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Percolation Theory and Ergodic Theory of Infinite Particle Systems pp 73–83Cite as

Recent Results for the Stepping Stone Model

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Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 8))

Abstract

Let G be a graph, and C = {0,1,...,κ-1} a set of colors. We want to discuss a continuous time random process ζt with state space CG = configurations of colors from C on the graph G. This system ζt, known as the stepping stone model, has very simple dynamics: in any state ζ and at any site x, the color at that site waits a mean 1/2 exponential holding time and then paints a randomly chosen neighboring vertex with its color. We will also be treating a companion process ξt on {subsets of G} called coalescing random walks. As the name implies, ξt consists of rate 1/2 continuous time random walks on G which coalesce when they collide. Let us write ξ At to denote the evolution of those walks which start on a subset A ⊂ G. The graphs of principal interest to us are:

  • GN = the complete graph on {0,1,...,N-1},

  • Z dN = the d-dimensional integers with period N,

  • Zd = the d-dimensional integers.

Tropical interpretations of the dimension d are good therapy for a Minnesota February: Baja California (d=1), Society Islands (d=2), Caribbean Marine World (d=3), and of course Mathematical Physics (d > 4). The cardinality κ of the color set C might be 2 or 32,768 or even ∞.

Partially supported by N.S.F. Grant MDS-841317

Partially supported by N.S.F. Grant MDS-830549

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References

  1. Andjel, E. and Kipnis, C. (1985) Pointwise ergodic theorems for nonergodic interacting particle systems. Preprint.

    Google Scholar 

  2. Arratia, R. (1982) Coalescing Brownian motions and the voter model on Z. Unpublished manuscript.

    Google Scholar 

  3. Bramson, M., Cox, J.T. and Griffeath, D. (1986) Consolidation rates for two interacting systems in the plane. To appear.

    Google Scholar 

  4. Bramson, M., Cox, J.T. and Griffeath, D. (1986) Occupation time large deviations of the voter model. To appear.

    Google Scholar 

  5. Bramson, M. and Griffeath, D. (1979) Renormalizing the 3-dimensional voter model. Annals of Probability 7, 418–432.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bramson, M. and Griffeath, D. (1980) Clustering and dispersion rates for some interacting particle systems on Z. Annals of Probability 8, 183–213.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bramson, M. and Griffeath, D. (1980) Asymptotics for interacting particle systems on Zd. Z. Wahrsch. verw. Gebiete 45, 183–196.

    Article  Google Scholar 

  8. Clifford, P. and Sudbury, A. (1973) A model for spatial conflict. Biometrika 60, 581–588.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cox, J.T. and Griffeath, D. (1983) Occupation time limit theorems for the voter model. Ann. Probability 11, 876–893.

    Article  MathSciNet  MATH  Google Scholar 

  10. Cox, J.T. and Griffeath, D. (1985) Large deviations for some infinite particle system occupation times. Contemporary Math. 41, 43–54.

    MathSciNet  Google Scholar 

  11. Cox, J.T. and Griffeath, D. (1986) Diffusive clustering in the two dimensional voter model. Ann. Probability 14, 347–370.

    Article  MathSciNet  MATH  Google Scholar 

  12. Cox, J.T. and Griffeath, D. (1987) Mean field asymptotics for the planar stepping stone model. In preparation.

    Google Scholar 

  13. Donnelly, P. (1984) The transient behaviour of the Moran model in population genetics. Proc. Cambridge Phil. Soc. 95, 349–358.

    Article  MathSciNet  MATH  Google Scholar 

  14. Erdös, P. and Taylor, S.J. (1960) Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungaricae 11, 137–162.

    Article  MATH  Google Scholar 

  15. Ethier, S. and Kurtz, T. (1981) The infinitely-many-neutral alleles diffusion model. Adv. in Appl. Probab. 13, 429–452.

    Article  MathSciNet  MATH  Google Scholar 

  16. Ewens, W.J. (1979) Mathematical Population Genetics. Springer-Verlag, New York.

    MATH  Google Scholar 

  17. Felsenstein, J. (1975) A pain in the torus: some difficulties with models of isolation by distance. American Naturalist 109, 359–368.

    Article  Google Scholar 

  18. Fisher, R.A. (1930) The Genetical Theory of Natural Selection. Clarendon Press, Oxford.

    MATH  Google Scholar 

  19. Griffiths, R.C. (1979) Exact sampling distributions from the infinite neutral alleles model. Adv. Appl. Prob. 11, 326–354.

    Article  MATH  Google Scholar 

  20. Holley, R. and Liggett, T.M. (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. Annals of Probability 3, 643–663.

    Article  MathSciNet  MATH  Google Scholar 

  21. Holley, R. and Stroock, D. (1979) Central limit phenomena of various interacting systems. Ann, Math. 110, 333–393.

    Article  MathSciNet  Google Scholar 

  22. Jacobi, C.G.J. (1829) Fundamenta Nova Theoriae Functionum Elipticarum, Regiomontis, fratum Borntraeger. Reprinted in Gesammelte Werke, vol. 1, Reimer, Berlin, 1881.

    Google Scholar 

  23. Kimura, M. (1953) “Stepping stone” model of population. Ann. Rept. Nat. Inst. Gentics, Japan 3, 62.

    Google Scholar 

  24. Kimura, M. and Weiss, G. (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 561–576.

    Google Scholar 

  25. Kingman, J.F.C. (1982) The coalescent. Stoch. Proc. Applns. 13, 235–248.

    Article  MathSciNet  MATH  Google Scholar 

  26. Liggett, T.M. (1985) Interacting Particle Systems. Springer-Verlag, New York.

    MATH  Google Scholar 

  27. Major, P. (1980) Renormalizing the voter model. Space and space-time renormalization. Studia Sci. Math. Hungar. 15, 321–341.

    MathSciNet  MATH  Google Scholar 

  28. Malécot, G. (1948) The Mathematics of Heredity. English translation, W.H. Freeman, San Francisco, 1969.

    Google Scholar 

  29. Moran, P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.

    MATH  Google Scholar 

  30. Presutti, E. and Spohn, H. (1983) Hydrodynamics of the voter model. Annals of Probability 11, 867–875.

    Article  MathSciNet  MATH  Google Scholar 

  31. Sawyer, S. (1976) Results for the stepping-stone model for migration in population genetics. Annals of Probability 4, 699–728.

    Article  MathSciNet  MATH  Google Scholar 

  32. Sawyer, S. (1977) Rates of consolidation in a selectively neutral migration model. Annals of Probability 5, 486–493.

    Article  MathSciNet  MATH  Google Scholar 

  33. Sawyer, S. (1979) A limit theorem for patch sizes in a selectively-neutral migration model. J. Appl. Probab. 16, 482–495.

    Article  MathSciNet  MATH  Google Scholar 

  34. Shiga, T. (1981) Diffusion processes in population genetics. J. Math. Kyoto Univ. 21, 133–151.

    MathSciNet  MATH  Google Scholar 

  35. Tavare, S. (1984) Line-of-descent and genealogical processes, and their applications in population genetics models. Theoretical Population Biology 26, 119–164.

    Article  MathSciNet  MATH  Google Scholar 

  36. Watterson, G.A. (1976) The stationary distribution of the infinitely-many neutral alleles diffusion model. J. Appl. Prob. 13, 639–651.

    Article  MathSciNet  MATH  Google Scholar 

  37. Williams, T. and Bjerknes, R. (1972) Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 19–21.

    Article  Google Scholar 

  38. Wright, S. (1931) Evolution in Mendelian populations. Genetics 16, 97–159.

    Google Scholar 

  39. Wright, S. (1943) Isolation by distance. Genetics 28, 114–138.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Syracuse University, Syracuse, NY, 13210, USA

    J. Theodore Cox

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

    David Griffeath

Authors
  1. J. Theodore Cox
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  2. David Griffeath
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Editor information

Editors and Affiliations

  1. Institute for Mathematics and Its Applications, Minneapolis, Minnesota, 55455, USA

    Harry Kesten

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© 1987 Springer-Verlag New York, Inc.

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Cox, J.T., Griffeath, D. (1987). Recent Results for the Stepping Stone Model. In: Kesten, H. (eds) Percolation Theory and Ergodic Theory of Infinite Particle Systems. The IMA Volumes in Mathematics and Its Applications, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8734-3_6

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  • DOI: https://doi.org/10.1007/978-1-4613-8734-3_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8736-7

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