  Random tensor networks are a powerful toy model for understanding the
entanglement structure of holographic quantum gravity. However, unlike
holographic quantum gravity, their entanglement spectra are flat. It has
therefore been argued that a better model consists of random tensor networks
with link states that are not maximally entangled, i.e., have nontrivial
spectra. In this work, we initiate a systematic study of the entanglement
properties of these networks. We employ tools from free probability, random
matrix theory, and one-shot quantum information theory to study random tensor
networks with bounded and unbounded variation in link spectra, and in cases
where a subsystem has one or multiple minimal cuts. If the link states have
bounded spectral variation, the limiting entanglement spectrum of a subsystem
with two minimal cuts can be expressed as a free product of the entanglement
spectra of each cut, along with a Marchenko-Pastur distribution. For a class of
states with unbounded spectral variation, analogous to semiclassical states in
quantum gravity, we relate the limiting entanglement spectrum of a subsystem
with two minimal cuts to the distribution of the minimal entanglement across
the two cuts. In doing so, we draw connections to previous work on split
transfer protocols, entanglement negativity in random tensor networks, and
Euclidean path integrals in quantum gravity.
