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 $β_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<β\leqβ_1^p$ and one-step replica symmetry breaking (1-RSB) on a nonempty interval immediately above $β_1^p$. In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature $β_2^p>β_1^p$ such that the Parisi measure is 1-RSB for $β_1^p<β\leqβ_2^p$, while for $β>β_2^p$, $\operatorname{supp}μ_β=\{0\}\cup[q_β,q'_β],$ for some $0<q_β<q'_β<1$, 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.