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.