On properties of Parisi measures

• 成果类型: Article
• 署名作者: Auffinger, Antonio; Chen, Wei-Kuo
• 署名单位: University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0563-y
• 发表日期: 2015
• 页码: 817-850
• 关键词: spin-glass model; FORMULA

摘要

We investigate the structure of Parisi measures, the functional order parameters of mixed -spin models in mean field spin glasses. In the absence of external field, we prove that a Parisi measure satisfies the following properties. First, at all temperatures, the support of any Parisi measure contains the origin. If it contains an open interval, then the measure has a smooth density on this interval. Next, we give a criterion on temperature parameters for which a Parisi measure is neither Replica Symmetric nor One Replica Symmetry Breaking. Finally, we show that in the Sherrington-Kirkpatrick model, slightly above the critical temperature, the largest number in the support of a Parisi measure is a jump discontinuity. An analogue of these results is discussed in the spherical mixed -spin models. As a tool to establish these facts and of independent interest, we study functionals of the associated Parisi PDEs and derive regularity properties of their solutions.