FKG inequality

Remarks

When $\mu$ is a probability measure, the inequality reads $\mathbb{E}[f g] \ge \mathbb{E}[f] \, \mathbb{E}[g]$: increasing functions are positively correlated. It follows from the Ahlswede–Daykin inequality. Adding a constant to $f$ or to $g$ does not change the difference of the two sides, so one may assume $f, g \ge 0$. Then the four functions $f \mu$, $g \mu$, $f g \mu$ and $\mu$ satisfy $f(x) \mu(x) \, g(y) \mu(y) \le f(x \vee y) g(x \vee y) \mu(x \vee y) \, \mu(x \wedge y)$, by monotonicity and log-supermodularity, and Ahlswede–Daykin with both sets equal to $L$ gives the result.

Example: a product measure on the lattice of subsets of $\set{1, \ldots, n}$ satisfies the hypothesis with equality, and the theorem becomes the Harris inequality. If $f$ is increasing and $g$ is decreasing, the inequality reverses. Log-supermodularity cannot be dropped. On $\set{0, 1}^2$, the measure giving mass $1/2$ to each of $(1, 0)$ and $(0, 1)$ is not log-supermodular. For it, the increasing indicators of $\set{x_1 = 1}$ and $\set{x_2 = 1}$ are negatively correlated.

Going deeper

For a student

The inequality was proved by Fortuin, Kasteleyn and Ginibre for models of statistical mechanics such as the Ising model with ferromagnetic interactions, whose Gibbs measures are log-supermodular. In combinatorics and probability it is used to compare probabilities of monotone events in random graphs and percolation. The Harris inequality is its product-measure case, and the Holley inequality is a two-measure generalization. The XYZ inequality for linear extensions of posets is proved with it.