Discoveries / mathematical-physics

The Gardner Transition in the Ising pure $p$-spin glass

For the Ising pure $p$-spin glass where $p \geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$, the Parisi measure is full replica symmetry breaking (FRSB). The earlier work by the author solved Part (1) and partially solved Part (2) when $\beta$ is sufficiently close to $\beta_p^1$, while this work solves Part (2) and Part (3) entirely.