We introduce and study a class of models of free fermions hopping between neighbouring sites with random Brownian amplitudes. These simple models describe stochastic, diffusive, quantum, unitary dynamics. We focus on periodic boundary conditions and derive the complete stationary distribution of the system. It is proven that the generating function of the latter is provided by the Harish-Chandra-Itzykson-Zuber integral which allows us to access all fluctuations of the system state. The steady state is characterized by non trivial correlations which have a topological nature. Diagrammatic tools appropriate for the study of these correlations are presented. In the thermodynamic large system size limit, the system approaches a non random equilibrium state plus occupancy and coherence fluctuations of magnitude scaling proportionally with the inverse of the square root of the volume. The large deviation function for those fluctuations is determined. Although decoherence is effective on the mean steady state, we observe that sub-leading fluctuating coherences are dynamically produced from the inhomogeneities of the initial occupancy profile.
We thank both referees for their useful comments.
Answer to referee 1:
1- We explain in the introduction that the model we consider may be viewed as coding for the long time dynamics of the Heisenberg XX spin chain with random dephasing. We clarified the reference to  by indicating the appropriate section in this reference.
2- We wrote in the text that the SSEP model is recovered if one only deals with the diagonal part of the average density matrix. The relation to SSEP is pointed at the top of p.3.
3- The other comments and request (mainly typos) have been taken care of.
Answer to referee 2:
1- Although these stochastic models are clearly in the same `universality class’ (in a vague sense) as random quantum circuits, to make the connection mathematically precise is beyond the scope of this paper.
We modified the sentence in the 3rd line on p.3. to take care of this comment. See also the reply to the first referee.
2- We clarify this point and added the suggested references.
3- Thank you. Done.
See also the modification in the abstract.
4- Thank you. Done.
5- One needs to be more than familiar with RMT to recognise these expectations as characteristic of RMT.
6- We clarified this point.
8- See answer to referee 1.
The relation to SSEP is pointed at the top of p.3.
9- We think that Section 5 is interested enough by itself to be in the main text.
10- Thank you for the comment. We added a sentence to make this point.
Of course, we don’t have (yet?) a proof of our conjecture.