  Model systems of self-propelled particles reproduce many phenomena observed
in laboratory active matter systems that defy our thermal equilibrium-based
intuition. In particular, in stationary states of self-propelled systems, it is
recognized that velocities of different particles exhibit non-trivial
equal-time correlations. Such correlations are absent in equivalent equilibrium
systems. Recently, researchers found that the range of the velocity
correlations increases with increasing persistence time of the self-propulsion
and can extend over many particle diameters. Here we review the initial studies
of long-ranged velocity correlations in solid-like systems of self-propelled
particles. Then, we demonstrate that the long-ranged velocity correlations are
also present in dense fluid-like systems. We show that the range of velocity
correlations in dense systems of self-propelled particles is determined by the
combination of the self-propulsion and the virial bulk modulus that originates
from repulsive interparticle interactions.
