Random walks in attractive potentials: the case of critical drifts. (English) Zbl 1477.60138

Ioffe, Dmitry; Velenik, Yvan

Actes Rencontres C.I.R.M. 2, No. 1, 11-13 (2010).

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.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43	Percolation

Keywords:

critical drifts; attractive potentials

Citations:

Zbl 0938.60098

References:

[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