Abstract
A precise description of the nontrivial upper invariant measure for λ>λc is still an open problem for the basic contact process, which is a self-dual, attractive, but nonreversible Markov process of an interacting particle system. By its self-duality, to identify the invariant measure is equivalent to determining the initial-state dependence of the survival probability of the process. A procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by Harris. Two new bounds are given, improving the simple branching-process bound. In the one-dimensional case, the present procedure can be viewed as a trial to make approximate measures by generalized Markov extensions.
Similar content being viewed by others
References
T. E. Harris, Contact interactions on a lattice,Ann. Prob. 2:969–988 (1974).
T. M. Liggett,Interacting Particle Systems (Springer-Verlag, New York, 1985).
D. Griffeath,Additive and Cancellative Interacting Particle Systems (Springer Lecture Notes in Mathematics, Vol. 724, Springer-Verlag, New York, 1979).
R. Durrett,Lecture Notes on Particle Systems and Percolation (Wadsworth and Brooks/Cole Advanced Books & Software, California, 1988).
D. Amati, M. Le Bellac, G. Marchesini, and M. Ciafaloni, Reggeon field theory forα(0)>1,Nucl. Phys. B 112:107–149 (1976).
R. C. Brower, M. A. Furman, and K. Subbarao, Quantum spin model for Reggeon field theory,Phys. Rev. D 15:1756–1771 (1977); R. C. Brower, M. A. Furman, and M. Moshe, Critical exponents for the Reggeon quantum spin model,Phys. Lett. 76B:213–219 (1978).
P. Grassberger and A. de la Torre, Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behavior,Ann. Phys. 122:373–396 (1979).
F. Schlögl, Chemical reaction models for non-equilibrium phase transitions,Z. Physik 253:147–161 (1972).
P. Grassberger and M. Scheunert, Fock-space methods for identical classical objects,Fortschr. Phys. 28:547–578 (1980).
T. E. Harris, On a class of set-valued Markov processes,Ann. Prob. 4:175–194 (1976).
R. Holley and T. M. Liggett, The survival of contact processes,Ann. Prob. 6:198–206 (1978).
R. H. Shonmann and M. E. Vares, The survival of the large dimensional basic contact process,Prob. Theory Rel. Fields 72:387–393 (1986).
R. Dickman, Nonequilibrium lattice model: Series analysis of steady states,J. Stat. Phys. 55:997–1026 (1989).
R. Dickman, Universality and diffusion in nonequilibrium critical phenomena,Phys. Rev. B 40:7005–7010 (1989); I. Jensen, H. C. Fogedby, and R. Dickman, Critical exponents for an irreversible surface reaction model,Phys. Rev. A 41:3411–3414 (1990); R. Dickman, On the nonequilibrium critical behavior of the triplet annihilation model,Phys. Rev. A, to appear.
N. Konno and M. Katori, Applications of the CAM based on a new decoupling procedure of correlation functions in the one-dimensional contact process,J. Phys. Soc. Jpn. 59:1581–1592 (1990).
D. Griffeath, Ergodic theorems for graph interactions,Adv. Appl. Prob. 7:179–194 (1975).
M. Katori and N. Konno, Correlation inequalities and lower bounds for the critical valueλ c of contact processes,J. Phys. Soc. Jpn. 59:877–887 (1990).
H. Ziezold and A. Grillenberger, On the critical infection rate of the one-dimensional contact process: Numerical results,J. Appl. Prob. 25:1–8 (1987).
A. G. Schlijper, On some variational approximations in two-dimensional classical lattice systems,J. Stat. Phys. 40:1–27 (1985).
M. Katori and N. Konno, Three-point Markov extension and an improved upper bound for survival probability of the one-dimensional contact process,J. Phys. Soc. Jpn. 60(2) (1991).
N. Konno and M. Katori, Applications of the Harris-FKG inequality to upper bounds for order parameters in the contact processes,J. Phys. Soc. Jpn. 60(2) (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Katori, M., Konno, N. Upper bounds for survival probability of the contact process. J Stat Phys 63, 115–130 (1991). https://doi.org/10.1007/BF01026595
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01026595