  We study systems of particles on a line which have a maximum, are locally
finite and evolve with independent increments. ``Quasi-stationary states'' are
defined as probability measures, on the \sigma-algebra generated by the gap
variables, for which joint distribution of gaps between particles is invariant
under the time evolution. Examples are provided by Poisson processes with
densities of the form \rho(dx)=e^{-sx}s dx, with s>0, and linear superpositions
of such measures. We show that, conversely, any quasi-stationary state for the
independent dynamics, with an exponentially bounded integrated density of
particles, corresponds to a superposition of Poisson processes with densities
\rho (dx)=e^{-sx}s dx with s>0, restricted to the relevant \sigma-algebra.
Among the systems for which this question is of some relevance are spin-glass
models of statistical mechanics, where the point process represents the
collection of the free energies of distinct ``pure states,'' the time evolution
corresponds to the addition of a spin variable and the Poisson measures
described above correspond to the so-called REM states.
