  We study thermalization of weakly nonintegrable nonlinear unitary lattice
dynamics. We identify two distinct thermalization regimes close to the
integrable limits of either linear dynamics or disconnected lattice dynamics.
For weak nonlinearity the almost conserved actions correspond to extended
observables which are coupled into a long-range network. For weakly connected
lattices the corresponding local observables are coupled into a short-range
network. We compute the evolution of the variance $\sigma^2(T)$ of finite time
average distributions for extended and local observables. We extract the
ergodization time scale $T_E$ which marks the onset of thermalization, and
determine the type of network through the subsequent decay of $\sigma^2(T)$. We
use the complementary analysis of Lyapunov spectra [M. Malishava and S. Flach,
Phys. Rev. Lett. 128, 134102 (2022)] and compare the Lyapunov time
$T_{\Lambda}$ with $T_E$. We characterize the spatial properties of the tangent
vector and arrive at a complete classification picture of weakly nonintegrable
macroscopic thermalization dynamics.
