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.