Journal of Theoretical Probability

, Volume 27, Issue 3, pp 1045–1057 | Cite as

Asymptotic Behaviour of the Critical Value for the Contact Process with Rapid Stirring



We study the behaviour of the contact process with rapid stirring on the lattice \(\mathbb Z ^d\) in dimensions \(d\ge 3.\) This process was studied earlier by Konno and Katori, who proved results for the speed of convergence of the critical value as the rate of stirring approaches infinity. In this article we improve the results of Konno and Katori and establish the sharp asymptotics of the critical value in dimensions \(d\ge 3.\)


Asymptotic behaviour Contact processes Rapid stirring  Interacting particle systems 

Mathematics Subject Classification (2010)

Primary 82C22 60K35 Secondary 60J80 



Both authors thank an anonymous referee for the careful reading of the manuscript and for a number of useful comments and suggestions that improved the exposition. Supported by the ISF grant 497/10.


  1. 1.
    Bramson, M., Durrett, R., Swindle, G.: Statistical mechanics of crab grass. Ann. Probab. 17, 444–481 (1989)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    DeMassi, A., Ferrari, P.A., Lebowitz, J.L.: Reaction-diffusion equations for interacting particle systems. J. Stat. Phys. 44(3), 589–644 (1986)CrossRefGoogle Scholar
  3. 3.
    Durret, R., Neuhauser, C.: Particle systems and reaction-diffusion equations. Ann. Probab. 2(1), 289–333 (1994)CrossRefGoogle Scholar
  4. 4.
    Durrett, R., Perkins, E.A.: Rescaled contact processes converge to super-Brownian motion for d\(\ge \)2. Probab. Theory Relat. Fields 114, 309–399 (1999)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Katori, M.: Rigorous results for the diffusive contact processes in \(d\ge 3\). J. Phys A Math. Gen. 27, 7327–7341 (1994)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Konno, N.: Asymptotic behavior of basic contact process with rapid stirring. J. Theor. Probab. 8(4), 833–876 (1995)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Liggett, M.L.: Interacting Particle Systems. Spinger, New York (1985)CrossRefMATHGoogle Scholar
  8. 8.
    Liggett, M.L.: Interacting Particle Systems: Contact, Voter and Exclusion Processes. Spinger, New York (1991)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Faculty of Industrial Engineering and Management TechnionIsrael Institute of TechnologyHaifaIsrael

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