  We study the scrambling of quantum information in local random unitary
circuits by focusing on the tripartite information proposed by Hosur et al. We
provide exact results for the averaged R\'enyi-$2$ tripartite information in
two cases: (i) the local gates are Haar random and (ii) the local gates are
dual-unitary and randomly sampled from a single-site Haar-invariant measure. We
show that the latter case defines a one-parameter family of circuits, and prove
that for a "maximally chaotic" subset of this family quantum information is
scrambled faster than in the Haar-random case. Our approach is based on a
standard mapping onto an averaged folded tensor network, that can be studied by
means of appropriate recurrence relations. By means of the same method, we also
revisit the computation of out-of-time-ordered correlation functions,
re-deriving known formulae for Haar-random unitary circuits, and presenting an
exact result for maximally chaotic random dual-unitary gates.
