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Phase transition in one-dimensional random walk with partially reflecting boundaries

Published online by Cambridge University Press:  01 July 2016

Ora E. Percus
Ora E. Percus*
Affiliation:
Courant Institute of Mathematical Sciences
*
∗Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA.
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Abstract

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).

The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

Keywords

ASYMMETRIC RANDOM WALKASYMPTOTIC BEHAVIOR
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 3 , September 1985 , pp. 594 - 606
Copyright
Copyright © Applied Probability Trust 1985 

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References

Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Kac, M. (1954) Random walk and the theory of Brownian motion. In Selected Papers on Noise and Stochastic Processes , ed. Wax, N.. Dover, New York.Google Scholar
Tákács, L. (1960) Stochastic Processes. Methuen, London.Google Scholar