  Two-sided infinite systems of Brownian particles with rank-dependent
dynamics, indexed by all integers, exhibit different properties from their
one-sided infinite counterparts, indexed by positive integers, and from finite
systems. Consider the gap process, which is formed by spacings between adjacent
particles. In stark contrast with finite and one-sided infinite systems,
two-sided infinite systems can have one- or two-parameter family of stationary
gap distributions, or the gap process weakly converging to zero as time goes to
infinity.
