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Actes des rencontres du CIRM

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Dmitry Ioffe; Yvan Velenik
Random Walks in Attractive Potentials: The Case of Critical Drifts
Actes des rencontres du CIRM, 2 no. 1: Déviations pour les temps locaux d’auto-intersections (2010), p. 11-13, doi: 10.5802/acirm.17
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Résumé - Abstract

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 $d\ge 2$ the transition is always of the first order. (Joint work with Y.Velenik)

Bibliographie

[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.
arXiv |  MR 2603219 |  Zbl pre05375174
[2] Martin P. W. Zerner. Directional decay of the Green’s function for a random nonnegative potential on ${\bf Z}^d$. Ann. Appl. Probab., 8(1):246–280, 1998.
Article |  MR 1620370 |  Zbl 0938.60098
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