  The commutator between operators at different space and time has been a
diagnostic for locality of unitary evolution. Most existing results are either
for specific tractable (random) Hamiltonians(Out-of-Time-Order-Correlators
calculations), or for worse case Hamiltonians (Lieb-Robinson-like bounds or
OTOC bounds). In this work, we study commutators in typical Hamiltonians. Draw
a sample from any zero-mean bounded independent random Hamiltonian ensemble,
time-independent or Brownian, we formulate concentration bounds in the spectral
norm and for the OTOC with arbitrary non-random state. Our bounds hold with
high probability and scale with the sum of interactions squared. Our Brownian
bounds are compatible with the Brownian limit while deterministic operator
growth bounds must diverge. We evaluate this general framework on short-ranged,
1d power-law interacting, and SYK-like k-local systems and the results match
existing lower bounds and conjectures. Our main probabilistic argument employs
a robust matrix martingale technique called uniform smoothness and may be
applicable in other settings.
