  We argue that in a large class of disordered quantum many-body systems, the
late time dynamics of time-dependent correlation functions is captured by
random matrix theory, specifically the energy eigenvalue statistics of the
corresponding ensemble of disordered Hamiltonians. We find that late time
correlation functions approximately factorize into a time-dependent piece,
which only depends on spectral statistics of the Hamiltonian ensemble, and a
time-independent piece, which only depends on the data of the constituent
operators of the correlation function. We call this phenomenon "spectral
decoupling," which signifies a dynamical onset of random matrix theory in
correlation functions. A key diagnostic of spectral decoupling is
$k$-invariance, which we refine and study in detail. Particular emphasis is
placed on the role of symmetries, and connections between $k$-invariance,
scrambling, and OTOCs. Disordered Pauli spin systems, as well as the SYK model
and its variants, provide a rich source of disordered quantum many-body systems
with varied symmetries, and we study $k$-invariance in these models with a
combination of analytics and numerics.
