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We study point processes on the real line whose configurations X can beordered decreasingly and evolve by increments which are functions of correlatedgaussian variables. The correlations are intrinsic to the points and quantifiedby 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 wasconjectured by Aizenman and co-authors that the matrix Q of robustlyquasi-stationary systems must exhibit a hierarchal structure. This wasestablished recently, up to a natural decomposition of the system, whenever theset S_Q of values assumed by q_ij is finite. In this paper, we study thegeneral case where S_Q may be infinite. Using the past increments of theevolution, we show that the law of robustly quasi-stationary systems must obeythe Ghirlanda-Guerra identities, which first appear in the study of spin glassmodels. This provides strong evidence that the above conjecture also holds inthe general case.

Competing Particle Systems and the Ghirlanda-Guerra Identities