Griffiths inequalities

Positivity and monotonicity properties of spin correlations for the finite-volume ferromagnetic Ising model.

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

H_\Lambda(\sigma) = -\sum_{\{i,j\}\subset \Lambda} J_{ij}\,\sigma_i\sigma_j - \sum_{i\in\Lambda} h_i\,\sigma_i,

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:

(Griffiths I) For every $A\subset\Lambda$ ,
$\langle \sigma_A\rangle_\Lambda \ge 0.$
(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).