A tame sequence of transitive Boolean functions

Abstract

Given a sequence of Boolean functions $ (f_{n})_{n \geq 1} $, $ f_{n} \colon \{ 0,1 \}^{n} \to \{ 0,1 \}$, and a sequence $ (X^{(n)})_{n\geq 1} $ of continuous time $ p_{n} $-biased random walks $ X^{(n)} = (X_{t}^{(n)})_{t \geq 0}$ on $ \{ 0,1 \}^{n} $, let $ C_{n} $ be the (random) number of times in $ (0,1) $ at which the process $ (f_{n}(X_{t}))_{t \geq 0} $ changes its value. In [7], the authors conjectured that if $ (f_{n})_{n \geq 1} $ is non-degenerate, transitive and satisfies $ \lim _{n \to \infty } \mathbb {E}[C_{n}] = \infty $, then $ (C_{n})_{n \geq 1} $ is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.