where A ¯ ¯ = A ¯ ¯ ( E α α ) and A 2 ¯ ¯ = A 2 ¯ ¯ ( E α α , E β β ) are smooth functions of energy, D = e s V is the many-body Hilbert space dimension, and R α α β β is a random variable with zero mean and unit variance. Conversely if a quantum many-body system satisfies the ETH, the matrix representation of any local operator in the energy eigen basis is expected to follow the above ansatz.