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Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems. (English) Zbl 1125.82302
Summary: We present and discuss some physical hypotheses on the decrease of truncated correlation functions and we show that they imply the analyticity of the thermodynamic limits of the pressure and of all correlation functions with respect to the reciprocal temperature β and the magnetic field h (or the chemical potential mu) at all (real) points (β 0 ,h 0 ) (or (β 0 ,μ 0 )) where they are supposed to hold. A decrease close to our hypotheses is derived in certain particular situations at the end.
MSC:
82B05Classical equilibrium statistical mechanics (general)
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
References:
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[2]Ruelle, D.: Statistical mechanics, rigourous results, pp. 90–92. New York: Benjamin 1969.
[3]Ruelle, D.: Statistical mechanics, rigourous results, p. 100. New York: Benjamin 1969.
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[9]We owe the ideas of this subsection to D. Ruelle.
[10]We owe this elegant proof to C. Billionnet and C. DeCalan.
[11]Arguments about this point have been given to us by D. Ruelle.
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[13]Dobrushin, R. I.: Gibbsian random fields for lattice systems with pairwise interactions. Funktsional’. Analiz i Ego Prilozhen,2 (4), 31 (1968) (russian).
[14]Ruelle, D.: Statistical mechanics, rigourous results. New York: Benjamin 1969.
[15]For Vitali’s theorem see for instance: Goluzin, G. M.: Geometrical theory of functions of a complex variable, p. 18. AMS, Providence, Rhode Island 1969.
[16]For Hartog’s theorem see for instance: Bochner, S., Martin, W. T.: Several complex variables. Princeton 1948, VII, 4, Th. 4.
[17]Lebowitz, J. L., Penrose, O.: Commun. math. Phys.11, 99 (1968). · Zbl 0162.59002 · doi:10.1007/BF01645899
[18]See for instance Camp, W. J., Fisher, M. E.: Phys. Rev. B,6 (2), 946 (1972) where references to the transfer matrix formalism will be found. · doi:10.1103/PhysRevB.6.946