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Thermalization of chaotic quantum many-body systems under unitary timeevolution is related to the growth in complexity of initially simple Heisenbergoperators. Operator growth is a manifestation of information scrambling and canbe diagnosed by out-of-time-order correlators (OTOCs). However, the behavior ofOTOCs of local operators in generic chaotic local Hamiltonians remains poorlyunderstood, with some semiclassical and large N models exhibiting exponentialgrowth of OTOCs and a sharp chaos wavefront and other random circuit modelsshowing a diffusively broadened wavefront. In this paper we propose a unifiedphysical picture for scrambling in chaotic local Hamiltonians. We construct arandom time-dependent Hamiltonian model featuring a large N limit where theOTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation andexhibits exponential growth and a sharp wavefront. We show that quantumfluctuations manifest as noise (distinct from the randomness of the couplingsin the underlying Hamiltonian) in the FKPP equation and that the noise-averagedOTOC exhibits a cross-over to a diffusively broadened wavefront. At small N wedemonstrate that operator growth dynamics, averaged over the random couplings,can be efficiently simulated for all time using matrix product statetechniques. To show that time-dependent randomness is not essential to ourconclusions, we push our previous matrix product operator methods to very largesize and show that data for a time-independent Hamiltonian model are alsoconsistent with a diffusively-broadened wavefront.

Locality, Quantum Fluctuations, and Scrambling