Griffiths inequality

Remarks

For the first inequality, expand $\exp(\sum_C J_C \sigma_C)$ as a power series in the $J_C$ with nonnegative coefficients. Each term is a product of spins, and summing $\sigma_A$ times such a product over all $\sigma$ gives either $2^{\abs{\Lambda}}$ or $0$. For the second, take two independent copies $\sigma, \sigma'$ and apply the same expansion in the variables $\sigma_i$ and $\sigma_i \sigma'_i$. The second inequality says that $\partial \langle \sigma_A \rangle / \partial J_B = \langle \sigma_A \sigma_B \rangle - \langle \sigma_A \rangle \langle \sigma_B \rangle \ge 0$: correlations increase when any coupling increases. The sign condition matters: with a single antiferromagnetic bond $J_{\set{1,2}} < 0$ on two sites, $\langle \sigma_1 \sigma_2 \rangle = \tanh J_{\set{1,2}} < 0$.

Example: for the Ising model with $J > 0$ and $h \ge 0$, the magnetization $\langle \sigma_0 \rangle$ is nonnegative and increases with $J$, with $h$, and, under free boundary condition, with the box.

Going deeper

For a student

The inequalities were proved by Robert Griffiths in 1967 for pair interactions and extended by Kelly and Sherman; they are also called the GKS inequalities. They imply that correlations of the Ising model increase with the volume under free boundary condition and decrease under plus boundary condition, so the infinite-volume limits exist. They also give monotonicity of the critical temperature in the couplings and comparison between lattices of different dimension. Related tools are the FKG inequality and the GHS inequality, which controls third-order correlations.