Abstract:
We consider the hypercubic model originally introduced by Parisi as a model for an array
of Josephson junctions. This is a model where the Hamiltonian is given by the discretized
Laplacian on a d-dimensional hypercube with U(1) gauge fields on the links but with a magnetic
flux of constant magnitude and random orientation through all faces. The spectral flow of
this model resembles that of the Maldacena-Qi model, and at zero flux, it coincides with the
Maldacena-Qi model at infinite coupling. It also has a ground state that is separated from the
rest of the spectrum by a gap, and a discrete symmetry in additional to the bipartite chiral
symmetry. As is the case for the SYK model, the spectral density of this model is given by
density function of the Q-Hermite polynomials. We analyze the spectral correlations of this model and find that the spectral form factor and the number variance are in the universality
class of the Gaussian Unitary Ensemble, while the eigenvalues near zero are described by the
chiral Gaussian Unitary Ensemble.
