The Order of Free Energy Fluctuations in the Critical Sherrington-Kirkpatrick Model Revisited

We study the fluctuations of the free energy of the Sherrington-Kirkpatrick model at the critical inverse temperature $\beta =1$. We prove that $$\mathrm{Var}(F_N(1))\le \frac{1}{4}\log N+C,$$ with $C$ independent of $N$. This gives a logarithmic upper bound, in agreement with the order predicted in the physics literature. We also prove that the critical variance diverges: more precisely, $$\mathrm{Var}(F_N(1))\ge \frac{1}{2}\log \log \log N-C\mathrm{.}$$