FKG inequality

Statement

Let $\Omega=\{-1,+1\}^\Lambda$ with $\Lambda$ finite, equipped with the product partial order $\sigma\le \eta$ iff $\sigma_i\le \eta_i$ for all $i\in\Lambda$.

A function $f:\Omega\to\mathbb{R}$ is increasing if $\sigma\le \eta$ implies $f(\sigma)\le f(\eta)$.

A probability measure $\mu$ on $\Omega$ satisfies the FKG lattice condition (log-supermodularity) if for all $\sigma,\eta\in\Omega$,

where $(\sigma\wedge\eta)_i=\min\{\sigma_i,\eta_i\}$ and $(\sigma\vee\eta)_i=\max\{\sigma_i,\eta_i\}$.

FKG inequality (positive association).
If $\mu$ satisfies the FKG lattice condition, then for all bounded increasing functions $f,g$,

where $\mathbb{E}_\mu[\cdot]$ denotes expectation (see expectation ).

Key hypotheses and conclusions

Hypotheses

• Finite partially ordered configuration space $\Omega=\{-1,+1\}^\Lambda$.
• A probability measure $\mu$ satisfying the FKG lattice condition.
• Observables $f,g$ are bounded and increasing.

Conclusions

• Nonnegative covariance of increasing observables: for increasing $f,g$, $\mathrm{Cov}_\mu(f,g)=\mathbb{E}_\mu[f g]-\mathbb{E}_\mu[f]\mathbb{E}_\mu[g]\ge 0.$
• Monotonicity and comparison: positive association is a key input for stochastic domination and monotone coupling arguments (often used to compare different boundary conditions or external fields in lattice systems).

Connection to lattice Gibbs measures

For the ferromagnetic Ising model (and more generally many attractive lattice systems), the finite-volume Gibbs measure satisfies the FKG lattice condition when the couplings are ferromagnetic ($J_{ij}\ge 0$). In that setting, the FKG inequality yields positivity of correlations for increasing observables, and it is a standard tool for proving existence and ordering of infinite-volume Gibbs states (see infinite-volume Gibbs measures and phase transitions ).

Proof idea / significance

The original proof proceeds by induction on $|\Lambda|$, reducing the inequality to a two-point inequality using conditional measures and the lattice condition. Conceptually, log-supermodularity is a discrete analogue of log-convexity that forces “alignment”: raising some coordinates makes increasing observables tend to increase together.

In statistical mechanics, FKG is one of the main correlation-inequality engines, complementary to the more model-specific GKS inequalities and Griffiths inequalities .