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 $c_N:=N^{1\mathrm{/}3}(1-{\beta }_N^2)$ is bounded away from $0$, we prove that Var$(F_N({\beta }_N))=-\frac{1}{2}\log (1-{\beta }_N^2)-{\beta }_N^2\mathrm{/}2+O(c_N^{-3\mathrm{/}2})\mathrm{.}$ As a consequence, we obtain Var$(F_N(1-c{N}^{-1\mathrm{/}3}))=\frac{1}{6}\log N+O(1)$ for any fixed constant $c\in (0,\mathrm{\infty })$. We also prove a Gaussian central limit theorem for the centered and scaled $F_N({\beta }_N)$.