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Abstract cluster expansion with applications to statistical mechanical systems

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Poghosyan, Suren and Ueltschi, Daniel. (2009) Abstract cluster expansion with applications to statistical mechanical systems. Journal of Mathematical Physics, Vol.50 (No.5). p. 3509. ISSN 0022-2488

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Official URL: http://dx.doi.org/10.1063/1.3124770

Abstract

We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Statistical mechanics -- Research, Cluster analysis -- Research, Statistical physics -- Research, Particles (Nuclear physics) -- Mathematical models, Quantum theory
Journal or Publication Title:	Journal of Mathematical Physics
Publisher:	American Institute of Physics
ISSN:	0022-2488
Date:	May 2009
Volume:	Vol.50
Number:	No.5
Page Range:	p. 3509
Identification Number:	10.1063/1.3124770
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	National Science Foundation (U.S.) (NSF)
Grant number:	DMS-0601075 (NSF)
References:	1. Abdesselam, A. and Rivasseau, V., “Trees, forests, and jungles: a botanical garden for cluster expansions,” Lect. Notes Phys. 446, 7 (1994) e-print arXiv:hep-th/9409094. 2. Battle, G. and Federbush, P., “A phase cell cluster expansion for Euclidean field theories,” Ann. Phys. 142, 95 (1982). 3. Bovier, A. and Zahradník, M., “A simple approach to the problem of convergence of cluster expansions of polymer models,” J. Stat. Phys. 100, 765 (2000). [Inspec] [ISI] 4. Brydges, D. C., “A short course on cluster expansions,” in Phénomènes critiques, systèmes aléatoires, théories de jauge, Les Houches, 1984 (Elsevier/North Holland, Amsterdam, 1986), pp. 129–183. 5. Brydges, D. C. and Federbush, P., “A new form of the Mayer expansion in classical statistical mechanics,” J. Math. Phys. 19, 2064 (1978)JMAPAQ000019000010002064000001. [ISI] 6. Brydges, D. C. and Kennedy, T., “Mayer expansions and the Hamilton-Jacobi equation,” J. Stat. Phys. 48, 19 (1987). [Inspec] 7. Dobrushin, R. L., “Estimates of semi-invariants for the Ising model at low temperatures,” in Topics in Statistical and Theoretical Physics, AMS Translational Series 2 (American Mathematical Society, Providence, RI, 1996), Vol. 177, pp. 59–81. 8. Faris, W. G., “A connected graph identity and convergence of cluster expansions,” J. Math. Phys. 49, 113302 (2008)JMAPAQ000049000011113302000001. 9. Fernández, R. and Procacci, A., “Cluster expansion for abstract polymer models. New bounds from an old approach,” Commun. Math. Phys. 274, 123 (2007). 10. Ginibre, J., “Reduced density matrices of quantum gases. III. Hard-core potentials,” J. Math. Phys. 6, 1432 (1965)JMAPAQ000006000009001432000001. [ISI] 11. Ginibre, J., “Some applications of functional integration in statistical mechanics,” in Mécanique statistique et théorie quantique des champs, Les Houches, 1970 (Gordon and Breach, New York, 1971), pp. 327–427. 12. Glimm, J., Jaffe, A., and Spencer, T., “The Wightman axioms and particle structure in the [script P](phi)2 quantum field model,” Ann. Math. 100, 585 (1974). [ISI] 13. Gruber, C. and Kunz, H., “General properties of polymer systems,” Commun. Math. Phys. 22, 133 (1971). [ISI] 14. Jackson, B., Procacci, A., and Sokal, A. D., “Complex zero-free regions at large \|q\| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights,” e-print arXiv:0810.4703. 15. Kotecký, R. and Preiss, D., “Cluster expansion for abstract polymer models,” Commun. Math. Phys. 103, 491 (1986). [Inspec] [ISI] 16. Malyshev, V. A., “Cluster expansions in lattice models of statistical physics and quantum field theory,” Russ. Math. Surveys 35, 1 (1980). 17. Minlos, R. A. and Poghosyan, S., “Estimates of Ursell functions, group functions, and their derivatives,” Theor. Math. Phys. 31, 408 (1977). [ISI] 18. Miracle-Solé, S., “On the convergence of cluster expansions,” Physica A 279, 244 (2000). [Inspec] 19. Penrose, O., Statistical Mechanics, Foundations and Applications (Benjamin, New York, 1967). 20. Pfister, Ch. -E., “Large deviations and phase separation in the two-dimensional Ising model,” Helv. Phys. Acta 64, 953 (1991). [Inspec] [ISI] 21. Poghosyan, S. and Zessin, H., “Decay of correlations of the Ginibre gas obeying Maxwell-Boltzmann statistics,” Markov Processes Relat. Fields 7, 581 (2001). 22. Procacci, A., “Abstract polymer models with general pair interactions,” e-print arXiv:0707.0016 (version 2 of 26 November 2008). 23. Ruelle, D., Statistical Mechanics: Rigorous Results (World Scientific, Singapore, 1999). 24. Seiringer, R. and Ueltschi, D., “Rigorous upper bounds on the critical temperature of dilute Bose gases,” e-print arXiv:0904.0050. 25. Sokal, A. D., “Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions,” Combinatorics, Probab. Comput. 10, 41 (2001). [ISI] 26. Ueltschi, D., “Cluster expansions and correlation functions,” Mosc. Math. J. 4, 511 (2004), e-print arXiv:math-ph/0304003.
URI:	http://wrap.warwick.ac.uk/id/eprint/2195