A BKR operation for events occurring for disjoint reasons with high probability

Abstract

Given events $A$ and $B$ on a product space $S=\prod_{i=1}^n S_i$, the set $A \Box B$ consists of all vectors ${\bf x}=(x_1,\ldots,x_n) \in S$ for which there exist disjoint coordinate subsets $K$ and $L$ of $\{1,\ldots,n\}$ such that given the coordinates $x_i, i \in K$ one has that ${\bf x} \in A$ regardless of the values of ${\bf x}$ on the remaining coordinates, and likewise that ${\bf x} \in B$ given the coordinates {$x_j, j \in L$}. For a finite product of discrete spaces endowed with a product measure, the BKR inequality $$ P(A \Box B) \le P(A)P(B) \quad (1) $$ was conjectured by van den Berg and Kesten [3] and proved by Reimer [13]. In [7] inequality (1) was extended to general product probability spaces, replacing $A \Box B$ by the set $A \Box_{11} B$ consisting of those outcomes ${\bf x}$ which only assure with probability one that ${\bf x} \in A$ and ${\bf x} \in B$ based only on the revealed coordinates in $K$ and $L$ as above. A strengthening of the original BKR inequality (1) results, due to the fact that $A \Box B \subseteq A \Box_{11} B$. In particular, it may be the case that $A \Box B$ is empty, while $A \Box_{11} B$ is not. We propose the further extension $A \Box_{st} B$ depending on probability thresholds $s$ and $t$, where $A \Box_{11} B$ is the special case where both $s$ and $t$ take the value one. The outcomes ${\bf x}$ in $A \Box_{st} B$ are those for which disjoint sets of coordinates $K$ and $L$ exist such that given the values of $\bf x$ on the revealed set of coordinates $K$, the probability that $A$ occurs is at least $s$, and given the coordinates of $\bf x$ in $L$, the probability of $B$ is at least $t$. We provide simple examples that illustrate the utility of these extensions.