In the talk, we discuss the probability of Boolean functions with small max influence to become constant under random restrictions. Let f be a Boolean function such that the variance of f is $\Omega(1)$ and all its individual influences are bounded by $\tau$. We show that when restricting all but a $\tilde{\Omega}((\log1/\tau)^{-1})$ fraction of the coordinates, the restricted function remains nonconstant with overwhelming probability. This bound is essentially optimal, as witnessed by the tribes function $AND_{n/C\log n} \circ OR_{C\log n}$.
We extend it to an anti-concentration result, showing that the restricted function has nontrivial variance with probability 1-o(1). This gives a sharp version of the ``it ain't over till it's over'' theorem due to Mossel, O'Donnell, and Oleszkiewicz.

Joint work with Avi Wigderson and Ronen Eldan


Video Recording