The geometry of the Gibbs measure of pure spherical spin glasses

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.