04-323 A. Ruzmaikina and M. Aizenman
Characterization of invariant measures at the leading edge for competing particle systems (87K, LaTex) Oct 12, 04
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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.

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