Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

At the critical inverse temperature $\beta=1$ in the Sherrington-Kirkpatrick spin glass model, Talagrand conjectured that the expected squared overlap of two independent Gibbs replicas has an exact $N^{-2/3}$ scaling: there exists a constant $a>0$ such that $$ \lim_{N\to\infty} N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle=a. $$ Du and Huang prove that this limit exists and is positive and finite. More strongly, they determine the full limiting quenched distribution of the rescaled overlap $N^{1/3}R_{1,2}$ in terms of the reflected $\mathrm{Airy}_1$ point process.

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.