Free energy fluctuations of the $2$-spin spherical SK model at critical temperature

We investigate the fluctuations of the free energy of the $2$-spin spherical Sherrington-Kirkpatrick model at critical temperature $beta_c = 1$. When $beta = 1$ we find asymptotic Gaussian fluctuations with variance $frac{1}{6N^2} log(N)$, confirming in the spherical case a physics prediction for the SK model with Ising spins. We furthermore prove the existence of a critical window on the scale $beta = 1 +alpha sqrt{ log(N) } N^{-1/3}$. For any $alpha in mathbb{R}$ we show that the fluctuations are at most order $sqrt{ log(N) } / N$, in the sense of tightness. If $ alpha to infty$ at any rate as $N to infty$ then, properly normalized, the fluctuations converge to the Tracy-Widom$_1$ distribution. If $ alpha to 0$ at any rate as $N to infty$ or $ alpha <0$ is fixed, the fluctuations are asymptotically Gaussian as in the $alpha=0$ case. In determining the fluctuations, we apply a recent result of Lambert and Paquette on the behavior of the Gaussian-$beta$-ensemble at the spectral edge.