The Annals of Probability

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

Anastasia Ruzmaikina and Michael Aizenman

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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 σ-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 ρ(dx)=esxsdx, 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 ρ(dx)=esxsdx with s>0, restricted to the relevant σ-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.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 82-113.

First available in Project Euclid: 11 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60G55: Point processes 62P35: Applications to physics

Stochastic processes Poisson processes invariant measures large deviations spin glasses REM states


Ruzmaikina, Anastasia; Aizenman, Michael. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005), no. 1, 82--113. doi:10.1214/009117904000000865.

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