Van den Berg–Kesten inequality

Van den Berg–Kesten inequality

Type	Theorem
Field	Probability theory
Symbolic statement	${\displaystyle \mathbb {P} (A\mathbin {\square } B)\leq \mathbb {P} (A)\mathbb {P} (B)}$
Conjectured by	van den Berg and Kesten
Conjectured in	1985
First proof by	Reimer

In probability theory, the van den Berg–Kesten (BK) inequality or van den Berg–Kesten–Reimer (BKR) inequality states that the probability for two random events to both happen, and at the same time one can find "disjoint certificates" to show that they both happen, is at most the product of their individual probabilities. The special case for two monotone events (the notion as used in the FKG inequality) was first proved by van den Berg and Kesten in 1985, who also conjectured that the inequality holds in general, not requiring monotonicity. Reimer later proved this conjecture.^{: 159 : 44} The inequality is applied to probability spaces with a product structure, such as in percolation problems.^: 829