I am very happy that after a real tour de force at the end of the last year, we managed to put out two different papers proposing a random matrix description of the volume-law phase of measurement-induced phase transitions. As it is well known, in MIPTs, the effect of measurements is extremely weak and the time scale for purification is exponentially long in the system size. If one is interested in studying the dynamics on these long timescales, the unitary dynamics has all the time to completely scrable the whole quantum system. For this reason, we proposed to replace the dynamics with an effective evolution controlled by random matrices. We explain in 2312.17744 that this idea naturally brings to a concept of universality controlling the evolution in time of entanglement observables (e.g. Renyi’s entropies). From the technical point of view, the calculation of the Von Neumann entropy took us some efforts because a non-trivial analytic continuation that needed to be performed involving sums over partitions of integers. I would be interested in exchanging more with mathematicians on this topic!

In 2401.00822, we considered a continuous-time version of the problem, where essentially at every time a random hermitian operator is weaky monitored. We were able to deduce evoution equations for the eigenvalues of the density matrix, which are analogous to the Dyson brownian motion valid for the standard orthogonal matrix ensembles. Within this model, we were able to characterised both short-time dynamics in terms of a Coulomb gas description and a large time one, where consistently with the generic universality, we re-obtained the same expressions of the other paper.

I am particularly proud of these results and I think that they might be relevant in other contexts where multiplicative multichannel noise plays a role as it amounts to multiplying random matrices!