  We consider multi-energy level distributions in the SYK model, and in
particular, the role of global fluctuations in the density of states of the SYK
model. The connected contributions to the moments of the density of states go
to zero as $N \to \infty$, however, they are much larger than the standard RMT
correlations. We provide a diagrammatic description of the leading behavior of
these connected moments, showing that the dominant diagrams are given by 1PI
cactus graphs, and derive a vector model of the couplings which reproduces
these results. We generalize these results to the first subleading corrections,
and to fluctuations of correlation functions. In either case, the new set of
correlations between traces (i.e. between boundaries) are not associated with,
and are much larger than, the ones given by topological wormholes. The
connected contributions that we discuss are the beginning of an infinite series
of terms, associated with more and more information about the ensemble of
couplings, which hints towards the dual of a single realization. In particular,
we suggest that incorporating them in the gravity description requires the
introduction of new, lighter and lighter, fields in the bulk with fluctuating
boundary couplings.
