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How does the matrix of a typical operator look in the basis of eigenstates of a chaotic Hamiltonian?

The Eigenstate Thermalization Hypothesis (ETH), developed by Berry, Deutsch, and especially Srednicki in the 1990s, predicts the level of randomness in its matrix elements. However, higher-order correlations between different matrix elements were not considered at the time. A prediction for these correlations became necessary when higher-order expectations emerged as important objects of study.

Introducing a “full” version of ETH, incorporating all possible correlations, led to a proliferation of terms—until the bookkeeping framework of Free Probability came to the rescue, re-establishing simplicity.