  Thermalization and scrambling are the subject of much recent study from the
perspective of many-body quantum systems with locally bounded Hilbert spaces
(`spin chains'), quantum field theory and holography. We tackle this problem in
1D spin-chains evolving under random local unitary circuits and prove a number
of exact results on the behavior of out-of-time-ordered commutators (OTOCs),
and entanglement growth in this setting. These results follow from the
observation that the spreading of operators in random circuits is described by
a `hydrodynamical' equation of motion, despite the fact that random unitary
circuits do not have locally conserved quantities (e.g., no conserved energy).
In this hydrodynamic picture quantum information travels in a front with a
`butterfly velocity' $v_{\text{B}}$ that is smaller than the light cone
velocity of the system, while the front itself broadens diffusively in time.
The OTOC increases sharply after the arrival of the light cone, but we do
\emph{not} observe a prolonged exponential regime of the form $\sim
e^{\lambda_\text{L}(t-x/v)}$ for a fixed Lyapunov exponent $\lambda_\text{L}$.
We find that the diffusive broadening of the front has important consequences
for entanglement growth, leading to an entanglement velocity that can be
significantly smaller than the butterfly velocity. We conjecture that the
hydrodynamical description applies to more generic ergodic systems and support
this by verifying numerically that the diffusive broadening of the operator
wavefront also holds in a more traditional non-random Floquet spin-chain. We
also compare our results to Clifford circuits, which have less rich
hydrodynamics and consequently trivial OTOC behavior, but which can
nevertheless exhibit linear entanglement growth and thermalization.
