On the Structure of Quasi-Stationary Competing Particle Systems

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 $Q = {q_{ij}}i,j}\in\mathbb{N}$ . A probabil...

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Bibliographic Details
Published in:The Annals of probability 2009-05, Vol.37 (3), p.1080-1113
Main Authors: Arguin, Louis-Pierre, Aizenman, Michael
Format: Article
Language:English
Subjects:
60G10
60G55
Correlation analysis
Covariance
Covariance matrices
Dirichlet problem
Equivalence relation
Ergodic theory
Evolution
Exact sciences and technology
General topics
Mathematical vectors
Mathematics
Matrices
Multivariate analysis
Normal distribution
Point processes
Poisson distribution
Poisson process
Probability
Probability and statistics
Probability theory and stochastic processes
Probability theory on algebraic and topological structures
Property partitioning
quasi-stationarity
Ruelle probability cascades
Sciences and techniques of general use
Spin glasses
Statistics
Stochastic processes
Studies
ultrametricity
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Summary: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 $Q = {q_{ij}}i,j}\in\mathbb{N}$ . 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 $q_{ij}$ 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.
ISSN:0091-1798
2168-894X
DOI:10.1214/08-AOP429