General Griffiths' Inequalities on Correlations in Ising Ferromagnets

Abstract

Let N = (1, 2, ⋯, n). For each subset A of N, let JA ≥ 0. For eachi∈N, let σi ± 1. For each subset A of N, define σA=∏i∈A σi. Let the Hamiltonian be − ΣACN JA σA. Then for each A, B⊂N, 〈σA〉≥0 and 〈σAσB〉−〈σA〉〈σB〉≥0. This weakens the hypothesis and widens the conclusion of a result due to Griffiths.

Correlation inequalities and the decay of correlations in ferromagnets

“…Such methods were introduced by Griffiths in his original series [18][19][20] and extended by Kelley and Sherman [25], Griffiths et al [22], and Newman [32]. In particular, our proof has elements in common with Griffiths' proof of his third inequality [20] and with Newman's proff of his inequality [32].…”

The (φ4)2 field theory as a classical Ising model

“…The first class includes the zero theorem of Lee and Yang [15,1] and the correlation inequalities of the Griffiths-Hurst-Sherman (GHS) type [8]. The second class includes the correlation inequalities of Griffiths, Kelly, and Sherman (GKS) [6,13] and of Fortuin, Kasteleyn, and Ginibre (FKG) [3]. (GKS inequalities were originally proven for classical models [6] but were eventually proven with many body interactions [13], higher spin [7] and arbitrary even spin distributions [4].…”

“…The second class includes the correlation inequalities of Griffiths, Kelly, and Sherman (GKS) [6,13] and of Fortuin, Kasteleyn, and Ginibre (FKG) [3]. (GKS inequalities were originally proven for classical models [6] but were eventually proven with many body interactions [13], higher spin [7] and arbitrary even spin distributions [4]. ) Recently, Guerra, Rosen and Simon [12] have shown how the P(φ) 2 Euclidean field theory [16,12] can be approximated by general Ising models.…”

Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States

“…Bricmont and Fontain derived the contour bound for the spin systems with the potential energy (4.1) with the help of the second Griffiths [23] and Jensen inequalities [24] (see also [25,26])…”