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 \cite{js2006}, 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.