Criticality of $AC^0$ formulae

Rossman [In \emph{Proc.\ $34$th Comput.\ Complexity Conf.}, 2019] introduced the notion of \emph{criticality}. The criticality of a Boolean function $f \colon \zo^n \to \zo$ is the minimum $\lambda \geq 1$ such that for all positive integers $t$, \[ \Pr_{\brho \sim \calR_p}\left[\DT_{\depth}(f|_{\brho}) \geq t\right] \leq (p\lambda)^t. \] \hastad's celebrated switching lemma shows that the criticality of any $k$-DNF is at most $O(k)$. Subsequent improvements to correlation bounds of $\AC^0$-circuits against parity showed that the criticality of any $\AC^0$-\emph{circuit} of size $S$ and depth $d+1$ is at most $O(\log S)^d$ and any \emph{regular} $\AC^0$-\emph{formula} of size $S$ and depth $d+1$ is at most $O(\frac1d \cdot \log S)^d$. We strengthen these results by showing that the criticality of \emph{any} $\AC^0$-formula (not necessarily regular) of size $S$ and depth $d+1$ is at most $O(\frac{\log S}{d})^d$, resolving a conjecture due to Rossman. This result also implies Rossman's optimal lower bound on the size of any depth-$d$ $\AC^0$-formula computing parity [Comput.\ Complexity, 27(2):209--223, 2018.]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved \#SAT algorithm for $\AC^0$-formulae.