Variational representations for the Parisi functional and the two-dimensional Guerra–Talagrand bound

Abstract

The validity of the Parisi formula in the Sherrington–Kirkpatrick model (SK) was initially proved by Talagrand [Ann. of Math. (2) 163 (2006) 221–263]. 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 Bovier–Klimovsky [Electron. J. Probab. 14 (2009) 161–241] and Auffinger–Chen [Comm. Math. Phys. 335 (2015) 1429–1444] 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 Chatterjee [Disorder chaos and multiple valleys in spin glasses. Preprint] and Chen [Ann. Probab. 41 (2013) 3345–3391]. In particular, we provide some examples of the models containing odd $p$-spin interactions.