  This paper is a physicist's review of the major conceptual issues concerning
the problem of spectral universality in quantum systems. Here we present a
unified, graph-based view of all archetypical models of such universality
(billiards, particles in random media, interacting spin or fermion systems). We
find phenomenological relations between the onset of ergodicity
(Gaussian-random delocalization of eigenstates) and the structure of the
appropriate graphs, and we construct a heuristic picture of summing
trajectories on graphs that describes why a generic interacting system should
be ergodic. We also provide an operator-based discussion of quantum chaos and
propose criteria to distinguish bases that can usefully diagnose ergodicity.
The result of this analysis is a rough but systematic outline of how ergodicity
changes across the space of all theories with a given Hilbert space dimension.
As a special case, we study the SYK model and report on the transition from
maximal to partial ergodicity as the disorder strength is decreased.
