Griffiths inequalities

Statement

Let $\Lambda$ be a finite set of sites with Ising spins $\sigma_i\in\{-1,+1\}$, and consider the finite-volume ferromagnetic Ising model with Hamiltonian

where the couplings satisfy $J_{ij}\ge 0$ (ferromagnetic) and the external fields satisfy $h_i\ge 0$.

Let $\mu_\Lambda$ be the associated finite-volume Gibbs measure and write $\langle \cdot\rangle_\Lambda$ for expectation (see expectation ). For $A\subset\Lambda$, define the spin monomial $\sigma_A := \prod_{i\in A}\sigma_i$ (with $\sigma_\varnothing=1$).

Griffiths inequalities:

1. (Griffiths I) For every $A\subset\Lambda$,

   $$\langle \sigma_A\rangle_\Lambda \ge 0.$$

2. (Griffiths II) For every $A,B\subset\Lambda$,

   $$\langle \sigma_A\sigma_B\rangle_\Lambda - \langle \sigma_A\rangle_\Lambda\,\langle \sigma_B\rangle_\Lambda \ge 0.$$

Equivalently, all covariances of spin monomials are nonnegative.

Key hypotheses and conclusions

Hypotheses

• Finite volume $\Lambda$.
• Ising spins $\sigma_i\in\{-1,+1\}$ (context: Ising model ).
• Ferromagnetic couplings $J_{ij}\ge 0$.
• Nonnegative external fields $h_i\ge 0$.
• Gibbs measure $\mu_\Lambda$ (context: lattice Hamiltonian , finite-volume Gibbs measure , partition function ).

Conclusions

• Positivity of moments: $\langle\sigma_A\rangle_\Lambda\ge 0$ for all $A$.
• Positive correlations: $\mathrm{Cov}_\Lambda(\sigma_A,\sigma_B)\ge 0$ for all $A,B$.
• Monotonicity (useful corollary): differentiating expectations with respect to fields/couplings yields covariances, so Griffiths II implies monotonicity of $\langle\sigma_A\rangle_\Lambda$ in the parameters (a basic “ferromagnets align more when you increase $J$ or $h$” principle).

Proof idea / significance

A common route is to express derivatives of the pressure $\log Z_\Lambda$ (see pressure ) in terms of truncated correlations and then prove these derivatives are nonnegative using ferromagnetic structure (e.g. random-current/high-temperature expansions or correlation-inequality arguments). In practice, Griffiths inequalities are often obtained as consequences of the stronger GKS inequalities (and, for monotone observables, the FKG inequality ).

They are foundational for:

• comparison/monotonicity arguments in boundary conditions and fields,
• establishing existence and order properties of infinite-volume limits (see infinite-volume Gibbs measures ),
• proving bounds on correlation functions (see two-point correlations ).