On the Gardner Transition in the Ising Pure $p$-Spin Glass II

Abstract

In previous work, we identified, for every $p\ge 3$, a unique first critical temperature ${\beta }_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<\beta \le {\beta }_1^p$ and one-step replica symmetry breaking (1-RSB) on a nonempty interval immediately above ${\beta }_1^p$. In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature ${\beta }_2^p>{\beta }_1^p$ such that the Parisi measure is 1-RSB for ${\beta }_1^p<\beta \le {\beta }_2^p$, while for $\beta >{\beta }_2^p$, $\mathrm{supp}{\mu }_{\beta }=\{0\}\cup [q_{\beta },q_{\beta }^{\prime }],$ for some $0<q_{\beta }<q_{\beta }^{\prime }<1$, and is therefore full replica symmetry breaking (FRSB). Combined with our earlier results, this establishes the two transitions predicted by Gardner for the Ising pure $p$-spin glass.