The geometry of the Gibbs measure of pure spherical spin glasses

Abstract

We analyze the statics for pure p-spin spherical spin glass models with $p\ge 3$, at low enough temperature. With $F_{N,\mathit{\beta }}$ denoting the free energy, we compute the (logarithmic) second order term of $N{F}_{N,\mathit{\beta }}$ and prove that, for an appropriate centering $c_{N,\mathit{\beta }},N{F}_{N,\mathit{\beta }}-c_{N,\mathit{\beta }}$ is a tight sequence. We further establish the absence of temperature chaos. 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 ‘pure states’. For the pure models, the latter makes precise the picture of ‘many valleys separated by high mountains’ and significant parts of the TAP analysis from the physics literature.