ON THE STRUCTURE OF QUASI-STATIONARY COMPETING PARTICLE SYSTEMS

• 成果类型: Article
• 署名作者: Arguin, Louis-Pierre; Aizenman, Michael
• 署名单位: Princeton University; Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP429
• 发表日期: 2009
• 页码: 1080-1113
• 关键词: ruelles probability cascades; MODEL

摘要

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.