Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature

We consider the Sherrington-Kirkpatrick spin glass model at the critical inverse temperature $\beta =1$ with zero external field. We prove that the free energy $F_N=F_{N,\beta =1}$ of this model has variance $$\mathrm{V}\mathrm{a}\mathrm{r}(F_N)=\frac{1}{6}\log N+O(1),$$ confirming a physics prediction of Aspelmeier \cite{aspelmeier2008free}, and that the centered and scaled $F_N$ satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving $$\mathbb{E}⟨R_{1,2}^2⟩\asymp N^{-2\mathrm{/}3},$$ as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for $N^{1\mathrm{/}3}\mathrm{\mid }{R}_{1,2}\mathrm{\mid }$. The key input is a critical reweighted moment method, in the spirit of the ``small subgraph conditioning’’ technique from probabilistic combinatorics, but capable of capturing diverging fluctuations. Through this reweighting, we relate the critical SK model to the BBP critical edge, which determines the overlap and fluctuation scales.