Abstract
We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages \(\bar X_\Lambda : = \frac{1}{{|\Lambda |}}\Sigma _{i \in \Lambda } X_i \), where the X i 's are copies of a self-adjoint element X (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of \(\frac{1}{{|\Lambda |}}\Sigma _{i \in \Lambda } \delta _{X_i } \)
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REFERENCES
W. Bryc, A remark on the connection between the large deviation principle and the central limit theorem, Stat. Prob. Lett. 18:253–256 (1993).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2 (Springer, Berlin, 1996).
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, (Springer-Verlag, New York, 1998).
Enter A. C. D. van, R. Fern´andez, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization group transformations: Scope and limitations of Gibbsian theory, J.Stat.Phys. 72:879–1167 (1993).
H.O. Georgii, Gibbs Measures and Phase Transitions(Walter de Gruyter & Co., Berlin, 1988).
R. S. Ellis, Large Deviations and Statistical Mechanics(Springer-Verlag, New York, 1985).
R. B. Israel, Convexity in the Theory of Lattice Gases(Princeton University Press, Princeton, 1979).
R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras(Academic Press, New York, London, 1983).
R. Kotecky and D. Preiss, Cluster expansion for abstract polymer models, Commun.Math.Phys. 103:491–498 (1986).
J. L. Lebowitz, M. Lenci and H. Spohn, Large deviations for ideal quantum systems, Math-Phys Archive 9906014.
G. Gallavotti, J. L. Lebowitz, and V. Mastropietro, Large deviations in rarefied quantum gases, J.Statist.Phys. 108:(5–6): 831–861 (2002).
S. Miracle-Sol´e, On the convergence of cluster expansions, Physica A 279:244–249 (2000).
S. Olla, Large deviations for Gibbs random fields, Prob.Theory.Related.Fields 77:343–357 (1988).
Y. M. Park, The cluster expansion for classical and quantum lattice systems, J.Stat.Phys. 27:553–576 (1982).
B. Simon, The Statistical Mechanics of Lattice Gases(Princeton University Press, Princeton, 1993).
D. Goderis and P. Vets P, Central limit theorem for mixing quantum systems and the CCR-algebra of fluctuations, Common.Math.Phys. 122:249–265 (1989).
D. Goderis, A. Verbeure, P. Vets, Noncommutative central limits, Prob.Theory Related Fields 82:527–544 (1989).
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Netočný, K., Redig, F. Large Deviations for Quantum Spin Systems. Journal of Statistical Physics 117, 521–547 (2004). https://doi.org/10.1007/s10955-004-3452-4
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DOI: https://doi.org/10.1007/s10955-004-3452-4