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

Abstract

In previous work, we identified, for every $p\geq3$, a unique first critical temperature $\beta_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<\beta\leq\beta_1^p$ 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 $\beta_1^p<\beta\leq\beta_2^p$, while for $\beta>\beta_2^p$, $\operatorname{supp}\mu_\beta=\{0\}\cup[q_\beta,q'_\beta],$ for some $0<q_\beta<q'_\beta<1$, 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.