  We present progress on the problem of asymptotically describing the adjacency
eigenvalues of random and complete uniform hypergraphs. There is a natural
conjecture arising from analogy with random matrix theory that connects these
spectra to that of the all-ones hypermatrix. Several of the ingredients along a
possible path to this conjecture are established, and may be of independent
interest in spectral hypergraph/hypermatrix theory. In particular, we provide a
bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show
that the spectrum of the complete \(k\)-uniform hypergraph for \(k=2,3\) is
close to that of an appropriately scaled all-ones hypermatrix.
