The geometry of the Gibbs measure of pure spherical spin glasses

TL;DR

This paper provides a detailed geometric and probabilistic analysis of pure spherical p-spin glass models at low temperatures, revealing the measure's structure, free energy fluctuations, and stability properties.

Contribution

It introduces a geometric description of the Gibbs measure as a union of infinitesimal bands around deep minima, confirming the 'many valleys' picture and analyzing chaos transition rates.

Findings

Computed the second order term of the free energy.

Proved the tightness of the centered free energy sequence.

Established the absence of temperature chaos and characterized disorder chaos transition.

Abstract

We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,\beta}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,\beta}$ and prove that, for an appropriate centering $c_{N,\beta}$, $NF_{N,\beta}-c_{N,\beta}$ is a tight sequence. We establish the absence of temperature chaos and analyze the transition rate to disorder chaos of the Gibbs measure and ground state. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical `bands' centered at deep minima, playing the role of so-called `pure states'. For the pure models, the latter makes precise the so-called picture of `many valleys separated by high mountains' and significant parts of the TAP analysis from the physics literature.