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.Comment: 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

Introduction to the SK model

“…There is a natural analogue of the cavity method (see Panchenko [75]), so proving the matching lower bound reduces to demonstrating the structure of Gibbs distribution predicted by Mézard and Parisi. The most recent results in this direction in Panchenko [79] are at the same stage as the SK model was after the work of Arguin, Aizenman [7] and Panchenko [69], namely, the Mézard-Parisi picture is proved in some generic sense (sufficient for proving the formula for the free energy) under the technical condition that the overlap takes finitely many values.…”

“…The measure G is a limiting analogue of the Gibbs distribution G N , and we will call it an asymptotic Gibbs distribution. Such definition of an asymptotic Gibbs distribution via the Dovbysh-Sudakov representation was first given by Arguin and Aizenman in [7] (see also [8]). …”

“…The first indication that ultrametricity could possibly be explained by the Ghirlanda-Guerra identities appeared in a seminal work of Arguin and Aizenman [7]. Instead of the Ghirlanda-Guerra identities they used a closely related property, called the Aizenman-Contucci stochastic stability [1] (see also [26]), and showed that, under a technical assumption that the overlap takes only finitely many values in the thermodynamic limit,…”

A phase transition behavior for Brownian motions interacting through their ranks

“…For more references in this direction, see Pal and Pitman [28]. Also, see the related discrete time models in statistical physics studied by Ruzmaikina and Aizenman [32], and Arguin and Aizenmann [2]. In particular see the article by Arguin [3] where he connects the Poisson-Dirichlet's with the competing particle models.…”

Construction of pure states in mean field models for spin glasses

“…In a some what different direction, Aizenman and Arguin [1] approached (and partially solved) the question of ultrametricity through "Competing particle systems". Rather than being directly related to an actual mean field model for spin glasses, competing particle systems model the expected behavior of such a system, and "the evolution of the Gibbs' state under the cavity dynamics".…”