Fluctuations for the Sherrington--Kirkpatrick spin glass model near the critical temperature

We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $\beta>0$. Let $F_N(\beta)$ be the corresponding log-partition function. Under the assumption that $c_N:=N^{1/3}(1-\beta_N^2)$ is bounded away from $0$, we prove that Var$(F_N(\beta_N)) = - \frac{1}{2} \log (1-\beta_N^2) -{\beta_N^2}/{2} + O( c_N^{-3/2}).$ As a consequence, we obtain Var$(F_N(1-c N^{-1/3})) = \frac16\log N + O(1)$ for any fixed constant $c\in(0,\infty)$. We also prove a Gaussian central limit theorem for the centered and scaled $F_N(\beta_N)$.