期刊
ANNALS OF PROBABILITY
卷 37, 期 3, 页码 1080-1113出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/08-AOP429
关键词
Point processes; quasi-stationarity; ultrametricity; Ruelle probability cascades; spin glasses
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 is an element of 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.
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