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random walk
A random walk is a mathematical formalization of a path that consists of a succession of random steps. For example, 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 can all be ''modeled'' as random walks, although they may not be truly random in reality. The term ''random walk'' was first introduced by Karl Pearson in 1905.〔Pearson, K. (1905). ''The Problem of the Random Walk.'' Nature. 72, 294.〕 Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology.〔)>Van Kampen N. G., Stochastic Processes in Physics and Chemistry, revised and enlarged edition (North-Holland, Amsterdam) 1992.〕〔)>Redner S., A Guide to First-Passage Process (Cambridge University Press, Cambridge, UK) 2001.〕〔)>Goel N. W. and Richter-Dyn N., Stochastic Models in Biology (Academic Press, New York) 1974.〕〔)>Doi M. and Edwards S. F., The Theory of Polymer Dynamics (Clarendon Press, Oxford) 1986〕〔)>De Gennes P. G., Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca and London) 1979.〕〔)>Risken H., The Fokker–Planck Equation (Springer, Berlin) 1984.〕〔)>.〕〔)>Cox D. R., Renewal Theory (Methuen, London) 1962.〕 Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity.
Various different types of random walks are of interest. Often, random walks are assumed to be Markov chains or Markov processes, but other, more complicated walks are also of interest. Some random walks are on graphs, others on the line, in the plane, in higher dimensions, or even curved surfaces, while some random walks are on groups. Random walks also vary with regard to the time parameter. Often, the walk is in discrete time, and indexed by the natural numbers, as in . However, some walks take their steps at random times, and in that case the position is defined for the continuum of times . Specific cases or limits of random walks include the Lévy flight. Random walks are related to the diffusion models and are a fundamental topic in discussions of Markov processes. Several properties of random walks, including dispersal distributions, first-passage times and encounter rates, have been extensively studied.
==Lattice random walk==
A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a simple random walk, the location can only jump to neighboring sites of the lattice, forming a lattice path. In simple symmetric random walk on a locally finite lattice, the probabilities of the location jumping to each one of its immediate neighbours are the same. The best studied example is of random walk on the ''d''-dimensional integer lattice (sometimes called the hypercubic lattice) .〔Révész Pal, Random walk in random and non random environments, World Scientific, 1990〕
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