Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound. (English) Zbl 1409.60139

Summary: The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by M. Talagrand [Ann. Math. (2) 163, No. 1, 221–263 (2006; Zbl 1137.82010)]. The central argument relied on a dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It is believed that this bound and its higher dimensional generalization are highly related to the conjectures of temperature chaos and ultrametricity in the SK model, but a complete investigation remains elusive. Motivated by A. Bovier and A. Klimovsky [Electron. J. Probab. 14, 161–241 (2009; Zbl 1205.60166)] and A. Auffinger and W.-K. Chen [Commun. Math. Phys. 335, No. 3, 1429–1444 (2015; Zbl 1320.82033)] the aim of this paper is to present a novel approach to analyzing the Parisi functional and the two-dimensional GT bound in the mixed \(p\)-spin models in terms of optimal stochastic control problems. We compute the directional derivative of the Parisi functional and derive equivalent criteria for the Parisi measure. We demonstrate how our approach provides a simple and efficient control for the GT bound that yields several new results on Talagrand’s positivity of the overlap and disorder chaos in [S. Chatterjee, “Disorder chaos and multiple valleys in spin glasses”, Preprint, arXiv:0907.3381] and W.-K. Chen [Ann. Probab. 41, No. 5, 3345–3391 (2013; Zbl 1303.60089)]. In particular, we provide some examples of the models containing odd \(p\)-spin interactions.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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