A Max-Flow Approach to Random Tensor Networks

The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. These can be regarded as specific probabilistic models for tensors with particular geometry dictated by a graph (or network) structure. First, we introduce a 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). The entanglement spectrum of the resulting random state is analyzed along a given bipartition of the local Hilbert spaces. The limiting eigenvalue distribution of the reduced density operator of the RTN state is provided in the limit of large local dimension. This limiting value is described through 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, an explicit formula for the limiting eigenvalue distribution is provided using classical and free multiplicative convolutions. The physical implications of these results are discussed, allowing the analysis to move beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN.