With my collaborator Amos Chan, we finalised a new paper about the behavior of the projected ensemble and deep thermalisation in random unitary circuits. The question about projected ensemble has emerged in the recent years: the question one asks is whether the behavior of a subsystem A conditioned to the outcomes of the measurement in its complementary B can be described by a simple statistical ensemble of pure states. This generalises the standard notion of thermalisation which is normally based on discarding the measurements on B, so that only the reduced density matrix of A is relevant. However, one can expect that for general chaotic quantum dynamics, a concept of deep thermalisation can emerge, so that the ensemble of states in A conditioned to the measurements in B is still described just by the value of the conserved quantities. With Amos, we analysed this question in the prototypical framework of a brick-wall random unitary circuit. We show the emergence of deep thermalisation. We also show that this problem displays many analogies with the calculation of entanglement, so that in practice to estimate the “deep thermalisation” time scales one can apply an argument similar to the membrane picture. In this way, we confirm the logarithmic scaling of the deep-thermalisation times (the times required to reach a k-design), originally found in a toy model by Ippoliti & Ho.