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On some growth models with a small parameter
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  • Published: December 1995

On some growth models with a small parameter

  • Harry Kesten1 &
  • Roberto H. Schonmann2 

Probability Theory and Related Fields volume 101, pages 435–468 (1995)Cite this article

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  • 17 Citations

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Summary

We consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models, when a certain parameter becomes small. The basic example treated is the variant of Richardson's growth model on ℤd in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate ɛ≦1 if it has exactly 1 occupied neighbor and, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this model has asymptotic speeds of growth in each direction (as time goes to infinity) and these speeds determine an asymptotic shape in the usual sense. It is proven that as ɛ tends to 0, the asymptotic speeds scale as ɛ1/d and the asymptotic shape, when renormalized by dividing it by ɛ1/d, converges to a cube. Other similar models which are partially oriented are also studied.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, 14853, Ithaca, NY, USA

    Harry Kesten

  2. Mathematics Department, University of California at Los Angeles, 90024, Los Angeles, CA, USA

    Roberto H. Schonmann

Authors
  1. Harry Kesten
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  2. Roberto H. Schonmann
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Additional information

The work of R.H.S. was supported by the N.S.F. through grant DMS 91-00725. In addition, both authors were supported by the Newton Institute in Cambridge. The authors thank the Newton Institute for its support and hospitality

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Kesten, H., Schonmann, R.H. On some growth models with a small parameter. Probab. Th. Rel. Fields 101, 435–468 (1995). https://doi.org/10.1007/BF01202779

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  • Received: 20 January 1994

  • Revised: 14 September 1994

  • Issue Date: December 1995

  • DOI: https://doi.org/10.1007/BF01202779

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Mathematics Subject Classification (1991)

  • Primary 60K35
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