  We study the finite-temperature scrambling behavior of a quantum system
described by a Hamiltonian chosen from a random matrix ensemble. This
effectively (0+1)-dimensional model admits an exact calculation of various
ensemble-averaged out-of-time-ordered correlation functions in the large-$N$
limit, where $N$ is the Hilbert space dimension. For a Hamiltonian drawn from
the Gaussian unitary ensemble, we calculate the ensemble averaged OTOC at all
temperatures. In addition to an early time quadratic growth of the averaged
out-of-time-ordered commutator (OTOC), we determine that the OTOC saturates to
its asymptotic value as a power-law in time, with an exponent that depends both
on temperature, and on one of four classes of operators appearing in the
correlation function, that naturally emerge from this calculation.
Out-of-time-ordered correlation functions of operators that are distributed
around the thermal circle take a time $t_{s}\sim \beta$ to decay at low
temperatures. We generalize these exact results, by demonstrating that
out-of-time-ordered correlation functions averaged over any ensemble of
Hamiltonians that are invariant under unitary conjugation $H \rightarrow {U} H
{U}^{\dagger}$, exhibit power-law decay to an asymptotic value. We argue that
this late-time behavior is a generic feature of unitary dynamics with energy
conservation. Finally, by comparing the OTOC with a commutator-anticommutator
correlation function, we examine whether there is a time window over which a
typical Hamiltonian behaves as a "coherent scrambler" in the language of Ref.
\cite{Kitaev_IAS, Kitaev_Suh}.

Strongly Correlated Electrons;Statistical Mechanics;High Energy Physics - Theory;Quantum Physics