  We extend random matrix theory to consider randomly interacting spin systems
with spatial locality. We develop several methods by which arbitrary
correlators may be systematically evaluated in a limit where the local Hilbert
space dimension $N$ is large. First, the correlators are given by sums over
'stacked' planar diagrams which are completely determined by the spectra of the
individual interactions and a dependency graph encoding the locality in the
system. We then introduce 'heap freeness' as a generalization of free
independence, leading to a second practical method to evaluate the correlators.
Finally, we generalize the cumulant expansion to a sum over 'dependency
partitions', providing the third and most succinct of our methods. Our results
provide tools to study dynamics and correlations within extended quantum
many-body systems which conserve energy. We further apply the formalism to show
that quantum satisfiability at large-$N$ is determined by the evaluation of the
independence polynomial on a wide class of graphs.
