A Useful Renormalization Argument

  • Maury Bramson
  • Lawrence Gray
Chapter
Part of the Progress in Probability book series (PRPR, volume 28)

Abstract

We define a collection of ‘generic’ population models for which we prove a survival criterion by using a renormalization argument. These models can be compared with other more familiar models, leading to simple proofs of various survival results. In particular, we prove a generalization of Toom’s Theorem concerning survival in multidimensional probabilistic cellular automata. Our technique should also be applicable to a variety of other discrete and continuous time models.

Keywords

Active Element lImax 

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References

  1. [1]
    M. Bramson, Survival of nearest particle systems with low birth rate, Ann. Probab. 17 (1989), 433–443MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    M. Bramson and R. Durrett, A simple proof of the stability criterion of Gray and Griffeath, Probab. Th. Rel. Fields 80 (1988), 293–298MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. Bramson, R. Durrett, and G. Swindle, The statistical mechanics of crabgrass, Ann. Probab. 17 (1989), 444–481MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    M. Bramson and L. Gray, The survival of branching annihilating random walk, Z. Wahrsch. Verw. Gebiete 68 (1985), 447–460MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    R. Durrett and L. Gray, Some peculiar properties of a particle system with sexual reproduction, preprint. Preliminary version in P. Tautu, Stochastic spatial processes, Lecture Notes in Mathematics No. 1212, Springer-Verlag, New York, 1986Google Scholar
  6. [6]
    R. Durrett and D. Griffeath, Supercritical contact processes on ℤ, Ann. Probab. 11 (1983), 1–15MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    L. Gray, Duality for general attractive spin systems, with applications in one dimension, Ann. Probab. 14 (1986), 371–396MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    L. Gray and D. Griffeath, A stability criterion for attractive nearest-neighbor spin systems on ℤ, Ann. Probab. 10 (1982), 67–85MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    T. Harris, Contact interactions on a lattice, Ann. Probab. 2 (1974), 969–988MATHCrossRefGoogle Scholar
  10. [10]
    R. Holley and T. Liggett, The survival of contact processes, Ann. Prob. 6 (1978), 198–206MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    R. Peierls, On Ising’s model of ferromagnetism, Proc. Cambridge Phil. Soc. 36 (1936), 477–481CrossRefGoogle Scholar
  12. [12]
    O. Stayskaya and I. Pyatetskii-Shapiro, On homogeneous nets of spontaneously active elements, Systems Theory Res. 20 (1971), 75–88Google Scholar
  13. [13]
    A. Toom, Nonergodic multidimensional systems of automata, Problems Inform. Transmission 10 (1974), 239–246MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Maury Bramson
    • 1
  • Lawrence Gray
    • 2
  1. 1.Dept. of MathematicsUniversity of WisconsinMadisonUSA
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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