Abstract
We consider perturbations of a massless Gaussian lattice field on ℤd,d≧3, which preserves the continuous symmetry of the Hamiltonian, e.g.,
It is known that for allT>0 the correlation functions in this model do not decay exponentially. We derive a power law upper bound for all (truncated) correlation functions. Our method is based on a combination of the log concavity inequalities of Brascamp and Lieb, reflection positivity and the Fortuin, Kasteleyn and Ginibre (F.K.G.) inequalities.
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References
Battle, G., Rosen, L.: J. Stat. Phys.22 128 (1980)
Brascamp, H. J., Lieb, E. H.: J. Funct. Anal.22, 366 (1976)
Brascamp, H. J., Lieb, E. H., Lebowitz, J. L.: Bull. Int. Statist. Inst.46, Invited Paper No. 62 (1975)
Brascamp, H. J., Lieb, E. H.: Lecture given at the Conference on Functional Integration, Cumberland Lodge, England, April 2–4, 1974.
Bricmont, J., Fontaine, J.R., Lebowitz, J.L., Spencer, T.: Commun. Math. Phys.78, 281–302 (1980)
Fortuin, C., Kasteleyn, P., Ginibre, J.: Commun. Math. Phys.22, 89 (1971)
Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976)
Fröhlich, J., Lieb, E. H.: Commun. Math. Phys.60, 233 (1978)
Fröhlich, J., Israël, R., Lieb, E. H., Simon, B.: Commun. Math. Phys.62, 1 (1978)
Fröhlich, J., Spencer, T.: On the statistical mechanics of classical goulomb and dipole gases. Preprint, IHES, 80 (to appear in J. Stat. Phys.)
Lebowitz, J.: Commun. Math. Phys.28, 313 (1972)
Park, Y. M.: Commun. Math. Phys.70, 161 (1979)
Schor, R.: Commun. Math. Phys.53, 213 (1978)
Simon, B.: Thep (φ)2 Euclidean (quantum) field theory. Princeton, N. J.: Princeton University Press, 1974.
Sokal, A.: In preparation
Mermin, N. D., Wagner, H.: Phys. Rev. Lett.17, 1133 (1966)
Hohenberg, P. C.: Phys. Rev.158, 383 (1967)
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Communicated by A. Jaffe
(J. B.) Supported by NSF Grant NrMCS78-01885 (J. L. L. and J. R. F.) Supported by NSF Grant NrPHY78-15320 (T. S.) Supported by NSF Grant NrDMR73-04355
On leave from: Institut de Physique Theorique, Universite de Louvain, Belgium
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Bricmont, J., Fontaine, JR., Lebowitz, J.L. et al. Lattice systems with a continuous symmetry. Commun.Math. Phys. 78, 363–371 (1981). https://doi.org/10.1007/BF01942329
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DOI: https://doi.org/10.1007/BF01942329