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 $β_c = 1$. When $β= 1$ 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 $β= 1 +α\sqrt{ \log(N) } N^{-1/3}$. For any $α\in \mathbb{R}$ we show that the fluctuations are at most order $\sqrt{ \log(N) } / N$, in the sense of tightness. If $ α\to \infty$ at any rate as $N \to \infty$ then, properly normalized, the fluctuations converge to the Tracy-Widom$_1$ distribution. If $ α\to 0$ at any rate as $N \to \infty$ or $ α<0$ is fixed, the fluctuations are asymptotically Gaussian as in the $α=0$ case. In determining the fluctuations, we apply a recent result of Lambert and Paquette on the behavior of the Gaussian-$β$-ensemble at the spectral edge.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 2.8