On the accuracy of the normal approximation for the free energy in the   REM
  In the present paper we consider the fluctuations of the free energy in the random energy model (REM) on a moderate deviation scale. We find that for high temperatures the normal approximation holds only in a narrow range of scalings away from the CLT. For scalings of higher order, probabilities of moderate deviations decay faster than exponentially. 
Raphael Meiners, Anselm Reichenbachs
math.PR