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

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 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)=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 ρ(dx)=e^{-sx}s dx 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.

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

Cited by in corpus (34)

• A branching random walk seen from the tip
• Statistics at the tip of a branching random walk and the delay of traveling waves
• On the structure of quasi-stationary competing particle systems
• A connection between the Ghirlanda--Guerra identities and ultrametricity
• Competing particle systems evolving by interacting Lévy processes
• Spin Glass Computations and Ruelle's Probability Cascades
• The branching Brownian motion seen from its tip
• The marginally stable Bethe lattice spin glass revisited
• Extrema of the two-dimensional Discrete Gaussian Free Field
• A unified stability property in spin glasses
• Concentration of measure for systems of Brownian particles interacting through their ranks
• The Extremal Process of Branching Brownian Motion
• Stationary systems of Gaussian processes
• Antiferromagnetic Potts model on the Erdos-Renyi random graph
• A Remark on the Infinite-Volume Gibbs Measures of Spin Glasses
• The infinite Atlas process: Convergence to equilibrium
• Contributions to Random Energy Models
• Last passage percolation and traveling fronts
• A phase transition behavior for Brownian motions interacting through their ranks
• Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction
• Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction (shortened version)
• On the REM approximation of TAP free energies
• Quasi-stationary Random Overlap Structures and the Continuous Cascades
• The Extended Variational Principle for Mean-Field, Classical Spin Systems
• Persistence of competing systems of branching random walks
• Convergence rates for rank-based models with applications to portfolio theory
• Large systems of diffusions interacting through their ranks
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Concentration for multidimensional diffusions and their boundary local times
• Stability properties and probability distribution of multi-overlaps in dilute spin glasses
• From Gumbel to Tracy-Widom
• A combinatorial analysis of interacting diffusions
• About Thinning Invariant Partition Structures
• Minimum reversion in multivariate time series