A Useful Renormalization Argument

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


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.


Generic Process Death Region Poisson Point Process Nonempty Intersection Contact Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations