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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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References:

[1] Bahadur, R. R. and Rao, R. R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015. · Zbl 0101.12603 · doi:10.1214/aoms/1177705674
[2] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276. · Zbl 0927.60071 · doi:10.1007/s002200050450
[3] Choquet, G. and Deny, J. (1960). Sur l’équation de convolution \(\mu * \sigma = \mu\). C. R. Acad. Sci. Paris Sér. I Math. 250 799–801. · Zbl 0093.12802
[4] Daley, D. J. and Vere-Jones, D. (1998). An Introduction to the Theory of Point Processes . Springer, Berlin. · Zbl 0657.60069
[5] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications , 2nd ed. Springer, New York. · Zbl 0896.60013
[6] Derrida, B. (1980). Random-energy model: Limit of a family of disordered models. Phys. Rev. Lett. 45 79–82. · doi:10.1103/PhysRevLett.45.79
[7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2 , 2nd ed. Wiley, New York. · Zbl 0219.60003
[8] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes , and Related Properties of Random Sequences and Processes . Springer, Berlin. · Zbl 0518.60021
[9] Liggett, T. (1979). Random invariant measures for Markov chains, and independent particle systems. Z. Wahrsch. Verw. Gebiete 45 297–854. · Zbl 0373.60076 · doi:10.1007/BF00537539
[10] Mezard, M., Parisi, G. and Virasoro, M. A. (1987). Spin Glass Theory and Beyond . World Scientific, Singapore. · Zbl 0992.82500
[11] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239. · Zbl 0617.60100 · doi:10.1007/BF01210613
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