GKS inequalities (Griffiths–Kelly–Sherman)

Statement

Consider the finite-volume Ising model on a finite $\Lambda$ with spins $\sigma_i\in\{-1,+1\}$ and Hamiltonian

with ferromagnetic couplings $J_{ij}\ge 0$ and nonnegative fields $h_i\ge 0$.

Let $Z_\Lambda(J,h)$ be the partition function and let $\langle\cdot\rangle_\Lambda$ denote expectation under the corresponding finite-volume Gibbs measure .

For $A\subset\Lambda$, write $\sigma_A=\prod_{i\in A}\sigma_i$.

GKS inequalities. For all $A,B\subset\Lambda$,

1. (GKS I) $\langle \sigma_A\rangle_\Lambda \ge 0$.
2. (GKS II) $\langle \sigma_A\sigma_B\rangle_\Lambda - \langle \sigma_A\rangle_\Lambda\,\langle \sigma_B\rangle_\Lambda \ge 0$.

In particular, all covariances of spin monomials are nonnegative.

A useful reformulation is in terms of the finite-volume pressure

(see pressure ): derivatives of $\log Z_\Lambda$ with respect to fields/couplings are (truncated) correlations, so GKS II asserts nonnegativity of the corresponding second derivatives.

Key hypotheses and conclusions

Hypotheses

• Finite volume $\Lambda$ and Ising spins.
• Ferromagnetic pair couplings $J_{ij}\ge 0$.
• Nonnegative external fields $h_i\ge 0$.
• Gibbs measure context: lattice Hamiltonian , finite-volume Gibbs measure .

Conclusions

• Positivity of correlations: all moments $\langle\sigma_A\rangle_\Lambda$ are nonnegative.
• Positive truncated correlations: $\mathrm{Cov}_\Lambda(\sigma_A,\sigma_B)\ge 0$.
• Monotonicity/convexity consequences: since
  • $\partial \log Z_\Lambda/\partial h_i = \langle\sigma_i\rangle_\Lambda$,
  • $\partial^2 \log Z_\Lambda/\partial h_i\partial h_j = \langle\sigma_i\sigma_j\rangle_\Lambda-\langle\sigma_i\rangle_\Lambda\langle\sigma_j\rangle_\Lambda$, nonnegativity of truncated correlations implies that magnetizations and many other observables are monotone in fields and that $\log Z_\Lambda$ is convex in the field variables.

Cross-links and relations

• The pair of inequalities above implies the Griffiths inequalities as a special case (many authors use the names interchangeably for the Ising ferromagnet).
• For increasing observables (not necessarily spin monomials), positivity of correlations can also be obtained from the FKG inequality when the measure is attractive.

Proof idea / significance

One strategy is to encode the Ising measure in an expansion with manifestly nonnegative weights (e.g. high-temperature expansion or random-current representations). Truncated correlations can then be expressed as sums of nonnegative terms, yielding GKS II.

GKS inequalities are central for:

• monotonicity arguments (comparison between fields and couplings),
• correlation bounds and control of response functions (see susceptibility ),
• structural results about phases and analyticity (often combined with the Lee–Yang circle theorem ).