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

• 成果类型: Article
• 署名作者: Bovier, A; Kurkova, I; Löwe, M
• 署名单位: Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Sorbonne Universite; Radboud University Nijmegen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 605-651
• 关键词: sherrington-kirkpatrick model; hopfield model; glass model

摘要

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, beta, smaller than a critical beta(p). We also show that beta(p) --> root2ln2 as p up arrow +infinity. Additionally, we study the formal p up arrow +infinity 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 beta < root2ln2, fluctuations are found at an exponentially small scale, with two distinct limit laws above and below a second critical value rootln2/2: for beta up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For beta larger than the critical root2ln2, the fluctuations of the logarithm of the partition function are on a scale of I and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1/2.