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Random walks in attractive potentials: the case of critical drifts. (English) Zbl 06938565
Summary: 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 [M. P. W. Zerner, Ann. Appl. Probab. 8, No. 1, 246–280 (1998; Zbl 0938.60098)]. Recently it was shown that super-critical drifts lead to a limiting speed. We shall explain that in dimensions \(d \geq 2\) the transition is always of the first order.
60Kxx Special processes
82Bxx Equilibrium statistical mechanics
82Bxx Equilibrium statistical mechanics
Zbl 0938.60098
Full Text: DOI Link
[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. | · Zbl 1255.60168
[2] Martin P. W. Zerner. Directional decay of the Green’s function for a random nonnegative potential on \(\mathbf{Z}^d\). \(Ann. Appl. Probab.\), 8(1):246-280, 1998. | | Copyright Cellule MathDoc 2018 · Zbl 0938.60098
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