Functional van den Berg-Kesten-Reimer Inequalities and their Duals, with Applications

The BKR inequality conjectured by van den Berg and Kesten in [11], and proved by Reimer in [8], states that for A and B events on S, a finite product of finite sets Si, i = 1, . . . , n, and P any product measure on S, P (A B) ≤ P (A)P (B), where the set A B consists of the elementary events which lie in both A and B for ‘disjoint reasons.’ Precisely, with n := {1, . . . , n} and K ⊂ n, for x ∈ S letting [x]K = {y ∈ S : yi = xi, i ∈ K}, the set A B consists of all x ∈ S for which there exist disjoint subsets K and L of n for which [x]K ⊂ A and [x]L ⊂ B. The BKR inequality is extended to the following functional version on a general finite product measure space (S, S) with product probability measure P , E  max K∩L=∅ K⊂n,L⊂n f K (X)g L (X)  ≤ E {f(X)} E {g(X)} , where f and g are non-negative measurable functions, f K (x) = ess infy∈[x]K f(y) and g L (x) = ess infy∈[x]L g(y). The original BKR inequality is recovered by taking f(x) = 1A(x) and g(x) = 1B(x), and applying the fact that in general 1A B ≤ maxK∩L=∅ fK(x)gL(x). Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth [6], are also considered. Applications include order statistics, assignment problems, and paths in random graphs.