×

zbMATH — the first resource for mathematics

Cluster expansion for abstract polymer models. (English) Zbl 0593.05006
Summary: A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or “combinatorics of trees”. A distinctive feature is that, at all steps, the considered clusters contain every polymer at most once.

MSC:
82B05 Classical equilibrium statistical mechanics (general)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82D60 Statistical mechanics of polymers
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Malyshev, V.A.: Cluster expansions in lattice models in statistical physics and quantum field theory. Usp. Mat. Nauk35, 3-53 (1980)
[2] Ruelle, D.: Statistical mechanics: Rigorous results. New York: Benjamin 1969 · Zbl 0177.57301
[3] Minlos, R.A., Sinai, Ya.G.: The phenomenon of ?phase separation? at low temperatures in some lattice models of a gas. I, II. Math. USSR-Sb.2, 335-395 (1967) and Trans. Mosc. Math. Soc.19, 121-196 (1968)
[4] Sinai, Ya.G.: Theory of phase transitions: rigorous results. Budapest and London: Akadémiai Kiadó and Pergamon Press 1982
[5] Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys.22, 133-161 (1971)
[6] Gallavotti, G., Martin-Löf, A., Miracle-Sole, S.: Some problems connected with the description of coexisting phases at low temperatures in the Ising model. Battelle Seattle 1971 Recontres. A. Lenard, (ed.). Lecture Notes in Physics, Vol. 20, pp. 162-204. Berlin, Heidelberg, New York: Springer 1973
[7] Seiler, E.: Gauge theories as a problem of constructive quantum field theory and statistical mechanics. Lecture Notes in Physics, Vol. 159. Berlin, Heidelberg, New York: Springer 1982
[8] Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Commun. Math. Phys.85, 517 (1982)
[9] Rushbrooke, G.S., Baker, G.A., Wood, P.J.: Heisenberg model, pp. 245-356. Domb, C.: Graph theory and embeddings, pp. 1-95. In: Phase transitions and critical phenomena, Vol. 3. Domb, C., Green, M.S. (eds.). London, New York, San Francisco: Academic Press 1974
[10] Mack, G.: Nonperturbative methods. In: Gange theories of the eighties. Raitio, R., Lindfors, J. (eds.). Lecture Notes in Physics, Vol. 181. Berlin, Heidelberg, New York: Springer 1983
[11] Guerra, F.: Gauge fields on a lattice: selected topics. In: Field theoretical methods in particle physics, pp. 41-65. Rühl, W. (ed.). New York, London: Plenum Press 1980
[12] Navrátil, J.: Contour models and unicity of random fields. (In Czech). Diploma thesis, Charles University, Prague (1982)
[13] Kotecký, R., Preiss, D.: Dobrushin unicity and its application to polymer models (in preparation) · Zbl 0593.05006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.