Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\mathrm{Airy}_1$ point process. The same limiting distribution and second-moment constant are obtained for the spherical SK model. This does not resolve the broader low-temperature overlap structure of the SK model.

Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: $N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\mathrm{Airy}_1$ point process. The same limiting distribution and second-moment constant are obtained for the spherical SK model. This does not resolve the broader low-temperature overlap structure of the SK model.