Summary
We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ d. Letη t denote this system, and letη N t denote a finite version ofη t defined on the torus [−N,N]d∩Z d. Ford≧3 we prove that for stationary, shift ergodic initial measures with density θ, that ifT(N)→∞ andT(N)/(2N+1)d →s∈[0,∞] asN→∞, then
{v θ}, θ≧0 is the set of extremal invariant measures for the infinite systemη t andQ s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.
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Aldous, D.J.: Stopping times and tightness II. Technical report no. 124. Statistics Department, University of Calif. Berkeley 1987
Athreya, K.B., Ney, P.E.: Branching processes. Berlin Heidelberg New York: Springer 1972
Bhattacharya, R.N., Rao, R.R.: Normal approximation and asymptotic expansions. New York: Wiley 1976
Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behaviors of stochastic dynamics: a pathwise approach. J. Stat. Phys.35, 606–628 (1984)
Comets, F.: Nucleation for a long range magnetic mode. Ann. Inst. Henri. Poincaré23, 135–178 (1987)
Cox, J.T. (1987): Coalescing random walks and voter model consensus times on the torus inR 3. Ann. Probab. (to appear)
Cox, J.T., Griffeath, D.: Diffusive clustering in the two dimensional voter model. Ann. Probab.14, 347–370 (1987)
Dawson, D.A., Gärtner, J.: Long-time behavior of interacting diffusions. Proceedings of the symposium on stochastic calculus, Cambridge, 1987. Pitman (to appear)
Dobrushin, R.L.: Markov processes with a large number of locally interacting components: existence of a limit process and its ergodicity. Probl. Inf. Trans.7, 149–164 (1971)
Donnelly, P., Welsh, D.: Finite particle systems and infection models. Math. Proc. Camb. Philos. Soc.94, 167–182 (1983)
Durrett, R.: An infinite particle system with additive interactions. Adv. Appl. Probab.11, 353–383 (1979)
Durrett, R.: On the growth of one dimensional contact processes. Ann. Probab.8, 890–907 (1980)
Durrett, R.: Oriented percolation in two dimensions. An. Probab.12, 999–1040 (1984)
Durrett, R.: Lecture Notes on Particle Systems and Percolation. Pacific Grove, Calif.: Wadsworth and Brooks 1988a
Durrett, R.: Crabgrass, measles, and gypsy moths: an introduction to interacting particle systems. Math. Intell.10, 37–47 (1988b)
Durrett, R., Liu, X.: The contact process on a finite set. Ann. Probab.16, 1158–1173 (1988)
Durrett, R., Schonmann, R.H.: The contact process on a finite set, II. (preprint 1988a)
Durrett, R., Schonmann, R.H.: The contact process on a finite set, III. (preprint 1988b)
Ethier, S.N., Kurtz, T.G.: Markov processes, characterization and convergence. New York: Wiley 1986
Fleischman, J.: Limiting distributions for branching random fields. Trans. Am. Math. Soc.239, 353–390 (1978)
Greven, A.: Large systems with locally interacting components. In: Syllabus of the Mark Kac Seminars, 85–87, Syllabus series CWI 1988, Amsterdam 1988
Griffeath, D.: Additive and cancellative interacting particle systems. (Lect. Notes Math. vol 724). Berlin Heidelberg New York: Springer 1979
Griffeath, D.: The basic contact process. Stochastic Processes Appl.11, 151–186 (1981)
Harris, T.E.: A correlation inequality for Markov processes in partiallys ordered state spaces. Ann. Probab.5, 451–454 (1977)
Harris, T.E.: Additive set-valued Markov processes. Ann. Probab.4, 969–988 (1978)
Kallenberg, O.: Stability of critical cluster fields. Math. Nachr.77, 7–43 (1977)
Lampertti, J.: The limit of a sequence of branching processes. Z. Wahrscheinlickeitstheor. Verw. Geb.7, 271–288 (1967)
Liggett, T.L.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985
Liggett, T.L., Spitzer, F.L.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 443–468 (1981)
Lindvall, T.: Convergence of critical Galton-Watson branching processes. J. Appl. Probab.9, 445–450 (1972)
Schonmann, R.: Metastability for the contact process. J. Stat. Phys.41, 445–464 (1985)
Tavaré, S.: Line-of-descent and genealogical processes, and their applications in population genetics models. Theor. Pop. Biol.26, 119–164 (1984)
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Work supported in part by the National Science Foundation under Grant DMS-8802055, by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by the Deutsche Forschungsgemeinschaft through the SFB 123 at the Universität Heidelberg
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Cox, J.T., Greven, A. On the long term behavior of some finite particle systems. Probab. Th. Rel. Fields 85, 195–237 (1990). https://doi.org/10.1007/BF01277982
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DOI: https://doi.org/10.1007/BF01277982