## The Annals of Probability

- Ann. Probab.
- Volume 37, Number 3 (2009), 1080-1113.

### On the structure of quasi-stationary competing particle systems

Louis-Pierre Arguin and Michael Aizenman

#### 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*={*q*_{ij}}_{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 *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.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 3 (2009), 1080-1113.

**Dates**

First available in Project Euclid: 19 June 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1245434029

**Digital Object Identifier**

doi:10.1214/08-AOP429

**Mathematical Reviews number (MathSciNet)**

MR2537550

**Zentralblatt MATH identifier**

1177.60050

**Subjects**

Primary: 60G55: Point processes

Secondary: 60G10: Stationary processes

**Keywords**

Point processes quasi-stationarity ultrametricity Ruelle probability cascades spin glasses

#### Citation

Arguin, Louis-Pierre; Aizenman, Michael. On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 (2009), no. 3, 1080--1113. doi:10.1214/08-AOP429. https://projecteuclid.org/euclid.aop/1245434029