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Characterization of invariant measures at the leading edge for competing particle systems. (English) Zbl 1096.60042

Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G70 Extreme value theory; extremal stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82C22 Interacting particle systems in time-dependent statistical mechanics
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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