How Far Can a Biased Random Walker Go?

Abstract

The random walk (RW) is a very important model in science and engineering researches. It has been studied over hundreds years. However, there are still some overlooked problems on the RW. Here, we study the mean absolute distance of an N-step biased random walk (BRW) in a d-dimensional hyper-cubic lattice. In this short paper, we report the exact results for d = 1 and approximation formulae for d ≥ 2.

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Yang, Z. and Yang, C. (2015) How Far Can a Biased Random Walker Go?. Journal of Applied Mathematics and Physics, 3, 1159-1167. doi: 10.4236/jamp.2015.39143.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Feynman, R.P., Leighton, R.B. and Sands, M. (1963) Feynman Lectures on Physic. Addison-Wesley, New York, Vol. 1, 6-5, 41-9.
[2] Whitney, C.A. (1990) Random Processes in Physical Systems. Wiley, New York, 37.
[3] Uhlenbeck, G.E. and Ornstein, L.S. (1930) On the Theory of the Brownian Motion. Physical Review Letters, 36, 823.
http://dx.doi.org/10.1103/PhysRev.36.823
[4] Grimmett, G. and Stirzaker, D. (2001) Probability and Random Processes. Oxford University Press, Oxford.
[5] van Kampen, N.G. (2007) Stochastic Processes in Physics and Chemistry. 3rd Edition, Elsevier, Amsterdam.
[6] Gardiner, C.W. (2004) Handbook of Stochastic Methods. 3rd Edition, Springer, Berlin.
http://dx.doi.org/10.1007/978-3-662-05389-8
[7] Borodin, A.N. and Salminen, P. (2002) Handbook of Brownian Motion—Facts and Formulae. 2nd Edition, Birkhauser, Basel.
http://dx.doi.org/10.1007/978-3-0348-8163-0
[8] Codling, E.A. (2003) Biased Random Walks in Biology. Ph.D. Thesis, University of Leeds, Leeds.
[9] Stauffer, D. (1985) Introduction to Percolation Theory. Talor & Francis, London.
http://dx.doi.org/10.4324/9780203211595
[10] Sun, T. and Yang, Z.J. (1992) How Far Can a Random Walker Go? Physica A: Statistical Mechanics and Its Applications, 182, 599-606.
http://dx.doi.org/10.1016/0378-4371(92)90024-K
[11] Wolfram, S. (2003) Mathematica. 5th Edition, Wolfram Media, Champaign.

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