the BK inequality
/ BK = van den Berg-Kesten /
The Harris-FKG inequality says increasing events are positively correlated — they help each other. The BK inequality captures the opposite phenomenon for DISJOINT occurrence: if two increasing events must happen 'using separate resources' (disjoint sets of open edges), then they hinder rather than help, and the probability of their disjoint occurrence is at most the product of the two probabilities. It is the natural upper-bound counterpart to FKG's lower bound, and the key tool for controlling events that need many independent open paths.
Make 'disjoint occurrence' precise. For a configuration omega and events A, B, say A and B occur disjointly, written A box B (the box operator), if there exist two disjoint sets of edges such that the values of omega on the first set already guarantee A, and on the second already guarantee B — each event is witnessed by its own private edges. Van den Berg and Kesten (1985) proved, for product measure and increasing events, that P(A box B) <= P(A) P(B). Reimer (1996) later proved the same bound P(A box B) <= P(A) P(B) for ALL events, not just increasing ones (the BKR inequality), a much deeper result. The intuition: requiring disjoint witnesses is a real constraint — the two events cannot share the helpful edges they would under FKG — so disjointness costs you, and the cost is exactly the independence-like product bound.
BK is indispensable whenever a configuration must contain several edge-disjoint structures, for instance two disjoint open paths between far-apart points, or k disjoint crossings. A classic use bounds the probability that a vertex lies on many disjoint open circuits, or controls the number of disjoint open clusters connecting a region to its complement, which appears in the proof of exponential decay and in the Burton-Keane uniqueness argument. The honest caveat is the word 'disjoint': BK bounds P(A box B), not P(A intersect B). The ordinary intersection of two increasing events goes the OTHER way (FKG gives a lower bound). Confusing intersection with disjoint occurrence is the standard error; the operator box is doing essential work.
Let A_k = {there exist k edge-disjoint open paths from the origin to distance n}. Then by iterating BK, P(A_k) <= P(one open path to distance n)^k, which decays geometrically in k. This kind of bound is exactly what shows a subcritical cluster cannot have too many independent long arms.
Disjoint open structures cost a product of probabilities — the upper-bound mirror of FKG.
BK bounds disjoint occurrence A box B, NOT the intersection A intersect B; for increasing events the intersection obeys the opposite (FKG lower) bound. Reimer's theorem removed the monotonicity hypothesis, but the disjoint-witness structure is still essential.