Entanglement Entropy Evolution and Random Matrices

We discuss the time-evolution of the entanglement entropy in an isolated quantum system prepared in a factorized state. We focus on random-matrix Hamiltonians from the GUE ensemble, and compare the entropy of the average density matrix to the probability distribution of the entropy over the ensemble of random Hamiltonians. While the first captures correctly the saturation value of the entropy only at the leading order, the second captures also sub-leading orders, fluctuations around the average and the short time behavior of the entanglement entropy. We discuss also the relation to other random matrix ensembles, such as GOE, and to SYK Hamiltonians.