We consider the global fluctuations in the density of states of the SYK model. The connected contributions to the moments are much larger than the standard RMT correlations. We provide a diagrammatic description of the leading behavior of these connected moments in terms of 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 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.