Holography has nowadays become one of the most popular research fields in quantum gravity and Random Tensor Networks (RTNs) have emerged as a minimal model for understanding the entanglement structure of holographic quantum systems. In this talk, I will give a review on the connections between RTNs and holography. I will start with an overview of topics in quantum many-body physics and move gradually towards some properties of RTNs (such as Ryu-Takayanagi formula) that let RTNs good toy models for AdS/CFT correspondence. I will explain how Schur-Weyl duality and Weingarten calculus turns the computations of entanglement measures of RTNs into graph combinatorial problems and, as examples, I will briefly introduce Marchenko-Pastur law and Ryu-Takayanagi formula for RTNs, which are two of the most well-known results in this field, together with some helpful combinatorial techniques that are used to obtain these results. Finally, I will give a quick look at our results which connects the asymptotic empirical eigenvalue distribution of RTNs with free probability theory..

https://indico.math.cnrs.fr/event/15253/