Artículos de revistas
Loss Network Representation Of Peierls Contours
Registro en:
Annals Of Probability. , v. 29, n. 2, p. 902 - 937, 2001.
911798
10.1214/aop/1008956697
2-s2.0-0035537297
Autor
Fernandez R.
Ferrari P.A.
Garcia N.L.
Institución
Resumen
We present a probabilistic approach for the study of systems with exclusions in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with "duplicated-variables expansion," used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques. 29 2 902 937 Athreya, K.B., Ney, P.E., (1972) Branching Processes, , Springer, New York Baddeley, A.J., Van Lieshout, M.N.M., Area-interaction point processes (1995) Ann. Inst. Statist. Math., 47, pp. 601-619 Bolthausen, E., On the central limit theorem for stationary mixing random fields (1982) Ann. Probab., 10, pp. 1047-1050 Borgs, C., Imbrie, J.Z., A unified approach to phase diagrams in field theory and statistical mechanics (1989) Comm. Math. Phys., 123, pp. 305-328 Bricmont, J., Kupiainen, A., High temperature expansions and dynamical systems (1996) Comm. Math. Phys., 178, pp. 703-732 Bricmont, J., Kupiainen, A., Infinite-dimensional SRB measures: Lattice dynamics (1997) Phys. D, 103, pp. 18-33 Brockmeyer, E., Halstrøm, H.L., Jensen, A., The life and works of A. K. Erlang (1948) Trans. Danish Acad. Tech. Sci. (1960) Acta Polytech. Scand., 287 Brydges, D.C., A short cluster in cluster expansions (1984) Critical Phenomena, Random Systems, Gauge Theories, pp. 129-183. , (K. Osterwalder and R. Stora, eds.) North-Holland, Amsterdam Dobhushin, R.L., Existence of a phase transition in the two-dimensional and three-dimensional Ising models (1965) Theory Probab. Appl., 10, pp. 193-213 Soviet Phys. Dok., 10, pp. 111-113 Dobrushin, R.L., Estimates of semiinvariants for the Ising model at low temperatures (1996) Topics in Statistics and Theoretical Physics. Amer. Math. Soc. Transl. Ser. 2, 177, pp. 59-81 Dobrushin, R.L., Perturbation methods of the theory of Gibbsian fields (1996) École d'Été de Probabilités de Saint-Flour XXIV. Lecture Notes in Math., 1648, pp. 1-66. , Springer, New York Von Dreifus, H., Klein, A., Perez, J.F., Taming Griffiths' singularities: Infinite differentiability of quenched correlation functions (1995) Comm. Math. Phys., 170, pp. 21-39 Durrett, R., Ten lectures on particle systems (1995) Lectures on Probability Theory, pp. 97-201. , Springer, Berlin Fernández, R., Ferrari, P.A., Garcia, N., Measures on contour, polymer or animal models: A probabilistic approach (1998) Markov Process. Related Fields, 4, pp. 479-497 Fernández, R., Ferrari, P.A., Garcia, N., (1999) Perfect Simulation of Fixed-routing Loss Networks: Application to the Low-temperature Ising Model, , Unpublished manuscript Ferrari, P.A., Garcia, N., One-dimensional loss networks and conditioned M/G/∞ queues (1998) J. Appl. Probab., 35, pp. 963-975 Griffiths, R.B., Peierls' proof of spontaneous magnetization in a two dimensional Ising ferromagnet (1964) Phys. Rev. A, 136, pp. 437-439 Hall, P., On continuum percolation (1985) Ann. Probab., 13, pp. 1250-1266 Hall, P., (1988) Introduction to the Theory of Coverage Processes, , Wiley, New York Harris, T.E., The theory of branching processes (1963) Grundlehren Math. Wiss., 119 Harris, T.E., Nearest-neighbor Markov interaction processes on multidimensional lattices (1972) Adv. in Math., 9, pp. 66-89 Kelly, F.P., Loss networks (1991) Ann. Appl. Probab., 1, pp. 319-378 Kendall, W.S., On some weighted Boolean models (1997) Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets, pp. 105-120. , World Scientific, River Edge, NJ Kendall, W.S., Perfect simulation for the area-interaction point process (1998) Probability Towards 2000, pp. 218-234. , Springer, New York Kotecký, R., Preiss, D., Cluster expansion for abstract polymer models (1986) Comm. Math. Phys., 103, pp. 491-498 Liggett, T.M., (1985) Interacting Particle Systems, , Springer, New York Liggett, T.M., Improved upper bounds for the contact process critical value (1995) Ann. Probab., 23, pp. 697-723 Malyshev, V.A., Cluster expansions in lattice models of statistical physics and quantum theory of fields (1980) Russian Math. Surveys, 35, pp. 1-62 Neveu, J., Processus ponctuels (1977) École d'Été de Probabilités de Saint-Flour VI. Lecture Notes in Math., 598, pp. 249-445. , Springer, Berlin Peierls, R., On Ising's model of ferromagnetism (1936) Math. Proc. Cambridge Philos. Soc., 32, pp. 477-481 Seiler, E., (1982) Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Phys., 159. , Springer, Berlin Zahradník, E., An alternate version of Pirogov-Sinai theory (1984) Comm. Math. Phys., 93, pp. 559-5581