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

Abstract: We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $β&gt;0$. Let $F_N(β)$ be the corresponding log-partition function. Under the assumption that $c_N:=N<sup>{1/3}(1-β_N<sup>2)$ is bounded away from $0$, we prove that Var$$(F_N(β_N)) = - \frac{1}{2} \log (1-β_N²⁾ -{β_N^2}/{2} + O( c_N<sup>{-3/2}).$$ 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 $F_N(β_N)$.