  We analyze the out-of-equilibrium behavior of exclusion processes where
agents interact with their nearest neighbors, and we study the short-range
correlations which develop because of the exclusion and other contact
interactions. The form of interactions we focus on, including adhesion and
contact-preserving interactions, is especially relevant for migration processes
of living cells. We show the local agent density and nearest-neighbor two-point
correlations resulting from simulations on two dimensional lattices in the
transient regime where agents invade an initially empty space from a source and
in the stationary regime between a source and a sink. We compare the results of
simulations with the corresponding quantities derived from the master equation
of the exclusion processes, and in both cases, we show that, during the
invasion of space by agents, a wave of correlations travels with velocity v(t)
~ t^(-1/2). The relative placement of this wave to the agent density front and
the time dependence of its height may be used to discriminate between different
forms of contact interactions or to quantitatively estimate the intensity of
interactions. We discuss, in the stationary density profile between a full and
an empty reservoir of agents, the presence of a discontinuity close to the
empty reservoir. Then, we develop a method for deriving approximate
hydrodynamic limits of the processes. From the resulting systems of partial
differential equations, we recover the self-similar behavior of the agent
density and correlations during space invasion.
