On properties of Parisi measures

TL;DR

This paper analyzes the structure and properties of Parisi measures in mean field spin glasses, revealing their support characteristics, regularity, and behavior near critical temperatures, with implications for different spin glass models.

Contribution

It provides new insights into the support, regularity, and phase transition behavior of Parisi measures in mixed p-spin models and the SK model.

Findings

Support of Parisi measures always contains the origin.

Measures with support on an interval have smooth densities.

Largest support point exhibits jump discontinuity above critical temperature.

Abstract

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.