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

ABSTRACT.   We consider the random fluctuations of the
free energy in the $p$-spin version of the Sherrington-Kirkpatrick 
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 $N^{-(p-2)/4}$ 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 
non-Gaussian 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 $1/2$.