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Selected Topics in Random Walks in Random Environment

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Book cover Topics in Percolative and Disordered Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 69))

Abstract

Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been introduced in a series of papers as a model of DNA chain replication and crystal growth (see Chernov [10] and Temkin [51, 52]), and also as a model of turbulent behavior in fluids through a Lorentz gas description (Sinai 1982 [42]). It is a simple but powerful model for a variety of complex large-scale disordered phenomena arising from fields such as physics, biology, and engineering. While the one-dimensional model is well-understood in the multidimensional setting, fundamental questions about the RWRE model have resisted repeated and persistent attempts to answer them. Two major complications in this context stem from the loss of the Markov property under the averaged measure as well as the fact that in dimensions larger than one, the RWRE is not reversible anymore. In these notes we present a general overview of the model, with an emphasis on the multidimensional setting and a more detailed description of recent progress around ballisticity questions.

A. F. Ramírez was partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1100298 and by Iniciativia Científica Milenio grant number NC130062.

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Notes

  1. 1.

    Note that, while the condition \({(\mathcal P)}_M\) of Definition 11 also is effective in the sense that it can be checked on finite boxes, the proof that it implies \((T')\) takes advantage of the effective criterion (cf. Definition 11 and Theorem 20)—we therefore do introduce this criterion here.

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Acknowledgement

The final version has benefitted from careful refereeing. We would also like to thank Gregorio Moreno for useful comments on the first draft of this text.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, Broadway 2990, New York, NY, 10027, USA

    Alexander Drewitz

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile

    Alejandro F. Ramírez

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  1. Alexander Drewitz
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Editor information

Editors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica, Chile, Santiago, Chile

    Alejandro F. Ramírez

  2. Department of Mathematics, New York University Courant Institute of Math. Sciences, New York, New York, USA

    Gérard Ben Arous

  3. Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina

    Pablo A. Ferrari

  4. Department of Mathematics, New York University Courant Institute of Math. Sciences, New York, New York, USA

    Charles M. Newman

  5. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Rio de Janeiro, Brazil

    Vladas Sidoravicius

  6. Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Rio de Janeiro, Brazil

    Maria Eulália Vares

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Drewitz, A., Ramírez, A. (2014). Selected Topics in Random Walks in Random Environment. In: Ramírez, A., Ben Arous, G., Ferrari, P., Newman, C., Sidoravicius, V., Vares, M. (eds) Topics in Percolative and Disordered Systems. Springer Proceedings in Mathematics & Statistics, vol 69. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0339-9_3

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