  We analyze the spectral properties of a $d$-dimensional HyperCubic (HC)
lattice model originally introduced by Parisi. The U(1) gauge links of this
model give rise to a magnetic flux of constant magnitude $\phi$ but random
orientation through the faces of the hypercube. The HC model, which also can be
written as a model of $2d$ interacting Majorana fermions, has a spectral flow
that is reminiscent of the Maldacena-Qi (MQ) model, and its spectrum at
$\phi=0$, actually coincides with the coupling term of the MQ model. As was
already shown by Parisi, at leading order in $1/d$ , the spectral density of
this model is given by the density function of the Q-Hermite polynomials, which
is also the spectral density of the double-scaled Sachdev-Ye-Kitaev model.
Parisi demonstrated this by mapping the moments of the HC model to $Q$-weighted
sums on chord diagrams. We point out that the subleading moments of the HC
model can also be mapped to weighted sums on chord diagrams, in a manner that
descends from the leading moments. The HC model has a magnetic inversion
symmetry that depends on both the magnitude and the orientation of the magnetic
flux through the faces of the hypercube. The spectrum for fixed quantum number
of this symmetry exhibits a transition from regular spectra at $\phi=0$ to
chaotic spectra with spectral statistics given by the Gaussian Unitary
Ensembles (GUE) for larger values of $\phi$. For small magnetic flux, the
ground state is gapped and is close to a Thermofield Double (TFD) state.

High Energy Physics - Theory;Disordered Systems and Neural Networks;Statistical Mechanics