On the structure of quasi-stationary competing particle systems

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

Published in at http://dx.doi.org/10.1214/08-AOP429 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Characterization of invariant measures at the leading edge for competing particle systems
• Spin Glass Computations and Ruelle's Probability Cascades
• About the cavity fields in mean field spin glass models

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• The Sherrington-Kirkpatrick model: an overview
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• Spectral gap estimates in mean field spin glasses
• Replica symmetry breaking in mean field spin glasses trough Hamilton-Jacobi technique
• Competing particle systems evolving by interacting Lévy processes
• On the TAP free energy in the mixed $p$-spin models
• Thouless-Anderson-Palmer equations for the generic p-spin glass model
• A connection between MAX $\kappa$-CUT and the inhomogeneous Potts spin glass in the large degree limit
• Structure of finite-RSB asymptotic Gibbs measures in the diluted spin glass models
• A unified stability property in spin glasses
• On spin distributions for generic p-spin models
• Stationary systems of Gaussian processes
• Exchangeable random measures
• A Remark on the Infinite-Volume Gibbs Measures of Spin Glasses
• Stochastic Stability: a Review and Some Perspectives
• Stability of the Spin Glass Phase under Perturbations
• Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction
• Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction (shortened version)
• On the REM approximation of TAP free energies