Pivotality versus noise stability for monotone transitive functions

TL;DR

This paper constructs a sequence of monotone, transitive Boolean functions that are both noise stable and volatile, demonstrating complex stability properties and the coexistence of pivotals with high probability.

Contribution

It provides the first example of a monotone, transitive Boolean function sequence that is both noise stable and volatile, with many pivotals.

Findings

Sequence is noise stable with many pivotals.

Sequence is also volatile, showing coexistence of properties.

Provides new insights into stability and volatility in Boolean functions.

Abstract

We construct a noise stable sequence of transitive, monotone increasing Boolean functions $f_n: \{-1,1\}^{k_n} \longrightarrow \{-1,1\}$ which admit many pivotals with high probability. We show that such a sequence is volatile as well, and thus it is also an example of a volatile and noise stable sequence of transitive, monotone functions.