  The Quantum Symmetric Simple Exclusion Process (Q-SSEP) is a model for
quantum stochastic dynamics of fermions hopping along the edges of a graph with
Brownian noisy amplitudes and driven out-of-equilibrium by injection-extraction
processes at a few vertices. We present a solution for the invariant
probability measure of the one dimensional Q-SSEP in the infinite size limit by
constructing the steady correlation functions of the system density matrix and
quantum expectation values. These correlation functions code for a rich
structure of fluctuating quantum correlations and coherences. Although our
construction does not rely on the standard techniques from the theory of
integrable systems, it is based on a remarkable interplay between the
permutation groups and polynomials. We incidentally point out a possible
combinatorial interpretation of the Q-SSEP correlation functions via a
surprising connexion with geometric combinatorics and the associahedron
polytopes.
