The Annals of Probability

On the structure of quasi-stationary competing particle systems

Louis-Pierre Arguin and Michael Aizenman

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We study point processes on the real line whose configurations X are locally finite, have a maximum and evolve through increments which are functions of correlated Gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q={qij}i, j∈ℕ. A probability measure on the pair (X, Q) is said to be quasi-stationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasi-stationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchically nested Poisson–Dirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi-stationary. The main result of this work is a proof of this conjecture for the case where qij assume only a finite number of values. The result is of relevance for mean-field spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchical organization of the Gibbs measure was first proposed as an ansatz.

Article information

Ann. Probab., Volume 37, Number 3 (2009), 1080-1113.

First available in Project Euclid: 19 June 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60G10: Stationary processes

Point processes quasi-stationarity ultrametricity Ruelle probability cascades spin glasses


Arguin, Louis-Pierre; Aizenman, Michael. On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 (2009), no. 3, 1080--1113. doi:10.1214/08-AOP429.

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