On the structure of quasi-stationary competing particle systems

TL;DR

This paper proves that, for finite correlation structures, the only robustly quasi-stationary point processes are those derived from hierarchical Poisson--Dirichlet processes, confirming a long-standing conjecture relevant to spin glass models.

Contribution

It proves the conjecture that only hierarchical Poisson--Dirichlet processes are robustly quasi-stationary when correlations take finitely many values.

Findings

Confirmed the conjecture for finite-valued correlation matrices.

Identified the unique structure of robustly quasi-stationary processes.

Relevance established for mean-field spin glass models.

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.