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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2007
- DOI:
- 10.48550/arXiv.0712.2338
- arXiv:
- arXiv:0712.2338
- Bibcode:
- 2007arXiv0712.2338A
- Keywords:
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- Mathematics - Probability;
- Condensed Matter - Disordered Systems and Neural Networks;
- Mathematical Physics;
- 60G55;
- 60G10;
- 82B44
- E-Print:
- 17 pages