  Thermalization of chaotic quantum many-body systems under unitary time
evolution is related to the growth in complexity of initially simple Heisenberg
operators. Operator growth is a manifestation of information scrambling and can
be diagnosed by out-of-time-order correlators (OTOCs). However, the behavior of
OTOCs of local operators in generic chaotic local Hamiltonians remains poorly
understood, with some semiclassical and large N models exhibiting exponential
growth of OTOCs and a sharp chaos wavefront and other random circuit models
showing a diffusively broadened wavefront. In this paper we propose a unified
physical picture for scrambling in chaotic local Hamiltonians. We construct a
random time-dependent Hamiltonian model featuring a large N limit where the
OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and
exhibits exponential growth and a sharp wavefront. We show that quantum
fluctuations manifest as noise (distinct from the randomness of the couplings
in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged
OTOC exhibits a cross-over to a diffusively broadened wavefront. At small N we
demonstrate that operator growth dynamics, averaged over the random couplings,
can be efficiently simulated for all time using matrix product state
techniques. To show that time-dependent randomness is not essential to our
conclusions, we push our previous matrix product operator methods to very large
size and show that data for a time-independent Hamiltonian model are also
consistent with a diffusively-broadened wavefront.
