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May 2009 On the structure of quasi-stationary competing particle systems
Louis-Pierre Arguin, Michael Aizenman
Ann. Probab. 37(3): 1080-1113 (May 2009). DOI: 10.1214/08-AOP429


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.


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Louis-Pierre Arguin. Michael Aizenman. "On the structure of quasi-stationary competing particle systems." Ann. Probab. 37 (3) 1080 - 1113, May 2009.


Published: May 2009
First available in Project Euclid: 19 June 2009

zbMATH: 1177.60050
MathSciNet: MR2537550
Digital Object Identifier: 10.1214/08-AOP429

Primary: 60G55
Secondary: 60G10

Keywords: Point processes , quasi-stationarity , Ruelle probability cascades , Spin glasses , Ultrametricity

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 3 • May 2009
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