Parisi's hypercube, Fock-space fluxes, and the microscopics of near-AdS$_2$/near-CFT$_1$ duality

Parisi's hypercube model describes a charged particle hopping on a $d$-dimensional hypercube with disordered background fluxes in the large $d$ limit. It was noted previously [Jia and Verbaarschot, J. High Energy Phys. 11 (2020) 154] that the hypercube model at leading order in $1/d$ has the same spectral density as the double-scaled Sachdev-Ye-Kitaev (DS-SYK) model. In this work we identify the set of observables that have the same correlation functions as the DS-SYK model, demonstrating that the hypercube model is an equally good microscopic model for near-AdS$_2$/near-CFT$_1$ holography. Unlike the SYK model, the hypercube model is not $p$-local. Rather, we note that the shared feature between the two models is that they both have a large amount of disordered but uniform fluxes on their Fock-space graphs, and we propose this is a broader characterization of near-CFT$_1$ microscopics. Moreover, we suggest that the hypercube model can be viewed as the operator growth model of the DS-SYK model. We explain some universality in subleading corrections and relate them to bulk vertices. Finally, we revise a claim made the aforementioned reference about the existence of a spectral gap.