Appears in collection : Inhomogeneous Random Systems

The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems, which can be understood by analogy with Random Matrix Theory. In fact within this ansatz these matrix elements are modelled as pseudo random variables. In this talk I will first describe how a complete description of high order thermal correlation functions requires correlations between matrix elements. I will then argue how, by analogy with Random Matrix Theory and "typicality arguments", one can assume a certain hierarchy between these correlations. The talk is based on our recent work in which we unveil a relation between ETH and Free Probability.