Competing Particle Systems and the Ghirlanda-Guerra Identities

TL;DR

This paper investigates point processes with correlated Gaussian increments, demonstrating that robustly quasi-stationary systems satisfy the Ghirlanda-Guerra identities, supporting a hierarchical structure conjecture in a general setting.

Contribution

It proves that robustly quasi-stationary systems obey the Ghirlanda-Guerra identities even when the correlation matrix has infinitely many values, extending previous finite cases.

Findings

Robustly quasi-stationary systems satisfy Ghirlanda-Guerra identities.

Supports the hierarchical structure conjecture in the general case.

Extends previous results from finite to infinite correlation structures.

Abstract

We study point processes on the real line whose configurations X can be ordered decreasingly and evolve by increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q={q_ij}. Quasi-stationary systems are those for which the law of (X,Q) is invariant under the evolution up to translation of X. It was conjectured by Aizenman and co-authors that the matrix Q of robustly quasi-stationary systems must exhibit a hierarchal structure. This was established recently, up to a natural decomposition of the system, whenever the set S_Q of values assumed by q_ij is finite. In this paper, we study the general case where S_Q may be infinite. Using the past increments of the evolution, we show that the law of robustly quasi-stationary systems must obey the Ghirlanda-Guerra identities, which first appear in the study of spin glass models. This provides strong evidence that the above conjecture also holds in the general case.