Duality for the multispecies stirring process with open boundaries

We study the stirring process with N − 1 species on a generic graph G = ( V , E ) with reservoirs. The multispecies stirring process generalizes the symmetric exclusion process, which is recovered in the case N = 2. We prove the existence of a dual process defined on an extended graph G ~ = ( V ~ , E ~ ) which includes additional sites in V ~ ∖ V where dual particles get absorbed in the long-time limit. We thus obtain a characterization of the non-equilibrium steady state of the boundary-driven system in terms of the absorption probabilities of dual particles. The process is integrable for the case of the one-dimensional chain with two reservoirs at the boundaries and with maximally one particle per site. We compute the absorption probabilities by relying on the underlying g l ( N ) symmetry and the matrix product ansatz. Thus one gets a closed-formula for (long-ranged) correlations and for the non-equilibrium stationary measure. Extensions beyond this integrable set-up are also discussed.