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 the transition is always of the first order. (Joint work with Y.Velenik)
@article{ACIRM_2010__2_1_11_0,
author = {Ioffe, Dmitry and Velenik, Yvan},
title = {Random {Walks} in {Attractive} {Potentials:} {The} {Case} of {Critical} {Drifts}},
journal = {Actes des rencontres du CIRM},
pages = {11--13},
publisher = {CIRM},
volume = {2},
number = {1},
year = {2010},
doi = {10.5802/acirm.17},
zbl = {06938565},
language = {en},
url = {http://www.numdam.org/articles/10.5802/acirm.17/}
}
TY - JOUR AU - Ioffe, Dmitry AU - Velenik, Yvan TI - Random Walks in Attractive Potentials: The Case of Critical Drifts JO - Actes des rencontres du CIRM PY - 2010 SP - 11 EP - 13 VL - 2 IS - 1 PB - CIRM UR - http://www.numdam.org/articles/10.5802/acirm.17/ DO - 10.5802/acirm.17 LA - en ID - ACIRM_2010__2_1_11_0 ER -
Ioffe, Dmitry; Velenik, Yvan. Random Walks in Attractive Potentials: The Case of Critical Drifts. Actes des rencontres du CIRM, Tome 2 (2010) no. 1, pp. 11-13. doi : 10.5802/acirm.17. http://www.numdam.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 . Ann. Appl. Probab., 8(1):246–280, 1998. | DOI | MR | Zbl
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