Combinatorics of the quantum symmetric simple exclusion process, associahedra and free cumulants

The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of quantum particles hopping on a finite interval and satisfying the exclusion principle. Recently Bernard and Jin have studied the fluctuations of the invariant measure for this process, when the number of sites goes to infinity. These fluctuations are encoded into polynomials, for which they have given equations and proved that these equations determine the polynomials completely. In this paper, I give an explicit combinatorial formula for these polynomials, in terms of Schr\"oder trees. I also show that, quite surprisingly, these polynomials can be interpreted as free cumulants of a family of commuting random variables.