Characterization of invariant measures at the leading edge for competing particle systems

Details Characterization of invariant measures at the leading edge for competing particle systems Characterization of invariant measures at the leading edge for competing particle systems

2004-10-11

arxiv.org/abs/math/0410276v3

Annals of Probability 2005, Vol. 33, No. 1, 82-113

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000865 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000865

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.

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