. New York: Gordon and Breach, 1970.Chandrasekhar, S. In Selected Papers on Noise and Stochastic Processes (Ed. N. Wax). New York: Dover, 1954.Doyle, P. G. and Snell, J. L. Random Walks and Electric Networks. Washington, DC: Math. Assoc. Amer, 1984.Dykin, E. B. and Uspenskii, V. A. Random Walks. New York: Heath, 1963.Erdős, P. and Révész, P. "Three Problems on the Random Walk in Z^d." Studia Sci. Math. Hung. 26, 309-320, 1991.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971.Gardner, M. "Random Walks and Gambling" and "Random Walks on the Plane and in Space." Chs. 6-7 in Mathematical Circus: More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments. Washington, DC: Math. Assoc. Amer., pp. 66-86, 1992.Hughes, B. D. Random Walks and Random Environments, Vol. 1: Random Walks. New York: Oxford University Press, 1995.Hughes, B. D. Random Walks and Random Environments, Vol. 2: Random Environments. New York: Oxford University Press, 1996.Lawler, G. F. Intersections of Random Walks. Boston, MA: Birkhäuser, 1996.Révész, P. Random Walks in Random and Non-Random Environments. Singapore: World Scientific, 1990.Spitzer, F. Principles of Random Walk, 2nd ed. New York: Springer-Verlag, 1976.Weiss, G. Aspects and Applications of the Random Walk. Amsterdam, Netherlands: North-Holland, 1994.Weisstein, E. W. "Books about Random Walks." http://www.ericweisstein.com/encyclopedias/books/RandomWalks.html.

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Random Walk

Cite this as:

Weisstein, Eric W. "Random Walk." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RandomWalk.html

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