Bound on Eigenstate Thermalization from Transport

We show that macroscopic thermalization and transport impose constraints on matrix elements entering the eigenstate thermalization hypothesis (ETH) ansatz and require them to be correlated. It is often assumed that the ETH reduces to random matrix theory (RMT) below the Thouless energy scale. We show that this conventional picture is not self-consistent. We prove that the energy scale at which the RMT behavior emerges has to be parametrically smaller than the inverse timescale of the slowest thermalization mode coupled to the operator of interest. We argue that the timescale marking the onset of the RMT behavior is the same timescale at which the hydrodynamic description of transport breaks down.