Competing particle systems and the Ghirlanda-Guerra identities

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 ex 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.