Abstract
A convergence criterion of cluster expansion is presented in the case of an abstract polymer system with general pair interactions (i.e. not necessarily hard core or repulsive). As a concrete example, the low temperature disordered phase of the BEG model with infinite range interactions, decaying polynomially as 1/r d+λ with λ>0, is studied.
Similar content being viewed by others
References
Battle, G.A., Federbush, P.: A phase cell cluster expansion for euclidean field theory. Ann. Phys. 142, 95–139 (1982)
Blume, M., Emery, V.J., Griffiths, R.B.: Ising model for the λ transition and phase separation in He3-He4 mixtures. Phys. Rev. A 4, 1071–1077 (1971)
Brydges, D., Federbush, P.: A new form of the Mayer expansion in classical statistical mechanics. J. Math. Phys. 19, 2064–2067 (1978)
Brydges, D.: A short course on cluster expansion. In: Osterwalder, K., Stora, R. (eds.) Proceedings of 1984 Les Houches Summer School. North-Holland, Amsterdam (1986)
Cammarota, C.: Decay of correlations for infinite range interactions in unbounded spin systems. Comm. Math. Phys. 85, 517–528 (1982)
Dinaburg, E.I., Sinai, Ya.G.: Contour models with interaction and their applications. Sel. Math. Sov. 7, 291–315 (1988)
Dobrushin, R.L.: Estimates of semi-invariants for the Ising model at low temperatures. In: Topics in Statistical and Theoretical Physics, American Mathematical Society Translations, Ser. 2, vol. 177, pp. 59–81 (1996)
Fernández, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Preprint, arxiv math-ph/0605041
Gruber, C., Kunz, H.: General properties of polymer systems. Comm. Math. Phys. 22, 133–161 (1971)
Imbrie, J.Z.: Decay of correlations in the one dimensional Ising model with J ij =|i−j|−2 decay of correlations. Comm. Math. Phys. 85, 491–515 (1982)
Kotecký, R., Preiss, D.: Cluster expansion for abstract polymer models. Comm. Math. Phys. 10, 491–498 (1986)
Miracle-Solé, S.: On the convergence of cluster expansions. Physica A 279, 244–249 (2000)
Park, Y.M.: Extension of Pirogov-Sinai theory of phase transitions to infinite range interactions. I. Cluster expansion. Comm. Math. Phys. 114, 187–218 (1988)
Penrose, O.: Convergence of fugacity expansions for classical systems. In: Bak, A. (ed.) Statistical Mechanics: Foundations and Applications. Benjamin, New York (1967)
Pfister, C.E.: Large deviation and phase separation in the two-dimensional Ising model. Helvetica Phys. Acta 64, 953–1054 (1991)
Procacci, A.: Cluster expansion methods in rigorous statistical mechanics. Preprint (www.mat.ufmg.br/aldo/papers/book.pdf)
Procacci, A., de Lima, B.N.B., Scoppola, B.: A remark on high temperature polymer expansion for lattice systems with infinite range pair interactions. Lett. Math. Phys. 45, 303–322 (1998)
Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics, vol. 159. Springer, Berlin (1982)
Sokal, A.: Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Comb. Probab. Comput. 10, 41–77 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/s10955-009-9753-x.
Rights and permissions
About this article
Cite this article
Procacci, A. Abstract Polymer Models with General Pair Interactions. J Stat Phys 129, 171–188 (2007). https://doi.org/10.1007/s10955-007-9378-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9378-x