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Inequalities for continuous-spin Ising ferromagnets

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Abstract

We investigate correlation inequalities for Ising ferromagnets with continuous spins, giving a simple unified derivation of inequalities of Griffiths, Ginibre, Percus, Lebowitz, and Ellis and Monroe. The single-spin measure and Hamiltonian for which an inequality may be proved become more restricted as the inequality becomes more complex. However, all results hold for a model with ferromagnetic pair interactions, positive (nonuniform) external field, and single-spin measureν eitherv(σ) = [1/(l + 1)] xΣ f=0/l δ(−l +2j +σ) (spinl/2) ordv(σ) = exp [−P(σ)]dσ, whereP is an even polynomial all of whose coefficients must be positive except the quadratic, which is arbitrary. The Lebowitz correlation inequality is a corollary of the Ellis-Monroe inequality. As an application, we generalize the method of van Beijeren to establish a sharp phase interface at low temperature in nearest neighbor ferromagnets of at least three dimensions with arbitrary (symmetric) single-spin measure.

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Authors and Affiliations

  1. Belfer Graduate School of Science, Yeshiva University, New York, New York

    Garrett S. Sylvester

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  1. Garrett S. Sylvester
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Additional information

Supported in part by the National Science Foundation under Grants MPS 73-05037 and MPS 75-20638. Much of this research was performed while the author was a student at the Massachusetts Institute of Technology and Harvard University, Cambridge, Masachusetts.

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Sylvester, G.S. Inequalities for continuous-spin Ising ferromagnets. J Stat Phys 15, 327–341 (1976). https://doi.org/10.1007/BF01023057

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  • DOI: https://doi.org/10.1007/BF01023057

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