  Quantum thermalization is well understood via the Eigenstate Thermalization
Hypothesis (ETH). The general form of ETH, describing all the relevant
correlations of matrix elements, may be derived on the basis of a `typicality'
argument of invariance with respect to local rotations involving nearby energy
levels. In this work, we uncover the close relation between this perspective on
ETH and Free Probability theory, as applied to a thermal ensemble or an energy
shell. This mathematical framework allows one to reduce in a straightforward
way higher-order correlation functions to a decomposition given by minimal
blocks, identified as free cumulants, for which we give an explicit formula.
This perspective naturally incorporates the consistency property that local
functions of ETH operators also satisfy ETH. The present results uncover a
direct connection between the Eigenstate Thermalization Hypothesis and the
structure of Free Probability, widening considerably the latter's scope and
highlighting its relevance to quantum thermalization.
