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

We prove the existence of one-step replica symmetry breaking (1RSB) for the mean field Ising spin glasses at finite temperature and identify the first critical temperature in Gardner transition. Specifically, Gardner conjectured that for Ising pure $p$-spin glasses with $p\geq 3,$ there are two phase transitions: the Parisi measure is Replica Symmetric (RS) at high temperatures, then transitions to One-Step Replica Symmetry Breaking (1RSB) at a lower temperature, and finally reaches Full Replica Symmetry Breaking (FRSB) at very low temperatures. Our main results verify the first part of the Gardner transition: for any $p \geq 3,$ there exists a unique $\beta^p_1>0$ such that the Parisi measure is RS for $0 < \beta \leq \beta^p_{1}$ and then 1RSB for $\beta^p_1 < \beta \leq \beta^p_1+\epsilon_p$ with some $\epsilon_p>0.$ We also provide a computational method to locate $\beta^p_1$ for any $p \geq 3.$