General Griffiths' Inequalities on Correlations in Ising Ferromagnets

Abstract

Let N = (1, 2, ⋯, n). For each subset A of N, let J_A ≥ 0. For each$\mathrm{i\in N}$ , let σ_i ± 1. For each subset A of N, define ${\sigma }^A{\mathrm{=\prod i\in A\hspace{0.17em}\sigma }}_i$ . Let the Hamiltonian be − Σ_ACN J_A σ^A. Then for each A, $\mathrm{B\subset N}$ , ${\mathrm{〈\sigma }}^A\text{〉≥0}$ and ${\mathrm{〈\sigma }}^A{\sigma }^B{\mathrm{〉-〈\sigma }}^A{\mathrm{〉〈\sigma }}^B\text{〉≥0}$ . This weakens the hypothesis and widens the conclusion of a result due to Griffiths.