  We investigate a two-dimensional system of active particles confined to a
narrow annular domain. Despite the absence of explicit interactions among the
velocities or the active forces of different particles, the system displays a
transition from a disordered and stuck state to an ordered state of global
collective motion where the particles rotate persistently clockwise or
anticlockwise. We describe this behavior by introducing a suitable order
parameter, the velocity polarization, measuring the global alignment of the
particles' velocities along the tangential direction of the ring. We also
measure the spatial velocity correlation function and its correlation length to
characterize the two states. In the rotating phase, the velocity correlation
displays an algebraic decay that is analytically predicted together with its
correlation length while in the stuck regime the velocity correlation decays
exponentially with a correlation length that increases with the persistence
time. In the first case, the correlation (and, in particular, its correlation
length) does not depend on the active force but the system size only. The
global collective motion, an effect caused by the interplay between
finite-size, periodicity, and persistent active forces, disappears as the size
of the ring becomes infinite, suggesting that this phenomenon does not
correspond to a phase transition in the usual thermodynamic sense.
