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• W2136571838 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, $beta$, smaller than a critical $beta_p$. We also show that $beta_p to sqrt{2 ln 2}$ as $p uparrow + infty$. Additionally, we study the formal $p uparrow + infty$ 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 $beta < sqrt{2 ln 2}$, fluctuations are found at an exponentially small scale, with two distinct limit laws above and below a second critical value $sqrt{ln 2/2:}$ 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 $sqrt{2 ln 2}$, the fluctuations of the logarithm of the partition function are on a scale of 1 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." @default.
• W2136571838 created "2016-06-24" @default.
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• W2136571838 date "2002-04-01" @default.
• W2136571838 modified "2023-10-18" @default.
• W2136571838 title "Fluctuations of the free energy in the REM and the $p$-spin SK models" @default.
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• W2136571838 doi "https://doi.org/10.1214/aop/1023481004" @default.
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