Competing Particle Systems and the Ghirlanda-Guerra Identities

Abstract

Competing particle systems are 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 hierarchical 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. In addition, it yields an alternative proof of a theorem of Ruzmaikina and Aizenman for independent increments.