  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.
