Fluctuations of the free energy in the rem and the p-spin SK models

Abstract

We consider the random fluctuations of the free energy in the p-spin version of the Sherrington-Kirkpatrick (SK) model in the high-temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the p-spin version, we prove that the random corrections to the free energy are on a scale N-(p-2)/2 only and, after proper rescaling, converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, β, smaller than a critical βp. We also show that βp √2ln2 as p ↑ +∞. Additionally, we study the formal p ↑ +∞ limit of these models, the random energy model. Here we compute the precise limit theorem for the (properly rescaled) partition function at all temperatures. For β