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Stochastic growth models: bounds on critical values

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

R. Durrett
R. Durrett*
Affiliation:
Cornell University
*
∗ Postal address: Department of Mathematics, Cornell University, White Hall, Ithaca, NY 14853–7901, USA.
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Abstract

We give upper bounds on the critical values for oriented percolation and some interacting particle systems by computing their behavior on small finite sets.

Keywords

ORIENTED PERCOLATIONCONTACT PROCESSPHASE TRANSITION

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B05: Classical equilibrium statistical mechanics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 11 - 20
Copyright
Copyright © Applied Probability Trust 1992 

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References

Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Brower, R. C, Furman, M. A. and Moshe, M. (1978) Critical exponents for the Reggeon quantum spin model. Phys. Lett. 76B, 213–219.Google Scholar
Durrett, R. (1980) On the growth of one dimensional contact processes. Ann. Prob. 8, 890–907.Google Scholar
Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.Google Scholar
Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth, Pacific Grove, CA.Google Scholar
Gray, L. and Griffeath, D. (1982) A stability criterion for nearest neighbor spin systems on ℤ. Ann. Prob. 10, 67–85.Google Scholar
Harris, T. E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969–988.Google Scholar
Holley, R. and Liggett, T. (1978) The survival of contact processes. Ann. Prob. 6, 198–206.Google Scholar
Kinzel, W. and Yeomans, J. (1981) Directed percolation: a finite-size renormalization approach. J. Phys. A14, L163–L168.Google Scholar
Liggett, T. (1985) Interacting Particle Systems. Springer-Verlag, New York.Google Scholar
Toom, A. (1968) A family of uniform nets of formal neurons. Soviet Math. Dokl. 9, 1338–1341.Google Scholar
Ziezold, H. and Grillenberger, A. (1988) On the critical infection rate of the one-dimensional contact process: numerical results. J. Appl. Prob. 25, 1–8.Google Scholar