The replica calculation of entanglement in random unitary circuit

We systematically study the entanglement growth of quantum chaotic evolution by a random unitary circuit using the replica trick. In the computation, the problem maps to the statistical mechanics of interacting random walks. This picture allows us to understand the deterministic linear growth of various R\'enyi entropies as well as the universal physics of the fluctuations. We find that the growth rate is corrected by the entropy of the random walks.