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

Abstract

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.

26 pages. Draft, comments welcome

References in corpus (4)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Free energy fluctuations and chaos in the Sherrington-Kirkpatrick model
• Spherical spin glass model with external field

Cited by in corpus (2)

• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with heavy-tailed interaction
• Finite-size relaxational dynamics of a spike random matrix spherical model