Skip to main content
SpringerLink
Log in

Ten lectures on particle systems

  • Chapter
  • First Online:
Lectures on Probability Theory

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 1608))

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 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
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

  • Andjel, E.D. T.M. Liggett, and T. Mountford (1992) Clustering in one dimensional threshold voter models. Stoch. Processes Appl. 42, 73–90

    Article  MathSciNet  MATH  Google Scholar 

  • Aronson, D.G. and H.F. Weinberger (1978) Multidimensional diffusion equations arising in population genetics. Advances in Math. 30, 33–76

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen, S. and N. Kaplan (1976) Branching random walks, I, Stoch. Processes Appl. 4, 1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Bezuidenhout, C. and L. Gray (1993) Critical attractive spin systems. Ann. Probab., to appear

    Google Scholar 

  • Bezuidenhout, C. and G. Grimmett (1990) The critical contact process dies out. Ann. Probab. 18, 1462–1482

    Article  MathSciNet  MATH  Google Scholar 

  • Bezuidenhout, C. and G. Grimmett (1991) Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19, 984–1009

    Article  MathSciNet  MATH  Google Scholar 

  • Boerlijst, M.C. and P. Hogeweg (1991) Spiral wave structure in pre-biotic evolution: hypercycles stable against parasites. Physica D 48, 17–28

    Article  MATH  Google Scholar 

  • Bramson, M. (1983) Convergence of solutions of the Kolmogorov equation to travelling waves. Memoirs of the AMS, 285

    Google Scholar 

  • Bramson, M. and R. Durrett (1988) A simple proof of the stability theorem of Gray and Griffeath. Probab. Th. Rel. Fields 80, 293–298

    Article  MathSciNet  MATH  Google Scholar 

  • Bramson, M., R. Durrett, and G. Swindle (1989) Statistical mechanics of Crabgrass. Ann. Prob. 17, 444–481

    Article  MathSciNet  MATH  Google Scholar 

  • Bramson, M. and L. Gray (1992) A useful renormalization argument. Pages??? in Random Walks, Brownian Motion, and Interacting Particle Systems, edited by R. Durrett and H. Kesten, Birkhauser, Boston

    Google Scholar 

  • Bramson, M. and D. Griffeath (1987) Survival of cyclic particle systems. Pages 21–30 in Percolation Theory and Ergodic Theory of Infinite Particle Systems edited by H. Kesten, IMA Vol. 8, Springer

    Google Scholar 

  • Bramson, M. and D. Griffeath (1989) Flux and fixation in cyclic particle systems Ann. Probab. 17, 26–45

    Article  MathSciNet  MATH  Google Scholar 

  • Bramson, M. and C. Neuhauser (1993) Survival of one dimensional cellular automata. Preprint

    Google Scholar 

  • Chen, H.N. (1992) On the stability of a population growth model with sexual reproduction in Z2. Ann. Probab. 20, 232–285

    Article  MathSciNet  Google Scholar 

  • Cox, J.T. (1988) Coalescing random walks and voter model consensus times on the torus in Z d. Ann. Probab.

    Google Scholar 

  • Cox, J.T. and R. Durrett (1988) Limit theorems for the spread of epidemics and forest fires. Stoch. Processes Appl. 30, 171–191

    Article  MathSciNet  MATH  Google Scholar 

  • Cox, J.T. and R. Durrett (1992) Nonlinear voter models. Pages 189–202 in Random Walks, Brownian Motion, and Interacting Particle Systems, edited by R. Durrett and H. Kesten, Birkhauser, Boston

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • DeMasi, A., P. Ferrari, and J. Lebowitz (1986) Reaction diffusion equations for interacting particle systems. J. Stat. Phys. 44, 589–644 *** DIRECT SUPPORT *** A00I6B63 00003

    Article  MathSciNet  MATH  Google Scholar 

  • DeMasi, A. and E. Presutti (1991) Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Math 1501, Springer, New York.

    Book  Google Scholar 

  • Durrett, R. (1980) On the growth of one dimensional contact processes. Ann. Probab. 8, 890–907.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. (1988) Lecture Notes On Particle Systems And Percolation, Wadsworth, Belmont, CA

    MATH  Google Scholar 

  • Durrett, R. (1991a) Stochastic models of growth and competition. Pages 1049–1056 in Proceedings of the International Congress of Mathematicians, Kyoto, Springer, New York.

    Google Scholar 

  • Durrett, R. (1991b) The contact process, 1974–1989. Pages 1–18 in Proceedings of the AMS Summer seminar on Random Media. Lectures in Applied Math 27, AMS, Providence, RI

    Google Scholar 

  • Durrett, R. (1991c) Some new games for your computer. Nonlinear Science Today Vol. 1, No. 4, 1–7.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. (1992a) Multicolor particle systems with large threshold and range. J. Theoretical Prob., 5 (1992), 127–152.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. (1992b) A new method for proving the existence of phase transitions. Pages 141–170 in Spatial Stochastic Processes, edited by K.S. Alexander and J. C. Watkins, Birkhauser, Boston

    Google Scholar 

  • Durrett, R. (1992c) Stochastic growth models: bounds on critical values. J. Appl. Prob. 29

    Google Scholar 

  • Durrett, R. (1992d) Probability: Theory and Examples. Wadsworth, Belmont, CA

    MATH  Google Scholar 

  • Durrett, R. (1993) Predator-prey systems. Pages 37–58 in Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, edited by K.D. Elworthy and N. Ikeda, Pitman Research Notes in Math 283, Longman, Essex, England.

    Google Scholar 

  • Durrett, R. and D. Griffeath (1993) Asymptotic behavior of excitable cellular automata. Preprint.

    Google Scholar 

  • Durrett, R. and S. Levin (1993) Stochastic spatial models: A user's guide to ecological applications. Phil. Trans. Roy. Soc. B, to appear

    Google Scholar 

  • Durrett, R. and A.M. Moller (1991) Complete convergence theorem for a competition model. Probab. Th. Rel. Fields 88, 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. and C. Neuhauser (1991) Epidemics with recovery in d=2. Ann. Applied Probab. 1, 189–206.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. and C. Neuhauser (1993) Particle systems and reaction diffusion equations. Ann. Probab., to appear

    Google Scholar 

  • Durrett, R. and R. Schinazi (1993) Asymptotic critical value for a competition model. Ann. Applied Probab., to apear

    Google Scholar 

  • Durrett, R. and J. Steif (1993) Fixation results for threshold voter models. Ann. Probab., to appear

    Google Scholar 

  • Durrett, R. and G. Swindle (1991) Are there bushes in a forest? Stoch. Proc. Appl. 37, 19–31.

    Article  MathSciNet  MATH  Google Scholar 

  • Durrett, R. and G. Swindle (1993) Coexistence results for catalysts. Preprint

    Google Scholar 

  • Eigen, M. and P. Schuster (1979) The Hypercycle: A Principle of Natural Self-Organization, Springer, New York.

    Book  Google Scholar 

  • Fife, P.C. and J.B. McLeod (1977) The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65, 335–361

    Article  MathSciNet  MATH  Google Scholar 

  • Fisch, R. (1990a) The one dimensional cyclic cellular automaton: a system with deterministic dynamics which emulates a particle system with stochastic dynamics. J. Theor. Prob. 3, 311–338.

    Article  MathSciNet  MATH  Google Scholar 

  • Fisch, R. (1990b) Cyclic cellular automata and related processes. Physica D 45, 19–25.

    Article  MATH  Google Scholar 

  • Fisch, R. (1992). Clustering in the one dimensional 3-color cyclic cellular automaton, Ann. Probab. 20, 1528–1548.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Fisch, R., J. Gravner and D. Griffeath (1992) Cyclic cellular automata in two dimensions. In Spatial Stochastic Processes edited by K. Alexander and J. Watkins, Birkhauser, Boston

    Google Scholar 

  • Fisch, R., J. Gravner, and D. Griffeath (1993) Metastability in the Greenberg Hastings model. Ann. Applied. Probab., to appear

    Google Scholar 

  • Grannan, E. and G. Swindle (1991) Rigorous results on mathematical models of catalyst surfaces. J. Stat. Phys. 61, 1085–1103

    Article  MathSciNet  MATH  Google Scholar 

  • Gravner, J. and D. Griffeath (1993) Threshold growth dynamics. Transactions A.M.S., to appear

    Google Scholar 

  • Gray, L. and D. Griffeath (1982) A stability criterion for attractive nearest neighbor spin systems on Z. Ann. Probab. 10, 67–85

    Article  MathSciNet  MATH  Google Scholar 

  • Gray, L. (1987) Behavior of processes with statistical mechanical properties. Pages 131–168 in Percolation Theory and Ergodic Theory of Infinite Particle Systems edited by H. Kesten, IMA Vol. 8, Springer

    Google Scholar 

  • Griffeath, D. (1979) Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math 724, Springer

    Google Scholar 

  • Harris, T.E. (1972) Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. in Math. 9, 66–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Harris, T.E. (1976) On a class of set valued Markov processes. Ann. Probab. 4, 175–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Hassell, M.P., H.N. Comins and R.M. May (1991) Spatial structure and chaos in insect population dynamics. Nature 353, 255–258.

    Article  Google Scholar 

  • Hirsch, M.W. and S. Smale (1974) Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York.

    MATH  Google Scholar 

  • Holley, R.A. (1972) Markovian interaction processes with finite range interactions. Ann. Math. Stat. 43, 1961–1967.

    Article  MathSciNet  MATH  Google Scholar 

  • Holley, R.A. (1974) Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227–231.

    Article  MathSciNet  Google Scholar 

  • Holley, R.A. and T.M. Liggett (1975) Ergodic theorems for weakly interacting systems and the voter model. Ann. Probab. 3, 643–663.

    Article  MathSciNet  MATH  Google Scholar 

  • Holley, R.A. and T.M. Liggett (1978) The survival of contact processes. Ann. Probab. 6, 198–206.

    Article  MathSciNet  MATH  Google Scholar 

  • Kinzel, W. and J. Yeomans (1981) Directed percolation: a finite size renormalization approach. J. Phys. A 14, L163–L168.

    Article  Google Scholar 

  • Liggett, T.M. (1985) Interacting Particle Systems. Springer, New York.

    Book  MATH  Google Scholar 

  • Liggett, T.M. (1993) Coexistence in threshold voter models. Ann. Probab., to appear

    Google Scholar 

  • Neuhauser, C. (1992) Ergodic theorems for the multitype contact process. Probab. Theory Rel. Fields. 91, 467–506

    Article  MathSciNet  MATH  Google Scholar 

  • Redheffer, R., R. Redlinger and W. Walter (1988) A theorem of La Salle-Lyapunov type for parabolic systems. SIAM J. Math. Anal. 19, 121–132

    Article  MathSciNet  MATH  Google Scholar 

  • Schonmann, R.H. and M.E. Vares (1986) The survival of the large dimensional basic contact process. Probab. Th. Rel. Fields 72, 387–393.

    Article  MathSciNet  MATH  Google Scholar 

  • Spohn, H. (1991) Large Scale Dynamics of Interacting Particle Systems, Springer, New York

    Book  MATH  Google Scholar 

  • Zhang, Yu (1992) A shape theorem for epidemics and forest fires with finite range interactions. Preprint.

    Google Scholar 

Download references

Authors
  1. Rick Durrett
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Pierre Bernard

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag

About this chapter

Cite this chapter

Durrett, R. (1995). Ten lectures on particle systems. In: Bernard, P. (eds) Lectures on Probability Theory. Lecture Notes in Mathematics, vol 1608. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0095747

Download citation

  • DOI: https://doi.org/10.1007/BFb0095747

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60015-2

  • Online ISBN: 978-3-540-49402-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation