Shattering Versus Metastability in Spin Glasses

Our goal in this work is to better understand the relationship between replica symmetry breaking, shattering, and metastability. To this end, we study the static and dynamic behaviour of spherical pure $p$-spin glasses above the replica symmetry breaking temperature $T_{s}$. In this regime, we find that there are at least two distinct temperatures related to non-trivial behaviour. First we prove that there is a regime of temperatures in which the spherical $p$-spin model exhibits a shattering phase. Our results holds in a regime above but near $T_s$. We then find that metastable states exist up to an even higher temperature $T_{BBM}$ as predicted by Barrat--Burioni--M\'ezard which is expected to be higher than the phase boundary for the shattering phase $T_d <T_{BBM}$. We develop this work by first developing a Thouless--Anderson--Palmer decomposition which builds on the work of Subag. We then present a series of questions and conjectures regarding the sharp phase boundaries for shattering and slow mixing.