  We study many-body quantum dynamics using Floquet quantum circuits in one
space dimension as simple examples of systems with local interactions that
support ergodic phases. Physical properties can be expressed in terms of
multiple sums over Feynman histories, which for these models are paths or
many-body orbits in Fock space. A natural simplification of such sums is the
diagonal approximation, where the only terms that are retained are ones in
which each path is paired with a partner that carries the complex conjugate
weight. We identify the regime in which the diagonal approximation holds, and
the nature of the leading corrections to it. We focus on the behaviour of the
spectral form factor (SFF) and of matrix elements of local operators, averaged
over an ensemble of random circuits, making comparisons with the predictions of
random matrix theory (RMT) and the eigenstate thermalisation hypothesis (ETH).
We show that properties are dominated at long times by contributions to orbit
sums in which each orbit is paired locally with a conjugate, as in the diagonal
approximation, but that in large systems these contributions consist of many
spatial domains, with distinct local pairings in neighbouring domains. The
existence of these domains is reflected in deviations of the SFF from RMT
predictions, and of matrix element correlations from ETH predictions;
deviations of both kinds diverge with system size. We demonstrate that our
physical picture of orbit-pairing domains has a precise correspondence in the
spectral properties of a transfer matrix that acts in the space direction to
generate the ensemble-averaged SFF. In addition, we find that domains of a
second type control non-Gaussian fluctuations of the SFF. These domains are
separated by walls which are related to the entanglement membrane, known to
characterise the scrambling of quantum information.
