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Random Walks in Attractive Potentials: The Case of Critical Drifts
Dmitry Ioffe1; Yvan Velenik1
1 Technion and Université de Genève
Actes des rencontres du CIRM, Volume 2 (2010) no. 1, pp. 11-13.

We consider random walks in attractive potentials - sub-additive functions of their local times. An application of a drift to such random walks leads to a phase transition: If the drift is small than the walk is still sub-ballistic, whereas the walk is ballistic if the drift is strong enough. The set of sub-critical drifts is convex with non-empty interior and can be described in terms of Lyapunov exponents (Sznitman, Zerner ). Recently it was shown that super-critical drifts lead to a limiting speed. We shall explain that in dimensions d≥2 the transition is always of the first order. (Joint work with Y.Velenik)

Published online:
DOI: 10.5802/acirm.17
Dmitry Ioffe 1; Yvan Velenik 1

1 Technion and Université de Genève 
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Dmitry Ioffe; Yvan Velenik. Random Walks in Attractive Potentials: The Case of Critical Drifts. Actes des rencontres du CIRM, Volume 2 (2010) no. 1, pp. 11-13. doi : 10.5802/acirm.17. https://pmb.cedram.org/articles/10.5802/acirm.17/

[1] Dmitry Ioffe and Yvan Velenik. Ballistic phase of self-interacting random walks. In Analysis and stochastics of growth processes and interface models, pages 55–79. Oxford Univ. Press, Oxford, 2008. | DOI | Zbl

[2] Martin P. W. Zerner. Directional decay of the Green’s function for a random nonnegative potential on Z d . Ann. Appl. Probab., 8(1):246–280, 1998. | DOI | MR | Zbl

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