Skip to main content
SpringerLink
Log in

Reverse Shapes in First-Passage Percolation and Related Growth Models

  • Chapter
Perplexing Problems in Probability

Part of the book series: Progress in Probability ((PRPR,volume 44))

Abstract

Over the past 40 years, it has been observed that many of the simplest random and deterministic local growth dynamics expand at a linear rate in each radial direction, and attain an asymptotic geometry. Shape theorems to this effect have been proved in several instances. In a similar manner, initially very large holes within supercritical local dynamics may be expected to attain a characteristic shape as they shrink, a while before disappearing. We describe a general theory of reverse shapes which formalizes this phenomenology, and then apply it to first-passage percolation and related deterministic and stochastic growth models. As an application, we analyze the last holes of such models started from sparse product measures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Aizenman and J. Lebowitz. Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen., 21 (1988), 3801–3813.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Andjel, T. Mountford, and R. Schonmann. Equivalence of exponential decay rates for bootstrap percolation like cellular automata. Ann. Inst. H. Poincaré, 31 (1995), 13–25.

    MathSciNet  MATH  Google Scholar 

  3. J. Biggins. The asymptotic shape of the branching random walk. Adv. Appl. Prob., 10 (1978), no. 1, 62–84.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Bohman. Discrete Threshold Growth dynamics are omnivorous for box neighborhoods. Trans. Amer. Math. Soc., 351 (1999), 947–983.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Bohman and J. Gravner. Random threshold growth models. Random Structures and Algorithms,1999, to appear.

    Google Scholar 

  6. T. Bohman, J. Gravner, and D. Griffeath. Asymptotic shapes for random threshold growth models, 1998, in preparation.

    Google Scholar 

  7. M. Bramson and D. Griffeath. On the Williams-Bjerknes Tumour Growth Model I. Ann. Probability, 9 (1981), 173–185.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bramson and D. Griffeath. On the Wiliams-Bjerknes Tumour Growth Model II. Math. Proc. Cambridge Philos. Soc., 88 (1980), 339–357.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Broadbent and J. Hammersley. Percolation processes. I. Crystals and mazes. Proc. Cambridge Philos. Soc., 53 (1957), 629–641.

    Article  MathSciNet  MATH  Google Scholar 

  10. J.T. Cox and R. Durrett. Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probability, 9 (1981), 583–603.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Durrett. Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Belmont, CA, 1988.

    MATH  Google Scholar 

  12. R. Durrett and D. Griffeath. Contact processes in several dimensions. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 59 (1982), 535–552.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Dobrushin, R. Kotecky, and S. Shlosman. Wulff Construction, A Global Shape from Local Interaction. American Mathematical Society, Providence, RI, 1992.

    MATH  Google Scholar 

  14. R. Durrett and T. Liggett. The shape of the limit set in Richardson’s growth model. Ann. Probability, 9 (1981), 186–193.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Eden. Page 359 of Symp. on Information Theory in Biology, H. Yockey, ed. Pergamon Press, New York, 1958.

    Google Scholar 

  16. R. Fisch, J. Gravner, and D. Griffeath. Threshold-range scaling of excitable cellular automata. Statistics and Computing, 1 (1991), 23–39.

    Article  Google Scholar 

  17. R. Fisch, J. Gravner, and D. Griffeath. Metastability in the Greenberg-Hastings model. Ann. Appl. Prob., 3 (1993), 935–967.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Gandolfi and H. Kesten. Greedy lattice animals. II. Linear growth. Ann. Appl. Prob., 4 (1994), 76–107.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Gravner. The boundary of iterates in Euclidean growth models. T.A.M.S., 348 (1996), 4549–4559.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Gravner. Recurrent ring dynamics in two-dimensional cellular automata. J. Appl. Prob.,1999, to appear.

    Google Scholar 

  21. J. Gravner and D. Griffeath. Threshold Growth dynamics. T.A.M.S., 340 (1993), 837–870.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Gravner and D. Griffeath. First-passage times for discrete Threshold Growth dynamics. Ann. Probability, 24 (1996), 1752–1778.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Gravner and D. Griffeath. Multitype Threshold Growth: convergence to Poisson-Voronoi tessellations. Ann. Appl. Prob., 7 (1997), 615–647.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Gravner and D. Griffeath. Cellular Automaton Grown on 7L2: theorems, examples and problems. Adv. Appl. Math., 21 (1998), 241–304.

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Griffeath. Primordial Soup Kitchen.http://psoup.math.wisc.edu/kitchen.html.

  26. J. Hammersley and D. Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin. (Statist. Lab., Univ. California, Berkeley). Springer-Verlag, New York, 1965, 61–110.

    Google Scholar 

  27. H. Kesten. First-passage percolation and a higher-dimensional generalization. In Particle Systems, Random Media and Large Deviations, R. Durrett, ed. American Mathematical Society, Providence, RI, 1984.

    Google Scholar 

  28. H. Kesten. On the speed of convergence in first-passage percolation. Ann. Appl. Prob., 4 (1994), 76–107.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Kesten and R. Schonmann. On some growth models with a small parameter. Pmbab. Th. Rel. Fields, 101 (1995), 435–468.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Krug and H. Spohn. Kinetic roughening of growing surfaces. In Solids Far from Equilibrium, C. Godreche, ed. Cambridge University Press, Cambridge, UK, 1992, 479–582.

    Google Scholar 

  31. T. Liggett. Interacting Particle Systems. Springer-Verlag, New York, 1985.

    MATH  Google Scholar 

  32. C. Newman and M. Piza. Divergence of shape fluctuations in two dimensions. Ann. Probability, 23 (1995), 977–1005.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Richardson. Random growth in a tesselation. Proc. Camb. Phil. Soc., 74 (1973), 515–528.

    Article  MATH  Google Scholar 

  34. T. Seppäläinen. Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Rel. Fields, 4 (1998), 1–26.

    MATH  Google Scholar 

  35. J. Taylor, J. Cahn, and C. Handwerker. Geometric models of crystal growth (Overview no. 98–1). Acta Met., 40 (1992), 1443–1474.

    Article  Google Scholar 

  36. S. Willson. On convergence of configurations. Discrete Math., 23 (1978), no. 3, 279–300.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of California, Davis, CA, 95616, USA

    Janko Gravner

  2. Mathematics Department, University of Wisconsin, Madison, WI, 53706, USA

    David Griffeath

Authors
  1. Janko Gravner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Griffeath
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Maury Bramson

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

    Rick Durrett

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Birkhäuser Boston

About this chapter

Cite this chapter

Gravner, J., Griffeath, D. (1999). Reverse Shapes in First-Passage Percolation and Related Growth Models. In: Bramson, M., Durrett, R. (eds) Perplexing Problems in Probability. Progress in Probability, vol 44. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2168-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2168-5_7

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-7442-1

  • Online ISBN: 978-1-4612-2168-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation