  We study the entanglement entropy of a random tensor network (RTN) using
tools from free probability theory. Random tensor networks are simple toy
models that help the understanding of the entanglement behavior of a boundary
region in the ADS/CFT context. One can think of random tensor networks are
specific probabilistic models for tensors having some particular geometry
dictated by a graph (or network) structure. We first introduce our model of
RTN, obtained by contracting maximally entangled states (corresponding to the
edges of the graph) on the tensor product of Gaussian tensors (corresponding to
the vertices of the graph). We study the entanglement spectrum of the resulting
random spectrum along a given bipartition of the local Hilbert spaces. We
provide the limiting eigenvalue distribution of the reduced density operator of
the RTN state, in the limit of large local dimension. The limit value is
described via a maximum flow optimization problem in a new graph corresponding
to the geometry of the RTN and the given bipartition. In the case of
series-parallel graphs, we provide an explicit formula for the limiting
eigenvalue distribution using classical and free multiplicative convolutions.
We discuss the physical implications of our results, allowing us to go beyond
the semiclassical regime without any cut assumption, specifically in terms of
finite corrections to the average entanglement entropy of the RTN.
