We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of AND, OR, and NOT gates.
Our hierarchy theorem says that for every d≥2, there is an explicit
n-variable Boolean function f, computed by a linear-size depth-d formula,
which is such that any depth-(d−1) circuit that agrees with f on (1/2+on(1)) fraction of all inputs must have size exp(nΩ(1/d)). This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions