# Random Walks in Attractive Potentials: The Case of Critical Drifts

Dmitry Ioffe^{[1]}; Yvan Velenik^{[1]}

- [1] Technion and Université de Genève

Actes des rencontres du CIRM (2010)

- Volume: 2, Issue: 1, page 11-13
- ISSN: 2105-0597

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topIoffe, Dmitry, and Velenik, Yvan. "Random Walks in Attractive Potentials: The Case of Critical Drifts." Actes des rencontres du CIRM 2.1 (2010): 11-13. <http://eudml.org/doc/115842>.

@article{Ioffe2010,

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)},

affiliation = {Technion and Université de Genève; Technion and Université de Genève},

author = {Ioffe, Dmitry, Velenik, Yvan},

journal = {Actes des rencontres du CIRM},

keywords = {self-attractive polymer; ballisticity; random walk; phase transition},

language = {eng},

month = {12},

number = {1},

pages = {11-13},

publisher = {CIRM},

title = {Random Walks in Attractive Potentials: The Case of Critical Drifts},

url = {http://eudml.org/doc/115842},

volume = {2},

year = {2010},

}

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

DA - 2010/12//

PB - CIRM

VL - 2

IS - 1

SP - 11

EP - 13

AB - 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)

LA - eng

KW - self-attractive polymer; ballisticity; random walk; phase transition

UR - http://eudml.org/doc/115842

ER -

## References

top- 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. Zbl1255.60168MR2603219
- 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. Zbl0938.60098MR1620370

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