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Phase Transition and Law of Large Numbers for a Non-Symmetric One-Dimensional Random Walk with Self-Interactions

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Franck Vermet
Franck Vermet*
Affiliation:
Université de Bretagne Occidentale
*
∗Postal address: Université de Bretagne Occidentale, Faculté des Sciences et Techniques, Département de Mathématiques, 6, av. Victor Le Gorgeu, BP 809, 29285 Brest Cedex, France.
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Abstract

We study a not necessarily symmetric random walk with interactions on ℤ, which is an extension of the one-dimensional discrete version of the sausage Wiener path measure. We prove the existence of a repulsion/attraction phase transition for the critical value λc ≡ −μ of the repulsion coefficient λ, where μ is a drift parameter. In the self-repellent case, we determine the escape speed, as a function of λ and μ, and we prove a law of large numbers for the end-point.

Keywords

Random walk with interactionsdiscrete version of the sausage Wiener path measurephase transitionlarge deviations

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 55 - 63
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Research partially supported by the European Contract CHRX-CT93-0411

References

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