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Random free fermions: An analytical example of eigenstate thermalization

Having analytical instances of the Eigenstate Thermalization Hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generically a hard task, due to the belief that non-linear interactions are basic ingredients of the thermalization mechanism. In this article we proof that random gaussian free fermions satisfy ETH in the multiparticle sector, by analytically computing the correlations and entanglement entropies of the theory. With the explicit construction at hand, we finally comment on the differences between fully random Hamiltonians and random Gaussian systems, and on the connection between chaotic energy spectra and ETH.