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Probability in Complex Physical Systems pp 195–223Cite as

Asymptotic Shape and Propagation of Fronts for Growth Models in Dynamic Random Environment

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Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 11))

Abstract

We survey recent rigorous results and open problems related to models of Interacting Particle Systems which describe the autocatalytic type reaction A+B→2B, with diffusion constants of particles being respectively D A ≥0 and D B ≥0. Depending on the choice of the values of D A and D B , we cover three distinct cases: the so called “rumor or infection spread” model (D A >0,D B >0); the Stochastic Combustion process (D A =0 and D B >0); and finally the “modified” Diffusion Limited Aggregation, which corresponds to the case D A >0, D B =0 with modified transition rule: A+B→2B occurs when an A- and a B-particles become nearest neighbors and the A-particle attempts to jump on a vertex where the B-particle is located. Then such jump is suppressed, and A-particle becomes B-particle.

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Acknowledgements

A.F.R and V.S. would like to thank the all organizers of both workshops and in particular to Prof. Wolfgang König for his hospitality. A.F.R. would like to thank the support of FONDECYT grant 1100298.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY, 14853-4201, USA

    Harry Kesten

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

    Alejandro F. Ramı́rez

  3. IMPA, Estr. Dona Castorina 110, Jardim Botanico, Rio de Janeiro, 22460-320, Brazil

    Vladas Sidoravicius

Authors
  1. Harry Kesten
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  2. Alejandro F. Ramı́rez
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  3. Vladas Sidoravicius
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Corresponding author

Correspondence to Harry Kesten .

Editor information

Editors and Affiliations

  1. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Jean-Dominique Deuschel

  2. Faculty of Mathematics, University of Bielefeld, P.O. Box 10 01 31, Bielefeld, 33501, Germany

    Barbara Gentz

  3. for Applied Analysis and Stochastics, Weierstrass Institute Berlin, Mohrenstr. 39, Berlin, 10117, Germany

    Wolfgang König

  4. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Max von Renesse

  5. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Michael Scheutzow

  6. Institute for Mathematical, Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8-10/105-1, Vienna, 1040, Austria

    Uwe Schmock

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Kesten, H., Ramı́rez, A.F., Sidoravicius, V. (2012). Asymptotic Shape and Propagation of Fronts for Growth Models in Dynamic Random Environment. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_8

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