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.

A named conjecture of Michel Talagrand from his standard monograph (Mean Field Models for Spin Glasses, vol. II, 2011), open fifteen years and quoted as open in the literature five months before this proof. Critical-point overlap scaling is a celebrated question inside spin-glass theory but specialist territory outside it: a technical conjecture from the book, not one of the field's headline problems. Placed with the strong specialist conjectures at 30, below the field-famous band of Krauth-Mezard and Feige at 35.