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
\[
\mathrm{Var}(F_N) = \frac16 \log N + O(1)\,,
\] 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
\[
\mathbb{E} \langle R_{1,2}^2\rangle \asymp N^{-2/3}\,,
\] as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for $N^{1/3} |R_{1,2}|$. The key input is a comparison between the Ising and spherical SK partition functions $Z_N$ and $Z^{\mathrm{sp}}_N$: if $X_N = Z_N / Z^{\mathrm{sp}}_N$, then $X_N = 1 + o(1)$ in $L^2$. Thus $Z^{\mathrm{sp}}_N$ captures the diverging critical fluctuations of $Z_N$ and serves as a tractable reweighting variable for estimating overlap moments.