Exact spectral gaps of the asymmetric exclusion process with open boundaries

Abstract

We derive the Bethe ansatz equations describing the complete spectrum of the transitionmatrix of the partially asymmetric exclusion process with the most general openboundary conditions. By analysing these equations in detail for the cases of totallyasymmetric and symmetric diffusion, we calculate the finite-size scaling of thespectral gap, which characterizes the approach to stationarity at large times. In thetotally asymmetric case we observe boundary induced crossovers between massive,diffusive and KPZ (Kardar–Parisi–Zhang) scaling regimes. We further study higherexcitations, and demonstrate the absence of oscillatory behaviour at large times on the‘coexistence line’, which separates the massive low and high density phases. In themaximum current phase, oscillations are present on the KPZ scale . While independent of the boundary parameters, the spectral gap as well as the oscillationfrequency in the maximum current phase have different values compared to the totallyasymmetric exclusion process with periodic boundary conditions. We discuss apossible interpretation of our results in terms of an effective domain wall theory.