### 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,j}εℕ. 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.

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

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## Cite this

*Annals of Probability*,

*37*(3), 1080-1113. https://doi.org/10.1214/08-AOP429