Discoveries / mathematical-physics

Full-RSB in the Sherrington–Kirkpatrick spin glass

For the Sherrington-Kirkpatrick model with no external field, at zero temperature $\beta=\infty$, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support $[0,1)$ - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows $q_\beta\to1$ as $\beta\to\infty$. The positive-temperature input is Lopatto's: for every $\beta>1$ the Parisi measure is supported on the closed interval $[0,q_\beta]$, with a smooth density on $[0,q_\beta)$ and a single atom at the right endpoint $q_\beta$. That endpoint atom is what the paper's own quantitative estimates target, so it is not incidental. This entry is cited as Theorem 1.1 rather than reproved. Remark 1.4 is worth reading beside the support claim: the half-open interval is essential, because the zero-temperature functional cannot see an endpoint atom at all, so there is no canonical mass there to converge to.