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

Abstract

We investigate the fluctuations of the free energy of the two-spin spherical Sherrington–Kirkpatrick model at critical temperature βc = 1. When β = 1, we find asymptotic Gaussian fluctuations with variance 16N2log(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+αlog(N)N−1/3. For any α∈R, we show that the fluctuations are at most order log(N)/N in the sense of tightness. If α → ∞ at any rate as N → ∞, then, properly normalized, the fluctuations converge to the Tracy–Widom1 distribution. If α → 0 at any rate as N → ∞ 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 G. Lambert and E. Paquette [“Strong approximation of Gaussian beta-ensemble characteristic polynomials: The edge regime and the stochastic airy function,” arXiv:2009.05003 (2020)] on the behavior of the Gaussian-β-ensemble at the spectral edge.