Abstract
Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the λ(φ4)2 quantum field model. We also establish new upper bounds on critical temperatures.
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Symanzik, K.: Euclidean quantum field theory. In: Local quantum theory. Jost, R. (ed.) New York, London: Academic Press 1969
Brydges, D., Federbush, P.: Commun. Math. Phys.62, 79 (1978) (also in unpublished work by J. Fröhlich and T. Spencer)
Durhuus, B., Fröhlich, J.: Commun. Math. Phys.75, 103 (1980)
Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Commun. Math. Phys.62, 1 (1978)
Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979
Lebowitz, J.L.: Commun. Math. Phys.28, 313 (1972). See also [5] p. 131
Newman, C.: Z. Warsch. verw. Gebiete33, 75 (1975)
Simon, B.: Commun. Math. Phys.77, 111 (1980)
Lieb, E.H.: Commun. Math. Phys.77, 127 (1980)
Rivasseau, V.: Commun. Math. Phys.77, 145 (1980)
Aizenman, M., Simon, B.: Commun. Math. Phys.77, 137 (1980)
Glimm, J., Jaffe, A., Spencer, T.: Commun. Math. Phys.45, 203 (1975)
Fröhlich, J., Lieb, E.H.: Commun. Math. Phys.60, 233 (1978)
Seiler, E., Simon, B.: Ann. Phys. (NY)97, 470 (1976)
Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976)
Lebowitz, J.L.: Commun. Math. Phys.28, 313 (1972)
McBryan, O., Spencer, T.: Commun. Math. Phys.53, 299 (1977)
Ellis, R.S., Monroe, J.L., Newman, C.M.: Commun. Math. Phys.46, 167 (1976)
Bricmont, J.: J. Stat. Phys.17, 289 (1977)
Ginibre, J.: Commun. Math. Phys.16, 310 (1970)
Glimm, J., Jaffe, A., Spencer, T.: Ann. Math.100, 585 (1974)
Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications of high temperature expansions. In: Constructive quantum field theory, Velo, G., Wightman, A.S. (eds.), p. 133. Berlin, Heidelberg, New York: Springer 1973
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Communicated by A. Jaffe
Partially supported by N.S.F. Grant No. 79-02490
On leave from the University of Virginia
Partially supported by N.S.F. Grant No. DMR 81-00417
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Brydges, D., Fröhlich, J. & Spencer, T. The random walk representation of classical spin systems and correlation inequalities. Commun.Math. Phys. 83, 123–150 (1982). https://doi.org/10.1007/BF01947075
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DOI: https://doi.org/10.1007/BF01947075