Partially Observed Boolean Sequences and Noise Sensitivity

Abstract

Let ${\mathcal H}$ 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 ${\mathcal H}$. 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.