the Harris-FKG inequality
/ FKG = Fortuin-Kasteleyn-Ginibre /
Percolation events are correlated in a particular, helpful direction: making more edges open helps every monotone-increasing connectivity event at once, so two such events tend to occur together. The Harris-FKG inequality is the precise statement of this positive association, and it is the single most-used tool in the subject. It is what lets us say things like 'the probability that the origin connects far in two different directions is at least the product of the two separate probabilities' — a correlation bound that, remarkably, always points the same way.
Call an event A increasing if adding open edges can only keep you in A (formally, A is closed under flipping closed edges to open); examples are 'x is connected to y', 'there is an open crossing of a box', '|C| >= k'. Harris (1960) proved, and Fortuin, Kasteleyn and Ginibre (1971) generalized, that for product measure (and more generally any measure whose density satisfies the FKG lattice condition / is log-supermodular), any two increasing events A and B satisfy P(A intersect B) >= P(A) P(B). Equivalently any two increasing functions f, g of the configuration are positively correlated: E[f g] >= E[f] E[g]. The clean way to see it for product measure is by induction on coordinates plus Chebyshev's sum inequality: conditioned on all but one edge, two increasing functions of that single Bernoulli variable are positively correlated, and one bootstraps up. By flipping signs, two DECREASING events are also positively correlated, while an increasing and a decreasing event are negatively correlated.
FKG is the workhorse for lower bounds: square-root tricks (if a box is crossed in some direction, by symmetry and FKG each specific crossing has probability at least 1 minus a root), gluing of crossings into a circuit, and the construction of infinite clusters from local connections all rely on it. It also drives the random-cluster model, where the FK measure is built precisely to satisfy the FKG lattice condition. The crucial caveat is monotonicity: FKG compares only events of the same monotone type. For an increasing and a decreasing event you get the opposite (negative) correlation, and for genuinely non-monotone events FKG says nothing — that gap is exactly where the BK inequality and more delicate tools take over.
Let A = {origin connects to the right side of a box} and B = {origin connects to the top}. Both are increasing, so FKG gives P(A and B) >= P(A)P(B): connecting up and connecting right are positively correlated, because the same open edges help both. This is the engine behind the 'square-root trick' used to lower-bound crossing probabilities.
Two increasing connectivity events helped by the same open edges are positively correlated.
FKG only relates events of the SAME monotonicity. An increasing event and a decreasing event are negatively correlated, and for non-monotone events FKG is silent — a common mistake is to apply it to 'A occurs and B does not'.