Decoupling weakly dependent events

In this note we discuss a probabilistic statement which conceptualizes arguments from recent papers [ST1, ST2]. Its framework appears to be sufficiently general to permit applications also beyond the original context. The setting is as follows: we have a sequence of events the probability of each of which is rather small and we would like to deduce that the probability of their intersection is very small. This is of course straightforward if the events are independent; our approach allows to obtain comparable upper bounds on probabilities when the dependence is not “too strong.” The precise formulation of the statement is somewhat technical, but its gist can be described as follows. Suppose that our probability space is a product space and that our events/sets are defined in terms of independent coordinates in a “local” way, i.e., while membership in each of the sets may depend on many or even all coordinates, it may be verified by checking a series of conditions each of which involves just a few coordinates (for example, to verify whether a sequence of vectors is orthogonal it is enough to look at just two elements of that sequence at a time). Then the probability of the intersection of these sets can “almost” be estimated as if they were independent. More precisely, the upper bound is not a product of their probabilities, but a homogeneous polynomial in the probabilities, the degree of which is high while the number of terms is controlled. Problems similar in spirit if not in details were considered by many authors in probabilistic combinatorics and theoretical computer science. See, for example, [JR1, JR2] and their references; particularly [J2] seems to exhibit many formal similarities to our setting. (We thank M. Krivelevich for helping us navigate the combinatorics literature.) Let us note, however, that while the results cited above have, as a rule, a “large deviation feel,” applications of our scheme go in the direction of “small ball” estimates, and