Concentration of Measure Phenomena in Random Tensor Networks and the Geometric Origin of the Ryu-Takayanagi Formula

The holographic principle, particularly as realized in the AdS/CFT correspondence, posits a profound connection between quantum entanglement and spacetime geometry. A central pillar of this connection is the Ryu-Takayanagi (RT) formula, which equates the entanglement entropy of a boundary subregion to the area of a minimal surface in the bulk. This work provides a rigorous mathematical framework for understanding the emergence of this geometric formula from the statistical properties of complex quantum systems, modeled by random tensor networks. We employ powerful tools from high-dimensional probability, specifically the phenomenon of concentration of measure, to analyze the entanglement structure of quantum states defined by networks of random tensors. Our main result demonstrates that for a large class of graphs, the entanglement entropy of a subregion concentrates sharply around a value proportional to the minimal cut in the graph that separates the subregion from its complement. This minimal cut serves as a discrete analogue of the minimal surface area. We establish a direct link between the spectral properties of reduced density matrices and the combinatorics of the underlying graph, showing that the generic entanglement structure of a random tensor network state is dictated by its connectivity. This provides a first-principles derivation of an area law and a concrete realization of the RT formula within a mathematically controllable setting, suggesting that holographic geometry is a generic feature of highly entangled quantum systems.