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Griffiths' inequalities for Ashkin‐Teller model

### Abstract

The two Griffiths' inequalities for the correlation functions of Ising ferromagnets and two others added by Kelly and Sherman and by Sherman are extended to what we call generalized Ashkin‐Teller model. In this model we consider a system of *N* particles; each can exist in *r* possible states. Let a collection of *pairs* of particles be represented by a graph with particles as vertices and *pairs* of particles as edges. The ``many‐body interaction'' among a cluster of particles represented by such a graph *G*(*A*) is ‐ *J*_{A} (*J*_{A} ≥ 0) when the particles in each connected component of *G*(*A*) all exist in the same state; it is 0 otherwise. For the special case with *r* = 2 and two‐body interactions only, the Ashkin‐Teller model is equivalent to Ising model. Therefore, what we present in this paper can be considered as yet another way of proving the original correlation inequalities for Ising ferromagnets with two‐body interactions. We have also discovered another new inequality, namely 〈δ^{ A }〉 〈δ^{ ABRS }〉 + 〈δ^{ AB }〉 〈δ^{ ARS }〉 ‐ 〈δ^{ AR }〉 〈δ^{ ABS }〉 ‐ 〈δ^{ AS }〉 〈δ^{ ABR }〉 ≥ 0.

© 1973 The American Institute of Physics

Received 26 March 1973
Published online 03 November 2003

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