Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T09:34:48.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of critical values in interacting particle systems

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Raúl Gouet ,
F. Javier López  and
Gerardo Sanz
Raúl Gouet*
Affiliation:
University of Chile
F. Javier López*
Affiliation:
University of Zaragoza
Gerardo Sanz*
Affiliation:
University of Zaragoza
*
∗Postal address: Dpto. Ingenierí a Matemática, Universidad de Chile, Santiago, Chile
∗∗Postal address: Dpto. Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
∗∗Postal address: Dpto. Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.

Keywords

Interacting particle systemscontact processcritical values

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 118 - 125
Copyright
Copyright © 2000 by The Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aukrust, T., Browne, D. A., and Webman, I. (1990). Critical behaviour of an autocatalytic reaction model. Phys. Rev. A 41, 5294–5301.Google Scholar
Buttell, L., Cox, J. T., and Durrett, R. (1993). Estimating the critical values of stochastic growth models. J. Appl. Prob. 30, 455–461.CrossRefGoogle Scholar
Dickman, R. (1989). Nonequilibrium lattice models: series analysis of steady states. J. Stat. Phys. 55, 997–1026.Google Scholar
Dickman, R., and Burschka, M. A. (1988). Nonequilibrium critical poisoning in a single-species model. Phys. Lett. A 127, 132–137.Google Scholar
Dickman, R. and Kamphorst Leal da Silva, J. (1998). Moment ratios for absorbing-state phase transition. Phys. Rev. E 58, 4266–4270.Google Scholar
Durrett, R. (1987). IPSmovies. Wadsworth, Belmont, California.Google Scholar
Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth, Belmont, California.Google Scholar
Grassberger, P. and de la Torre, A. (1979). Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour. Ann. Phys. 122, 373–396.Google Scholar
Harris, T. E. (1974). Contact interactions on a lattice. Ann. Prob. 2, 969–988.Google Scholar
Holley, R. A., and Liggett, T. M. (1978). The survival of contact processes. Ann. Prob. 6, 198–206.Google Scholar
Jensen, I. (1994). Critical exponents for branching annihilating random walks with an even number of offspring. Phys. Rev. E 50, 3623–3633.Google Scholar
Jensen, I. (1996). Low-density series expansions for directed percolation on square and triangular lattices. J. Phys. A 29, 7013–7040.CrossRefGoogle Scholar
Jensen, I., and Dickman, R. (1993). Nonequilibrium phase transitions in systems with infinitely many absorbing states. Phys. Rev. E 48, 1710–1725.Google Scholar
Jensen, I., and Dickman, R. (1993). Time-dependent perturbation theory for nonequilibrium lattice models. J. Stat. Phys. 71, 89–127.Google Scholar
Karlin, S., and Taylor, H. M. (1975). A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
Liggett, T. M. (1995). Improved upper bounds for the contact process critical value. Ann. Prob. 23, 697–723.Google Scholar
Liggett, T. M. (1997). Stochastic models of interacting systems [1996 Wald memorial lectures]. Ann. Prob. 25, 1–29.CrossRefGoogle Scholar
Moreira, A. G., and Dickman, R. (1996). Critical dynamics of the contact process with quenched disorder. Phys. Rev. E 54, R3090-R3093.Google Scholar
Ziezold, H., and Grillenberger, C. (1988). On the critical infection rate of the one-dimensional basic contact process: numerical results. J. App. Prob. 25, 1–8.Google Scholar