Abstract
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,jεℕ. 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.
Original language | English (US) |
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Pages (from-to) | 1080-1113 |
Number of pages | 34 |
Journal | Annals of Probability |
Volume | 37 |
Issue number | 3 |
DOIs | |
State | Published - May 2009 |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
Keywords
- Point processes
- Quasi-stationarity
- Ruelle probability cascades
- Spin glasses
- Ultrametricity