Abstract
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 → ∞, 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.
Highlights
The Sachdev-Ye-Kitaev (SYK) model [1,2,3] is a simple tractable model of quantum chaos and holography
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
In the double scaled limit p ∝ N, the model has a Schwarzian regime but it is related to a q-deformation of random matrix theory [15], and the full large N density of states is given by the q-Gaussian distribution [7, 10, 16, 17]
Summary
The Sachdev-Ye-Kitaev (SYK) model [1,2,3] is a simple tractable model of quantum chaos and holography. The precise RMT ensemble (GOE, GUE, or GSE) depends on the values N mod 8 and p mod 4, which give rise to different particle-hole symmetries, and have been completely classified in [9, 17,18,19,20,21] This level spacing statistics results in a universal RMT contribution to the spectral form factor which is dominant at exponentially late times [17]. Apart from these universal contributions, there are global fluctuations of the spectrum which are less suppressed in the large N limit [10, 17, 23]. We note that this problem can be mapped to the study of the cummulants of q-Brownian motion [15, 41]
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