A. Ruzmaikina and M. Aizenman
Characterization of invariant measures at the leading edge for competing particle systems
(87K, LaTex)
ABSTRACT. 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 the joint distribution of
the gaps is invariant under the time evolution. Examples are
provided by Poisson processes with densities of the form,
$\rho(dx) \ =\ e^{- s x} \, s\, dx$, 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 the above described probability
measures, 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.