2015/01/27 by Wei-Kuo Chen, Wei‐Kuo Chen, Chen, Wei-Kuo · 2 citations
Mathematics · Physics and Astronomy · #60K35 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #advanced mathematical theories #math-ph #math.MP #math.PR #msc:60K35 #msc:82B44
paper · pdf · doi:10.48550/arxiv.1501.06635
31 pages. Major revisions on Sections 1, 2 and 5
openalex publication_date 2015/01/27 · arxiv created 2016/05/13 · arxiv updated 2016/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [18]. The central argument therein relied on a very 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 [2] and Auffinger-Chen [3], 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 [20,Section 14.12] and disorder chaos in Chatterjee [5] and Chen [6]. In particular, we provide some examples of the models containing odd p-spin interactions.