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 tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.