I will present a class of models for quantum chaos in a spatially extended many-body system. It consists of a chain of sites with nearest-neighbour coupling under Floquet time evolution. Quantum states at each site span a q-dimensional Hilbert space and the time evolution is specified as a random circuit, whose local gates are random in space but periodic in time (Floquet). I will discuss a diagrammatic formalism useful to average over realisations of the random circuit. This approach leads to exact expressions in the large-q limit and sheds light on the universality of random matrices in many-body quantum spectra and the ubiquitous entanglement growth in out-of-equilibrium dynamics. I will also discuss universal behaviour which goes beyond random matrix theory and the manifestation of ergodicity breaking which can emerge at finite q.
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