Competing Particle Systems and the Ghirlanda-Guerra Identities

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.