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Limit theorems for the empirical vector of the Curie-Weiss-Potts model. (English) Zbl 0705.60027

Limit theorems for the empirical vector \(L_ n\) of spin random variables with a parameter of dependence \(\beta\) in the Curie-Weiss-Potts-Ising model of statistical mechanics in both non-critical and critical cases are obtained, including some earlier results of R. S. Ellis, C. M. Newman and J. S. Rosen [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 44, 117-139 (1978; Zbl 0364.60120), and ibid. 51, 153-169 (1980; Zbl 0404.60096)].
The limit theorems state the law of large numbers \(L_ n\to \delta_{\nu^ 0}\), \(n\to \infty\), with \(\nu^ 0=(q^{-1},...,q^{- 1})\), and the number of spin states q, in the sub-critical case \(\beta <\beta_ C\), and its breakdown in critical and super-critical cases \(\beta \geq \beta_ C\), and the central limits for (\(\sqrt{n}L_ n)\) compensated.
Reviewer: E.I.Trofimov

MSC:

60F05 Central limit and other weak theorems
82B05 Classical equilibrium statistical mechanics (general)
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[1] Comets, F.; Gidas, B., Asymptotics of maximum likelihood estimators for the Curie-Weiss model, (1988), Brown Univ Providence, RI, Preprint · Zbl 0749.62018
[2] De Coninck, J., On limit theorems for the bivariate (magnetization, energy) variable at the critical point, Comm. math. phys., 109, 191-205, (1987) · Zbl 0619.60096
[3] Eisele, T.; Ellis, R.S., Symmetry breaking and random waves for magnetic systems on a circle, Z. wahrsch. verw. gebiete, 63, 297-348, (1983) · Zbl 0494.60097
[4] Ellis, R.S., Entropy, large deviations, and statistical mechanics, (1985), Springer New York · Zbl 0566.60097
[5] Ellis, R.S., A unified approach to large deviations for Markov chains and applications to statistical mechanics, ()
[6] Ellis, R.S.; Newman, C.M., Limit theorems for sums of dependent random variables occurring in statistical mechanics, Z. wahrsch. verw. gebiete, 44, 117-139, (1978) · Zbl 0364.60120
[7] Ellis, R.S.; Newman, C.M.; Rosen, J.S., Limit theorems for sums of dependent random variables occurring in statistical mechanics, II: conditioning, multiple phases, and metastability, Z. wahrsch. verw. gebiete, 51, 153-169, (1980) · Zbl 0404.60096
[8] Kesten, H.; Schonmann, R.H., Behavior in large dimensions of the Potts model and Heisenberg models, Rev. math. phys., 1, 2-3, (1990)
[9] Orey, S., Large deviations for the empirical field of Curie-Weiss models, Stochastic, 25, 3-14, (1988) · Zbl 0657.60040
[10] Pearce, P.A.; Griffiths, R.B., Potts model in the many-component limit, J. phys. A, 13, 2143-2148, (1980)
[11] Wu, F.Y., The Potts model, Rev. modern phys., 54, 235-268, (1982)
[12] Wu, F.Y., Potts model of magnetism, J. appl. phys., 55, 2421-2425, (1984)
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