  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 this
conventional picture is not self-consistent. We prove that 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 hydrodynamic description of transport breaks down.
