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We find the exact solution for the stationary state measure of the partiallyasymmetric exclusion process on a ring with multiple species of particles. Thesolution is in the form of a matrix product representation where the matricesfor a system of N species are defined recursively in terms of the matrices fora system of N-1 species. A complete proof is given, based on the quadraticrelations verified by these matrices. This matrix product construction isinterpreted in terms of the action of a transfer matrix.

Matrix product solution of the multispecies partially asymmetric exclusion process