Partially observed Boolean sequences and noise sensitivity

TL;DR

This paper investigates how observing a larger set of elements influences the probability of a certain subset property in Boolean sequences, revealing connections to near-critical percolation and noise sensitivity.

Contribution

It introduces a framework to quantify the impact of partial observations on Boolean set properties and links this to noise sensitivity in percolation models.

Findings

For every r > 1/2, bond percolation on the r-present edges is almost surely noise sensitive at criticality.

The work generalizes previous results by Benjamini, Kalai, and Schramm.

Establishes a connection between partial observations and near-critical regimes in percolation.

Abstract

Let $\mathcal{H}$ denote a collection of subsets of $\{1,2,\ldots,n\}$, 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.