2013/03/14 by Antonio Auffinger, Wei-Kuo Chen, Auffinger, Antonio +1 · 2 citations
Mathematics · Physics and Astronomy · #60K35 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:82B44
paper · pdf · doi:10.48550/arxiv.1303.3573
31 pages, 2 figures
arxiv created 2013/03/14 · arxiv updated 2013/03/18
We investigate the structure of Parisi measures, the functional order parameters of mixed p-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 p-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.