 00290 A. Bovier, I. Kurkova, M. Loewe
 Fluctuations of the free energy in the REM and the $p$spin SK models
(226K, PS)
Jul 11, 00

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Abstract. We consider the random fluctuations of the
free energy in the $p$spin version of the SherringtonKirkpatrick
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 (for $p$ even)
the random corrections to the
free energy are on a scale $N^{(p2)/4}$ only, and after proper
rescaling converge to a standard Gaussian random variable. This is
shown to hold for all values of the inverse temperature, $\b$, smaller than
a critical $\b_p$. We also show that $\b_p\rightarrow \sqrt{2\ln 2}$ 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 partition
function at {\it all}
temperatures. For $\b<\sqrt{2\ln2}$, fluctuations are found
at an {\it exponentially small} scale, with two distinct limit laws
above and below a second critical value $\sqrt{\ln 2/2}$: For $\b$ up to that
value the rescaled fluctuations are Gaussian, while below that there are
nonGaussian fluctuations driven by the Poisson process of
the extreme values of the random energies.
For $\b$ larger than the critical $\sqrt{2\ln 2}$, the fluctuations of the
logarithm of the partition function are on scale one and are expressed in terms
of the Poisson process of extremes. At the critical temperature,
the partition function divided by its expectation converges to $1/2$.
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