We show that in the asymmetric simple exclusion process (ASEP) on a ring, conditioned on carrying a large flux, the particles experience an effective long-range potential which is similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. We obtain the quasi stationary measure, transition probabilities and spatio-temporal correlations using the Bethe ansatz and determinantal free fermion techniques. (joint work with V. Popkov, D. Simon)