Editing Random walk (section)

==Applications==
{{More citations needed section|date=February 2013}}
[[File:Antony Gormley Quantum Cloud 2000.jpg|thumb|[[Antony Gormley]]'s ''[[Quantum Cloud]]'' sculpture in London was designed by a computer using a random walk algorithm]]
As mentioned, the range of natural phenomena which have been subject to attempts at description by some flavour of random walks is considerable. This is  particularly the case in the fields of physics,<ref name=[5]>Risken H. (1984) ''The Fokker–Planck Equation''. Springer, Berlin.</ref><ref name=[4c]>De Gennes P. G. (1979) ''Scaling Concepts in Polymer Physics''. Cornell University Press, Ithaca and London.</ref>chemistry,<ref name=[1]>Van Kampen N. G. (1992) ''Stochastic Processes in Physics and Chemistry'', revised and enlarged edition. North-Holland, Amsterdam.</ref> [[materials science]],<ref name=[6]>{{cite book | last = Weiss | first = George H. | author-link = George Herbert Weiss | isbn = 978-0-444-81606-1 | mr = 1280031 | publisher = North-Holland Publishing Co., Amsterdam | series = Random Materials and Processes | title = Aspects and Applications of the Random Walk | year = 1994}}</ref><ref name=[4]>Doi M. and Edwards S. F. (1986) ''The Theory of Polymer Dynamics''. Clarendon Press, Oxford</ref> and biology.<ref name=[3]>Goel N. W. and [[Nira Dyn|Richter-Dyn N.]] (1974) ''Stochastic Models in Biology''. Academic Press, New York.</ref><ref name=[2]>Redner S. (2001) ''A Guide to First-Passage Process''. Cambridge University Press, Cambridge, UK.</ref><ref name=[7]>Cox D. R. (1962) ''Renewal Theory''. Methuen, London.</ref> 

The following are some specific applications of random walks:
*In [[financial economics]], the [[random walk hypothesis]] is used to model shares prices and other factors.<ref>David A. Kodde and Hein Schreuder (1984), Forecasting Corporate Revenue and Profit: Time-Series Models versus Management and Analysts, Journal of Business Finance and Accounting, vol. 11, no 3, Autumn 1984</ref> Empirical studies found some deviations from this theoretical model, especially in short term and long term correlations. See [[share price]]s.
*In [[population genetics]], random walk describes the statistical properties of [[genetic drift]] or long-term randomly fluctuating [[Natural selection|selection.]]<ref>{{Cite journal |last1=Hansen |first1=Thomas F. |last2=Martins |first2=Emília P. |date=August 1996 |title=Translating Between Microevolutionary Process and Macroevolutionary Patterns: The Correlation Structure of Interspecific Data |url=https://academic.oup.com/evolut/article/50/4/1404/6870754 |journal=Evolution |language=en |volume=50 |issue=4 |pages=1404–1417 |doi=10.1111/j.1558-5646.1996.tb03914.x |pmid=28565714 |issn=0014-3820}}</ref>
*In [[physics]], random walks are used as simplified models of physical Brownian motion and diffusion such as the [[random]] [[Motion (physics)|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 [[semiconductor manufacturing]], random walks are used to analyze the effects of thermal treatment at smaller nodes. It is applied to understand the diffusion of [[dopants]], [[Product defect|defects]], [[impurities]] etc., during critical fabrication steps. Random walk treatments are also used to study the diffusion of reactants, products and plasma during [[chemical vapor deposition]] processes. Continuum diffusion has been used to study the flow of gases, at macroscopic scales, in CVD reactors. However, smaller dimensions and increased complexity has forced us to treat them with random walk. This allows for accurate analysis of stochastic processes, at molecular level and smaller, in semiconductor manufacturing.
*In [[theoretical biology|mathematical ecology]], random walks are used to describe individual animal movements, to empirically support processes of [[diffusion|biodiffusion]], and occasionally to model [[population dynamics]].
*In [[polymer physics]], random walk describes an [[ideal chain]]. It is the simplest model to study [[polymers]].<ref>{{cite book|last1=Jones|first1=R.A.L.|title=Soft condensed matter|url=https://archive.org/details/softcondensedmat00jone|url-access=limited|date=2004|publisher=Oxford Univ. Pr.|location=Oxford [u.a.]|isbn=978-0-19-850589-1|pages=[https://archive.org/details/softcondensedmat00jone/page/n89 77]–78|edition=Reprint.}}</ref>
*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 [[Mathematical analysis|analysis]] and [[combinatorics]].
* In [[computer science]], random walks are used to estimate the size of the [[www|Web]].<ref name="Bar-Yossef Gurevich 2008 pp. 1–74">{{cite journal | last1=Bar-Yossef | first1=Ziv | last2=Gurevich | first2=Maxim | title=Random sampling from a search engine's index | journal=Journal of the ACM | publisher=Association for Computing Machinery (ACM) | volume=55 | issue=5 | year=2008 | issn=0004-5411 | doi=10.1145/1411509.1411514 | pages=1–74}}</ref>
* In [[Segmentation (image processing)|image segmentation]], random walks are used to determine the labels (i.e., "object" or "background") to associate with each pixel.<ref>{{cite journal|pmid=17063682|url=http://cns-web.bu.edu/~lgrady/grady2006random.pdf|year=2006|last1=Grady|first1=L|title=Random walks for image segmentation|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|volume=28|issue=11|pages=1768–83|doi=10.1109/TPAMI.2006.233|citeseerx=10.1.1.375.3389|s2cid=489789|access-date=2 November 2016|archive-date=5 July 2017|archive-url=https://web.archive.org/web/20170705061011/http://cns-web.bu.edu/~lgrady/grady2006random.pdf}}</ref> This algorithm is typically referred to as the [[random walker (computer vision)|random walker]] segmentation algorithm.
*In [[human brain|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.<ref>{{cite journal |pmid=25698649|pmc=4385455|year=2015|last1=Rucci|first1=M|title=The unsteady eye: An information-processing stage, not a bug|journal=Trends in Neurosciences|volume=38|issue=4|pages=195–206|last2=Victor|first2=J. D.|doi=10.1016/j.tins.2015.01.005}}</ref> According to some authors, [[fixational eye movements]] in general are also well described by a random walk.<ref>{{cite journal |doi= 10.1073/pnas.1102730108|title= An integrated model of fixational eye movements and microsaccades|journal= Proceedings of the National Academy of Sciences|volume= 108|issue= 39|pages= E765-70|year= 2011|last1= Engbert|first1= R.|last2= Mergenthaler|first2= K.|last3= Sinn|first3= P.|last4= Pikovsky|first4= A.|pmid=21873243|pmc=3182695|bibcode= 2011PNAS..108E.765E|doi-access= free}}</ref>
*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.<ref>{{cite journal|pmid=9127583 |url=http://oz.ss.uci.edu/237/readings/EBRW_nosofsky_1997.pdf |archive-url=https://web.archive.org/web/20041210231937/http://oz.ss.uci.edu/237/readings/EBRW_nosofsky_1997.pdf |archive-date=2004-12-10 |year=1997 |last1=Nosofsky |first1=R. M. |title=An exemplar-based random walk model of speeded classification |journal=Psychological Review |volume=104 |issue=2 |pages=266–300 |last2=Palmeri |first2=T. J. |doi=10.1037/0033-295x.104.2.266 }}</ref>
*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.{{citation needed|date=April 2012}} In [[computer science]], this method is known as [[Markov chain Monte Carlo|Markov Chain Monte Carlo]] (MCMC).
*In [[wireless networking]], a random walk is used to model node movement.{{citation needed|date=April 2012}}
*[[bacterial motility|Motile bacteria]] engage in [[biased random walk (biochemistry)|biased random walks]].<ref>{{cite journal|last1=Codling|first1=E. A|last2=Plank|first2=M. J|last3=Benhamou|first3=S.|title=Random walk models in biology|journal=Journal of the Royal Society Interface|date=6 August 2008|volume=5|issue=25|pages=813–834|doi=10.1098/rsif.2008.0014|pmid=18426776|pmc=2504494}}</ref>
*In physics, random walks underlie the method of [[Fermi estimation]].{{citation needed|date=April 2012}}
*On the web, the Twitter website uses random walks to make suggestions of whom to follow<ref name="twitterwtf">Gupta, Pankaj et al. [http://dl.acm.org/citation.cfm?id=2488433 WTF: The who-to-follow system at Twitter], Proceedings of the 22nd international conference on World Wide Web</ref>
*[[Dave Bayer]] and [[Persi Diaconis]] have proven that 7 [[shuffle|riffle shuffles]] are sufficient to mix a deck 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.