  We study a precise and computationally tractable notion of operator
complexity in holographic quantum theories, including the ensemble dual of
Jackiw-Teitelboim gravity and two-dimensional holographic conformal field
theories. This is a refined, "microcanonical" version of K-complexity that
applies to theories with infinite or continuous spectra (including quantum
field theories), and in the holographic theories we study exhibits exponential
growth for a scrambling time, followed by linear growth until saturation at a
time exponential in the entropy $\unicode{x2014}$a behavior that is
characteristic of chaos. We show that the linear growth regime implies a
universal random matrix description of the operator dynamics after scrambling.
Our main tool for establishing this connection is a "complexity renormalization
group" framework we develop that allows us to study the effective operator
dynamics for different timescales by "integrating out" large K-complexities. In
the dual gravity setting, we comment on the empirical match between our version
of K-complexity and the maximal volume proposal, and speculate on a connection
between the universal random matrix theory dynamics of operator growth after
scrambling and the spatial translation symmetry of smooth black hole interiors.
