Partially Observed Boolean Sequences and Noise Sensitivity

Let H denote a collection of subsets of {1, 2, . . . , n}, and assign independent random variables uniformly distributed over [0, 1] to the n elements. Declare an element p-present if its corresponding value is at most p. In this paper, we quantify how much the observation of the r-present (r > p) set of elements affects the probability that the set of p-present elements is contained in H. In the context of percolation, we find that this question is closely linked to the near-critical regime. As a consequence, we show that for every r > 1/2, bond percolation on the subgraph of the square lattice given by the set of r-present edges is almost surely noise sensitive at criticality, thus generalizing a result due to Benjamini, Kalai and Schramm.